#### Map of mathematics 0.5 # **A Comprehensive Taxonomy of Mathematical Topics** ## **1\. Introduction** ### **Overview of Mathematics** Mathematics constitutes a vast and multifaceted field dedicated to the exploration of properties and interactions concerning idealized objects.1 These objects range from abstractions derived from observations of the natural world to purely abstract entities defined by a set of foundational properties known as axioms.2 Initially developed as a practical tool for calculation and tabulation, mathematics has evolved into an exceptionally rich and diverse collection of methodologies, terminologies, and conceptual approaches, spanning the spectrum from the purely abstract to the profoundly utilitarian.1 Its historical roots lie primarily in arithmetic (the manipulation of numbers) and geometry (the study of shapes and space).2 The 16th and 17th centuries witnessed a significant expansion with the introduction of algebra, focusing on formulas and structures, and infinitesimal calculus, studying continuous change.2 This fourfold division—arithmetic, geometry, algebra, calculus—persisted until the late 19th century.2 Subsequently, driven by investigations into its own foundations and the systematization of the axiomatic method, mathematics experienced an explosion of new fields.2 Defining mathematics precisely remains elusive, with no single definition universally accepted.3 Various philosophical perspectives, notably logicism, intuitionism, and formalism, offer distinct characterizations.3 Logicism, championed by figures like Bertrand Russell, posits that all mathematics is essentially symbolic logic.3 Intuitionism, associated with L.E.J. Brouwer, emphasizes mental constructions allowed by intuition.3 Formalism, represented by Haskell Curry, views mathematics as the science of formal systems.3 Other definitions highlight its role in drawing necessary conclusions (Benjamin Peirce) 3, studying patterns (Walter Warwick Sawyer) 3, classifying structures (Oxford English Dictionary) 3, or serving as the foundation for exact sciences (David Hilbert).1 Despite this definitional diversity, the core activity involves rigorous deduction from axioms and established results to prove properties of abstract objects.2 ### **Purpose and Scope of the Report** This report endeavors to compile the most extensive list feasible of topics within the discipline of mathematics, encompassing its principal branches, significant sub-disciplines, foundational concepts, fundamental theorems, and specific theories, thereby addressing both breadth and depth as requested by the user query. The objective is to provide a structured and comprehensive map of the mathematical landscape. Each identified topic is accompanied by a concise, single-sentence explanation capturing its central idea, derived from authoritative mathematical resources including online encyclopedias like Wikipedia 2 and Wolfram MathWorld 1, academic glossaries 5, and established classification systems.6 The scope includes pure mathematics (developed independently of applications), applied mathematics (developed in close correlation with applications), foundational areas like logic and set theory, historical developments, and contemporary research frontiers.2 ### **Methodology and Structure** The compilation is organized logically using a hierarchical structure based on the primary branches of mathematics, such as Algebra, Analysis, Geometry, and Number Theory, as identified in standard references.2 To ensure comprehensiveness and alignment with established mathematical taxonomy, the structure and content have been cross-referenced with the Mathematics Subject Classification (MSC) 2020\.7 The MSC is a scheme collaboratively maintained by Mathematical Reviews and zbMATH, used widely by journals, databases, and researchers to categorize mathematical literature.7 Relevant MSC codes are indicated throughout the report to facilitate further exploration and provide context within this standard framework. A significant challenge in creating such a taxonomy lies in the inherently interconnected nature of mathematics. While classification systems like the MSC provide an invaluable organizational framework, facilitating literature searches and structuring understanding 14, they necessarily simplify a reality where boundaries between fields are often fluid and permeable.9 Many areas draw heavily upon techniques and concepts from others; for instance, number theory frequently employs methods from analysis, algebra, and geometry 9, while fields like homological algebra explicitly bridge topology and algebra.9 The need for periodic revisions of the MSC itself, reflecting new discoveries and evolving perspectives, underscores this interconnectedness and the dynamic nature of the discipline.7 Therefore, while this report utilizes a hierarchical classification for clarity and structure, it is important to recognize that this represents a practical organization of a deeply interwoven and constantly evolving body of knowledge. The historical development from ancient arithmetic and geometry 2 through the Renaissance additions of algebra and calculus 2 to the proliferation of fields following the late 19th-century foundational inquiries 2 illustrates this continuous growth and reorganization. The MSC 2020 now lists 63 top-level areas 2, reflecting the immense expansion and specialization within mathematics, influenced increasingly by factors like computation and data analysis.12 **Table 1: Main Report Sections and Corresponding MSC 2020 Codes** | Main Branch (Report Section) | Primary MSC 2020 Codes | | :---- | :---- | | 2\. Foundations of Mathematics | 03 (Mathematical logic and foundations), 18 (Category theory; homological algebra) | | 3\. Algebra | 06 (Order, lattices, ordered algebraic structures), 08 (General algebraic systems), 12 (Field theory and polynomials), 13 (Commutative algebra), 15 (Linear and multilinear algebra; matrix theory), 16 (Associative rings and algebras), 17 (Nonassociative rings and algebras), 19 (K-theory), 20 (Group theory and generalizations) | | 4\. Analysis | 26 (Real functions), 28 (Measure and integration), 30 (Functions of a complex variable), 31 (Potential theory), 32 (Several complex variables and analytic spaces), 33 (Special functions), 34 (Ordinary differential equations), 35 (Partial differential equations), 37 (Dynamical systems and ergodic theory), 39 (Difference and functional equations), 40 (Sequences, series, summability), 41 (Approximations and expansions), 42 (Harmonic analysis on Euclidean spaces), 43 (Abstract harmonic analysis), 44 (Integral transforms, operational calculus), 45 (Integral equations), 46 (Functional analysis), 47 (Operator theory), 49 (Calculus of variations and optimal control; optimization), 58 (Global analysis, analysis on manifolds) | | 5\. Geometry and Topology | 51 (Geometry), 52 (Convex and discrete geometry), 53 (Differential geometry), 54 (General topology), 55 (Algebraic topology), 57 (Manifolds and cell complexes) | | 6\. Discrete Mathematics | 05 (Combinatorics), 68 (Computer science \- theoretical aspects), 94 (Information and communication theory, circuits \- theoretical aspects) | | 7\. Number Theory | 11 (Number theory) | | 8\. Probability Theory and Statistics | 60 (Probability theory and stochastic processes), 62 (Statistics) | | 9\. Applied Mathematics & Interdisciplinary Areas | 00 (General), 01 (History and biography), 65 (Numerical analysis), 70 (Mechanics of particles and systems), 74 (Mechanics of deformable solids), 76 (Fluid mechanics), 78 (Optics, electromagnetic theory), 80 (Classical thermodynamics, heat transfer), 81 (Quantum Theory), 82 (Statistical mechanics, structure of matter), 83 (Relativity and gravitational theory), 85 (Astronomy and astrophysics), 86 (Geophysics), 90 (Operations research, mathematical programming), 91 (Game theory, economics, finance, social/behavioral sciences), 92 (Biology and other natural sciences), 93 (Systems theory; control), 97 (Mathematics education) | ## **2\. Foundations of Mathematics (MSC 03, 18\)** ### **Overview** The foundations of mathematics constitute a field dedicated to investigating the logical underpinnings and philosophical basis of mathematical reasoning and knowledge.1 This area critically examines the nature of mathematical objects, the validity of proofs, and the consistency and completeness of mathematical systems.15 Its modern development was significantly spurred by the foundational crisis at the turn of the 20th century, marked by the discovery of paradoxes (like Russell's paradox) within naive set theory, which challenged the perceived certainty of mathematics.2 This period saw the emergence of major schools of thought aiming to secure reliable foundations: logicism (Frege, Russell), proposing logic as the ultimate basis; formalism (Hilbert), focusing on the consistency of axiomatic systems treated as formal symbol manipulation; and intuitionism (Brouwer), emphasizing constructive mental processes.3 While set theory, particularly Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), currently serves as a widely accepted foundational framework for much of mathematics 2, alternative approaches like category theory also offer powerful perspectives on mathematical structures.18 The quest for foundations has not merely been an exercise in securing existing knowledge but has actively driven mathematical innovation, leading to the development of fields like modern mathematical logic, computability theory, and abstract set theory, which in turn provide essential tools and language for contemporary mathematics.2 This demonstrates how foundational inquiries act as both a product of mathematical evolution and a catalyst for further development. ### **Mathematical Logic (MSC 03\)** Mathematical logic is the subfield exploring the application of formal logic to mathematics, aiming for a precise understanding of mathematical proof and reasoning.5 * **03A: Philosophical and critical aspects:** This area addresses the philosophy of mathematics, examining the nature of mathematical truth, objects, and knowledge, including perspectives like intuitionism and finitism.3 * **03B: General logic:** This encompasses the study of various logical systems, including classical propositional and predicate logic, as well as non-classical systems like modal, temporal, intuitionistic, and many-valued logics, focusing on the formal structure of valid inference.11 * **03C: Model theory:** Model theory investigates the relationship between formal languages and their interpretations (models) within mathematical structures, studying classes like groups, fields, and graphs from a logical viewpoint.5 * **Non-standard models (MSC 03H):** These are models of a theory (e.g., arithmetic) that contain elements not present in the intended standard model, such as infinitesimals or non-standard integers.11 * **03D: Computability and recursion theory:** This branch studies the theoretical limits of computation, defining what functions are algorithmically computable (e.g., via Turing machines) and classifying problems based on their computability.5 * **03E: Set theory:** Set theory is the study of sets, or collections of objects, providing a fundamental language and framework for expressing most mathematical concepts.2 * **Axiomatic Set Theory:** Formal systems like Zermelo-Fraenkel set theory (ZF) and ZFC (ZF with the Axiom of Choice) provide rigorous foundations.26 * **Descriptive Set Theory:** Studies the properties of definable sets in Polish spaces.27 * **Large Cardinals:** Investigates axioms asserting the existence of very large infinite sets, used to measure the consistency strength of set theories.26 * **03F: Proof theory and constructive mathematics:** Proof theory analyzes the structure of mathematical proofs themselves as formal objects, while constructive mathematics focuses on methods that provide explicit constructions for mathematical objects.5 * **03G: Algebraic logic:** This area employs algebraic structures, such as Boolean algebras and lattices, to study logical systems and reasoning.11 * **Abstract Algebraic Logic (AAL):** Focuses on the general process of associating classes of algebras with logical systems and the relationship between logical properties and algebraic properties.28 ### **Category Theory (MSC 18\)** Category theory provides a high-level framework for studying mathematical structures and their relationships by abstracting them in terms of "objects" and "arrows" (morphisms) between them.18 It emphasizes relationships and transformations rather than the internal structure of objects, offering a unifying language across diverse mathematical fields like algebra, topology, and logic.18 Key concepts include categories, functors (structure-preserving maps between categories), natural transformations (maps between functors), limits, colimits, and adjoint functors.11 It originated partly from algebraic topology.32 * **18A: General theory of categories and functors:** Develops the basic language and concepts of categories, functors, and natural transformations.11 * **18B: Special categories:** Studies specific types of categories, such as the category of sets, groups, topological spaces, or topoi (categories behaving like the category of sets, important in logic).11 * **18C: Categories and theories:** Explores connections between category theory and logical theories, including categorical logic.5 * **18D: Categories with structure:** Investigates categories equipped with additional structure, such as monoidal categories (categories with a tensor product).11 * **18E: Abelian categories:** Studies categories that share key properties with the category of abelian groups, providing a framework for homological algebra.11 * **18G: Homological algebra:** Uses tools like chain complexes, homology, and cohomology to study algebraic objects, often within the framework of abelian or derived categories.2 It has deep roots and applications in algebraic topology.9 * **18M: Monoidal categories and operads (New in MSC2020):** Focuses on categories with tensor products and algebraic structures called operads that encode types of operations.12 * **18N: Higher categories and homotopical algebra (New in MSC2020):** Generalizes category theory to include higher-dimensional morphisms (2-categories, ∞-categories) and connections to homotopy theory.12 ## **3\. Algebra (MSC 06, 08, 12-17, 18, 19-20)** ### **Overview** Algebra, in its broad sense, is the branch of mathematics concerned with algebraic structures—sets equipped with operations satisfying specific axioms—and the manipulation of symbols and formulas within these systems.2 It evolved historically from the methods used to solve equations and manipulate numerical expressions 2 into the abstract study of fundamental structures like groups, rings, fields, vector spaces, and lattices.18 This move towards abstraction, focusing on axiomatic definitions rather than specific examples like numbers, allows algebra to achieve remarkable generality.5 By identifying common patterns and properties across different mathematical contexts, algebra develops powerful tools and concepts applicable throughout mathematics and its applications. Universal algebra and category theory represent further levels of abstraction, studying properties common to broad classes of structures or the relationships between structures themselves.18 This abstract approach unifies disparate areas and reveals deep underlying connections, such as the link between field extensions and group theory established by Galois theory.31 ### **Core Algebraic Structures and Subfields** * **08: General Algebraic Systems (Universal Algebra):** This field studies the properties shared by diverse algebraic structures, focusing on general concepts like algebras, operations, identities, varieties, clones, and congruence lattices, independent of the specific type (e.g., group, ring) being considered.5 * **06: Order, Lattices, Ordered Algebraic Structures:** This area investigates sets equipped with order relations (partially ordered sets or posets) and specialized structures derived from them, notably lattices (where pairs of elements have unique least upper bounds and greatest lower bounds) and Boolean algebras (lattices modeling logical operations).18 * **12: Field Theory and Polynomials:** Field theory studies fields, which are algebraic structures where addition, subtraction, multiplication, and division (except by zero) are well-defined and behave as they do for rational or real numbers.5 This includes the study of field extensions (embedding one field within a larger one) and polynomials defined over fields.11 * **Galois Theory (part of 12F):** Establishes a fundamental connection between field extensions and group theory, relating symmetries of the roots of polynomials to the structure of groups.27 * **13: Commutative Algebra:** This major area focuses on commutative rings—rings where the multiplication operation is commutative (e.g., integers, polynomials)—and modules over such rings.5 It provides the algebraic foundation for algebraic geometry and algebraic number theory.35 Key topics include ideals, prime ideals, localization, integral extensions, and dimension theory.35 * **15: Linear and Multilinear Algebra; Matrix Theory:** Linear algebra deals with vector spaces (spaces where vectors can be added and scaled) and linear transformations between them, which can be represented by matrices.5 It encompasses the study of systems of linear equations, determinants, eigenvalues, eigenvectors, vector spaces, and matrix operations.11 Multilinear algebra extends these concepts to tensors and operations involving multiple vector spaces.11 It is a foundational tool across mathematics, science, and engineering. * **16: Associative Rings and Algebras:** This field studies rings where multiplication is associative but not necessarily commutative, such as rings of matrices.11 It also includes associative algebras, which are rings that are also vector spaces (or modules over a commutative ring) with compatible scalar multiplication.11 Important sub-areas include representation theory (studying how algebraic structures act on vector spaces), the theory of modules, radical theory, and the study of specific ring classes like division rings and rings with polynomial identities.11 * **17: Nonassociative Rings and Algebras:** This area investigates algebraic structures where the multiplication operation is not required to be associative.11 Prominent examples include: * **Lie Algebras (MSC 17B):** Algebras satisfying anticommutativity (\[x,y\] \= \-\[y,x\]) and the Jacobi identity, arising naturally from associative algebras via the commutator bracket and fundamental to the study of Lie groups and symmetries in physics.11 * **Jordan Algebras (MSC 17C):** Commutative algebras satisfying the Jordan identity (\[x²,y,x\]=0), related to quantum mechanics and formally real algebras.11 * **Alternative Algebras (part of 17D):** Algebras satisfying the weaker alternative laws (x(xy)=(xx)y and y(xx)=(yx)x), such as the octonions.11 * **18: Category Theory; Homological Algebra:** (See Section 2\) Category theory provides abstract tools applicable to algebra, while homological algebra uses chain complexes to study algebraic structures.5 * **19: K-Theory:** K-theory assigns sophisticated algebraic invariants (K-groups) to mathematical objects like rings, operator algebras, or geometric spaces, often related to vector bundles or algebraic cycles.5 It has connections to algebraic topology, algebraic geometry, and number theory.11 * **20: Group Theory and Generalizations:** Group theory is the study of groups—sets with an associative binary operation, an identity element, and inverses for all elements—which mathematically capture the concept of symmetry.5 It includes the study of finite groups (e.g., classification of finite simple groups), infinite groups (e.g., free groups, geometric group theory), permutation groups, group representations, and generalizations like semigroups (associative operation, no inverses required) and groupoids.11 ## **4\. Analysis (MSC 26, 28, 30-35, 37, 39-47, 49, 58\)** ### **Overview** Analysis is the branch of mathematics originating from the rigorous development of calculus, concerned primarily with concepts arising from limits, continuity, and infinite processes.2 It encompasses the study of differentiation, integration, measure, infinite series, and analytic functions, applied to both real and complex numbers, as well as more abstract spaces like function spaces and manifolds.2 The core theme unifying the diverse subfields of analysis is the precise mathematical treatment of continuous change, approximation, and the infinite.2 From the foundations of calculus in real analysis to the complexities of partial differential equations modeling physical phenomena, or the infinite-dimensional spaces of functional analysis underpinning quantum mechanics, analysis provides the tools to understand and quantify continuous systems and limiting behaviors. ### **Subfields of Analysis** * **26: Real Functions (Real Analysis):** This field provides the rigorous foundation for calculus involving real numbers, dealing with sequences, series, limits, continuity, differentiation (derivatives), and integration (Riemann and Lebesgue integrals) of functions of one or several real variables.5 It also includes the study of inequalities.11 * **28: Measure and Integration:** Measure theory generalizes notions like length, area, and volume to more abstract sets, leading to the powerful Lebesgue integral, which extends the Riemann integral and is crucial for probability theory and functional analysis.18 This area also includes ergodic theory.5 * **30: Functions of a Complex Variable (Complex Analysis):** This branch extends the concepts of calculus and analysis to functions whose domain and/or range are complex numbers.5 It reveals remarkable properties unique to complex differentiability (holomorphicity or analyticity), such as the deep connections between derivatives and integrals expressed by Cauchy's integral theorem and formula.5 * **31: Potential Theory:** Potential theory focuses on the study of harmonic functions, which are solutions to Laplace's equation (Δu \= 0), and related concepts like subharmonic functions.5 It originated from the study of gravitational and electrostatic potentials in physics.44 In two dimensions, it is intimately connected with complex analysis.44 * **32: Several Complex Variables and Analytic Spaces:** This area generalizes complex analysis to functions of multiple complex variables.5 It exhibits phenomena distinct from the single-variable case, such as the Hartogs' extension theorem and the importance of domains of holomorphy.48 It connects with complex geometry and algebraic geometry.48 * **33: Special Functions:** Special functions are particular functions (like the Gamma function, Bessel functions, Legendre polynomials, hypergeometric functions) that arise frequently as solutions to differential equations or in specific applications in mathematical physics, number theory, and statistics.5 Their properties, integral representations, and interrelations are studied extensively.49 * **34: Ordinary Differential Equations (ODEs):** ODEs are equations involving unknown functions of a single independent variable (often time) and their derivatives.5 They are fundamental for modeling systems that evolve continuously.53 Topics include existence and uniqueness of solutions, methods of solution (analytical and numerical), qualitative behavior (stability, oscillations), boundary value problems, and spectral theory.11 * **35: Partial Differential Equations (PDEs):** PDEs are equations involving unknown functions of multiple independent variables and their partial derivatives.5 They are essential for modeling diverse phenomena in physics, engineering, biology, and finance, including waves, heat diffusion, fluid flow, and quantum mechanics.56 Key types include elliptic (e.g., Laplace's equation), parabolic (e.g., heat equation), and hyperbolic (e.g., wave equation) PDEs.58 * **37: Dynamical Systems and Ergodic Theory:** Dynamical systems theory studies the long-term evolution of systems governed by rules, often expressed via ODEs (continuous time) or iterated maps (discrete time).59 It investigates concepts like stability, periodicity, bifurcations, and chaos.5 Ergodic theory specifically studies the statistical properties of dynamical systems that preserve a measure, analyzing long-term averages and mixing behavior.5 * **39: Difference and Functional Equations:** Functional equations are equations where the unknown is a function, relating its values at different points (e.g., f(x+y)=f(x)+f(y)).5 Difference equations relate terms of a sequence or values of a function at discrete points (often called recurrence relations).11 Functional differential equations involve derivatives at different argument values (e.g., delay equations).65 * **40: Sequences, Series, Summability:** This area focuses on the convergence properties of infinite sequences and series of numbers or functions, and methods (summability methods) for assigning values to divergent series.11 * **41: Approximations and Expansions (Approximation Theory):** Approximation theory investigates how well functions can be approximated by simpler functions, such as polynomials, rational functions, or trigonometric series, and analyzes the error involved.5 It includes topics like interpolation, best approximation, and asymptotic expansions.11 * **42: Harmonic Analysis on Euclidean Spaces:** This field deals with the representation of functions on Euclidean spaces (like R^n) as superpositions of basic waves, generalizing classical Fourier series and Fourier transforms.5 It includes the study of function spaces (like L^p spaces), singular integrals, and wavelets.11 * **43: Abstract Harmonic Analysis:** Extends the ideas of harmonic analysis from Euclidean spaces to more general settings like topological groups (especially locally compact groups) and related structures.5 * **44: Integral Transforms, Operational Calculus:** Studies transformations that map functions to other functions via integration with a kernel function, such as the Laplace transform, Fourier transform, Mellin transform, and Hilbert transform.5 These are often used to convert differential or integral equations into simpler algebraic problems.67 * **45: Integral Equations:** Integral equations are equations where the unknown function appears within an integral.5 Major types include Fredholm equations (fixed integration limits) and Volterra equations (variable integration limit), further classified into equations of the first and second kind depending on whether the unknown function appears only inside or also outside the integral.73 * **46: Functional Analysis:** Functional analysis extends methods from linear algebra and analysis to study infinite-dimensional vector spaces, particularly spaces of functions (e.g., Banach spaces, Hilbert spaces).5 It investigates linear operators between these spaces, duality, and topological properties, providing essential tools for PDEs, quantum mechanics, and probability theory.5 * **47: Operator Theory:** A branch of functional analysis focusing specifically on the properties of linear operators acting on function spaces.5 Key areas include spectral theory (generalizing eigenvalues and eigenvectors to infinite dimensions), operator algebras (algebras whose elements are operators), and the study of specific classes of operators (e.g., compact, self-adjoint, Fredholm operators).11 * **49: Calculus of Variations and Optimal Control; Optimization:** Calculus of variations seeks functions that maximize or minimize functionals (mappings from functions to real numbers), typically expressed as integrals involving the function and its derivatives.11 Optimal control theory extends this to finding control strategies for dynamical systems to optimize some performance measure.11 Optimization (or Mathematical Programming, MSC 90\) is the broader field of finding the best element from a set according to some criterion.80 * **58: Global Analysis, Analysis on Manifolds:** This field applies the tools of analysis (calculus, differential equations, functional analysis) to geometric objects called manifolds (spaces that locally resemble Euclidean space).11 It studies global properties of manifolds and mappings between them, including topics like differential topology, Morse theory, index theory, and geometric analysis.82 ## **5\. Geometry and Topology (MSC 51-55, 57\)** ### **Overview** Geometry and topology are branches of mathematics concerned with the study of space and shape. Geometry traditionally focuses on properties like distance, size, angle, and relative position of figures 2, originating from practical needs like land measurement 85 and modeling the physical world.87 Topology, often described informally as "rubber-sheet geometry" 87, studies more fundamental properties of shapes that remain unchanged under continuous deformations like stretching or bending, but not tearing or gluing.2 These include properties like connectedness, compactness, and the number of "holes".87 Both fields have evolved significantly from their classical roots in Euclidean geometry, branching into numerous subfields that employ different tools (algebra, calculus, combinatorics) and focus on different aspects or abstractions of spatial structure.2 Differential geometry applies calculus to study smooth shapes, algebraic geometry uses algebraic equations, discrete geometry examines finite arrangements, and algebraic topology uses algebraic invariants to classify topological spaces.2 Together, these fields provide a multi-faceted exploration of "space," examining its properties at varying levels of rigidity and abstraction, from precise metric details to fundamental connectivity. ### **Geometry (MSC 51, 52, 53\)** * **51: Geometry:** This broad category encompasses foundational and classical geometric systems. * **Euclidean Geometry:** The classical geometry based on Euclid's axioms, dealing with points, lines, planes, circles, polygons, congruence, and similarity in flat space.87 It proceeds logically from axioms to theorems (synthetic geometry).92 * **Projective Geometry:** Studies properties invariant under projective transformations, effectively adding "points at infinity" where parallel lines meet, simplifying incidence properties.2 * **Affine Geometry:** Focuses on properties preserved by affine transformations, primarily parallelism, while disregarding concepts of distance and angle.2 * **Non-Euclidean Geometries:** Geometries resulting from modifying or negating Euclid's parallel postulate, such as hyperbolic geometry and elliptic (or spherical) geometry.97 * **Finite Geometry (MSC 51E):** Studies geometric structures built from a finite number of points and lines, often related to combinatorial designs.11 * **Metric Geometry (MSC 51F):** Focuses on spaces equipped with a distance function (metric).11 * **Ordered Geometry (MSC 51G):** Incorporates notions of order or "betweenness" into geometric axioms.11 * **52: Convex and Discrete Geometry:** This area deals with specific types of geometric objects and configurations. * **Convex Geometry:** The study of convex sets (sets containing the line segment between any two of their points) and related notions like polytopes, convex hulls, and their properties.2 It has strong connections to optimization.2 * **Discrete Geometry (Combinatorial Geometry):** Investigates combinatorial properties of discrete collections of geometric objects like points, lines, circles, and polygons, including problems of packing, covering, tiling, and incidence structures.2 * **53: Differential Geometry:** Applies the tools of calculus and linear algebra to study the geometry of smooth curves, surfaces, and higher-dimensional manifolds.2 * **Classical Differential Geometry (MSC 53A):** Studies curves and surfaces in Euclidean 3-space.11 * **Riemannian Geometry (part of 53C, 53K):** Studies Riemannian manifolds, which are smooth manifolds equipped with a Riemannian metric that allows defining lengths, angles, areas, volumes, and curvature intrinsically.2 Fundamental to general relativity.90 * **Symplectic Geometry (MSC 53D):** Studies symplectic manifolds, which provide the mathematical framework for classical Hamiltonian mechanics.11 * **Complex Geometry (MSC 53C, 32Q):** Studies manifolds with complex structures, such as complex manifolds and Kähler manifolds.2 ### **Topology (MSC 54, 55, 57\)** * **54: General Topology (Point-Set Topology):** Establishes the foundational concepts and language of topology, defining topological spaces via open sets, and studying fundamental properties like continuity, convergence, connectedness, compactness, and separation axioms (e.g., Hausdorff spaces).11 * **55: Algebraic Topology:** Uses algebraic tools, primarily from group theory and homological algebra, to study and classify topological spaces.2 It assigns algebraic invariants to spaces that capture their topological structure, often up to homotopy equivalence (a more flexible notion than homeomorphism).91 * **Homotopy Theory (MSC 55P, 55Q, 55U):** Studies topological spaces by considering continuous deformations (homotopies) between maps. Key objects include homotopy groups (like the fundamental group, which captures information about loops).11 * **Homology Theory (MSC 55N):** Associates sequences of abelian groups (homology groups) to topological spaces, capturing information about "holes" of various dimensions.11 * **Cohomology Theory (MSC 55N):** A dual theory to homology, often providing richer algebraic structure (e.g., a ring structure) to associate with topological spaces.11 * **57: Manifolds and Cell Complexes:** This area focuses on specific, well-behaved types of topological spaces. * **Manifolds:** Topological spaces that locally resemble Euclidean space of some fixed dimension n.2 Examples include curves (1D), surfaces (2D, like spheres and tori), and the spacetime of general relativity (4D).108 * **Differential Topology (MSC 57R):** Studies smooth (differentiable) manifolds and smooth maps between them, focusing on properties invariant under diffeomorphism (smooth homeomorphism).11 It bridges topology and differential geometry but ignores metric properties.90 * **Geometric Topology (MSC 57M, 57N):** Focuses particularly on the topology of low-dimensional manifolds (dimensions 2, 3, and 4), where unique phenomena occur.11 * **Knot Theory (MSC 57K, formerly 57M):** Studies mathematical knots, which are embeddings of circles into 3-dimensional space, classifying them up to continuous deformation.11 * **Cell Complexes (e.g., CW Complexes):** Methods for constructing topological spaces by systematically gluing together simple building blocks (cells) of various dimensions.11 ## **6\. Discrete Mathematics (MSC 05, 68, 94\)** ### **Overview** Discrete mathematics is the study of mathematical structures that are fundamentally discrete, meaning they consist of separate, distinct parts, rather than continuous.2 This contrasts with fields like calculus and analysis which deal with continuous quantities and smooth change.111 It encompasses topics where objects are countable, often finite, such as integers, graphs, logical statements, and finite sets.110 The rise of digital computers, which operate on discrete data, has significantly amplified the importance and application of discrete mathematics, making it the foundational mathematical language for computer science.110 Core areas include combinatorics (the science of counting and arrangement), graph theory (the study of networks), logic, and theoretical computer science.2 These fields provide the essential theoretical framework for understanding algorithms, data structures, computation limits, network design, cryptography, and information coding.110 ### **Core Areas of Discrete Mathematics** * **05: Combinatorics:** The branch of mathematics concerned with the study of finite or countable discrete structures, including aspects of enumeration (counting), existence, construction, and optimization of configurations.2 * **05A: Enumerative Combinatorics:** Focuses on counting the number of combinatorial objects of a given type or size.5 Techniques include generating functions, recurrence relations, and bijective proofs.27 * **05B: Designs and Configurations:** Studies arrangements of elements into subsets (blocks) with specific intersection properties, such as block designs, finite geometries, and Latin squares.11 Has applications in statistics (design of experiments), coding theory, and cryptography.98 * **05C: Graph Theory:** The study of graphs, which are mathematical structures used to model pairwise relations between objects, consisting of vertices (nodes) connected by edges (links).2 Includes topics like graph coloring, connectivity, network flows, planar graphs, and spectral graph theory.5 * **05D: Extremal Combinatorics:** Investigates the maximum or minimum possible size of a combinatorial structure satisfying certain properties.8 Includes extremal set theory (e.g., Sperner's theorem, Erdős–Ko–Rado theorem) and Ramsey theory (finding order in large structures).31 * **05E: Algebraic Combinatorics:** Uses techniques from abstract algebra (like group theory, representation theory, commutative algebra) to solve combinatorial problems, and applies combinatorial methods to algebra.5 Topics include symmetric functions, Young tableaux, association schemes, and combinatorial commutative algebra.122 * **05F: Matroid Theory (formerly 05B35):** Abstractly generalizes the notion of linear independence from vector spaces and cycle structure in graphs.2 Matroids can be defined via independent sets, bases, circuits, rank functions, or closure operators.124 Includes the study of matroid representations over fields.125 * **Partition Theory (related to 05A, 11P):** Studies the ways integers can be represented as sums of positive integers.98 * **Probabilistic Combinatorics:** Employs methods from probability theory to prove the existence of combinatorial structures or analyze properties of random structures. * **Topological Combinatorics:** Applies topological methods (e.g., from algebraic topology) to problems in combinatorics.102 * **68: Computer Science (Theoretical Aspects):** This field provides the mathematical foundations for computation and computer systems.8 * **68Q: Theory of Computation:** Includes automata theory, formal languages, computability theory, and computational complexity theory.11 * **Automata Theory:** Studies abstract machines (like finite automata, pushdown automata, Turing machines) and the computational problems they can solve.31 * **Formal Languages:** Investigates formal systems for describing sets of strings (languages) and their relationship to automata.112 * **Computability Theory (also 03D):** Determines which problems are solvable by algorithms in principle.5 * **Complexity Theory:** Classifies computational problems based on the resources (like time and memory) required to solve them (e.g., P vs NP problem).112 * **68R: Discrete Mathematics in Relation to Computer Science:** Covers topics like graph algorithms, combinatorial algorithms, and data structures.11 * **68W: Algorithms:** Focuses on the design, analysis, and implementation of efficient algorithms for solving computational problems.8 * **94: Information and Communication Theory, Circuits (Theoretical Aspects):** Deals with the mathematical principles underlying information processing and transmission. * **94A: Communication, Information:** Includes information theory, which quantifies information (using concepts like entropy) and studies the fundamental limits of data compression and reliable communication over noisy channels.8 * **94B: Coding Theory:** Develops methods (codes) for representing data efficiently (data compression) and for detecting and correcting errors introduced during transmission or storage.2 Closely related to combinatorics and finite geometry.98 * **94A60: Cryptography (also 68P25):** The study and practice of techniques for secure communication in the presence of adversaries, including encryption and authentication protocols.112 Relies heavily on number theory, algebra, complexity theory, and probability. ## **7\. Number Theory (MSC 11\)** ### **Overview** Number theory, often lauded by Carl Friedrich Gauss as the "queen of mathematics" 8, is the branch of pure mathematics primarily dedicated to the study of the integers (..., \-2, \-1, 0, 1, 2,...) and integer-valued functions.2 While its objects of study appear elementary, the questions it poses—particularly concerning prime numbers, divisibility, and solutions to equations in integers (Diophantine equations)—are often profound and challenging, requiring sophisticated tools from across the mathematical spectrum.8 This deep interconnectedness is a defining characteristic; solving problems about integers frequently involves techniques from analysis (analytic number theory), abstract algebra (algebraic number theory), geometry (geometry of numbers, Diophantine geometry), and combinatorics (combinatorial number theory).9 Thus, number theory serves as a central crossroads where diverse mathematical ideas converge and interact, illustrating the unity of the discipline. ### **Subfields of Number Theory** * **11A: Elementary Number Theory:** Investigates properties of integers using methods that do not rely on advanced theories from other branches of mathematics.11 Topics include divisibility, prime factorization, congruences, modular arithmetic, primitive roots, quadratic residues, and simple Diophantine equations. * **11B: Sequences and Sets:** Studies properties of specific sequences (like Fibonacci numbers, Bernoulli numbers) and sets of integers, often involving combinatorial aspects.11 * **11C: Polynomials and Matrices:** Considers number-theoretic questions related to polynomials and matrices with integer or rational entries.11 * **11D: Diophantine Equations:** Focuses on finding integer or rational solutions to polynomial equations with integer coefficients.11 This includes famous problems like Fermat's Last Theorem. * **11E: Forms and Linear Algebraic Groups:** Studies quadratic forms (homogeneous polynomials of degree 2\) and bilinear forms over fields and rings, and their connections to linear algebraic groups.11 * **11F: Discontinuous Groups and Automorphic Forms:** Investigates functions, such as modular forms and Maass forms, defined on the complex upper half-plane (or generalizations) that exhibit specific transformation properties under the action of discrete groups (like SL(2,Z)).11 This field has deep connections to elliptic curves, Galois representations, and L-functions (e.g., the Modularity Theorem). * **11G: Arithmetic Algebraic Geometry (Diophantine Geometry):** Applies techniques from algebraic geometry to study Diophantine equations and arithmetic properties of algebraic varieties defined over number fields or finite fields.11 A central topic is the study of elliptic curves and their arithmetic.27 * **11H: Geometry of Numbers:** Uses geometric concepts, particularly the study of lattices and convex bodies in Euclidean space, to address problems in number theory, especially concerning Diophantine approximation and quadratic forms.11 * **11J: Diophantine Approximation:** Concerns the approximation of real numbers by rational numbers, including measures of irrationality and transcendence theory.11 * **Transcendental Number Theory (part of 11J):** Studies numbers that are not roots of any non-zero polynomial equation with integer coefficients (transcendental numbers), such as π and e.31 * **11K: Probabilistic Number Theory; Distribution Modulo 1; Metric Theory of Algorithms:** Applies methods from probability theory to study the distribution of arithmetic sequences and properties of integers.11 Also includes the study of uniform distribution of sequences and metric properties related to algorithms like continued fractions. * **11L: Exponential Sums and Character Sums:** Studies sums involving exponential functions or group characters, which are important tools in analytic number theory for problems like counting solutions to congruences or analyzing the distribution of primes.11 * **11M: Zeta and L-functions: Analytic Theory:** Focuses on the analytic properties (e.g., location of zeros, functional equations) of the Riemann zeta function and more general L-functions, which encode deep arithmetic information.11 Includes the study of the distribution of prime numbers (Prime Number Theorem). * **11N: Multiplicative Number Theory:** Deals with the distribution of prime numbers and properties of multiplicative functions (functions f such that f(mn)=f(m)f(n) for coprime m, n).11 Includes sieve methods. * **11P: Additive Number Theory; Partitions:** Studies representations of integers as sums of elements from given sets (e.g., Waring's problem, Goldbach conjecture) and the theory of integer partitions (ways of writing an integer as a sum of positive integers).5 * **11R: Algebraic Number Theory: Global Fields:** Investigates algebraic number fields (finite extensions of the rational numbers Q) and their rings of integers, ideal class groups, units, and connections to Galois theory.5 Includes class field theory, which describes abelian extensions of number fields.27 * **11S: Algebraic Number Theory: Local and p-adic Fields:** Studies completions of number fields, particularly p-adic fields (completions with respect to a p-adic valuation), and their algebraic properties.11 Local methods are often crucial for solving global problems (Hasse principle). * **11T: Finite Fields and Commutative Rings (Number-Theoretic Aspects):** Deals with the structure and arithmetic of finite fields and related commutative rings, with applications in coding theory and cryptography.11 * **11U: Connections with Logic:** Explores interactions between number theory and mathematical logic, such as decidability questions for number-theoretic statements (e.g., Hilbert's tenth problem).11 * **11Y: Computational Number Theory:** Focuses on the development and analysis of algorithms for solving number-theoretic problems, including primality testing, integer factorization, and discrete logarithms.11 Essential for modern cryptography. * **11Z: Miscellaneous:** Topics not fitting elsewhere. ## **8\. Probability Theory and Statistics (MSC 60, 62\)** ### **Overview** Probability theory and statistics are closely related disciplines concerned with understanding randomness and extracting information from data.129 Probability theory provides the formal mathematical framework for modeling uncertainty and random phenomena, treating probability rigorously through axioms and probability spaces.129 Statistics applies these probabilistic models to collect, analyze, interpret, and present data, aiming to make inferences about larger populations based on samples and to make decisions in the face of uncertainty.130 While probability deduces outcomes from theoretical models, statistics induces general statements from specific data.130 This symbiotic relationship is fundamental: probability provides the theoretical underpinnings for statistical methods, while statistical applications often motivate new developments in probability theory. Together, they are essential tools in science, engineering, finance, social sciences, and many other fields involving data analysis and risk assessment.129 ### **Probability Theory (MSC 60\)** Probability theory is the mathematical study of chance events and their likelihood.2 * **60A: Foundations of Probability Theory:** Deals with the axiomatic basis of probability (e.g., Kolmogorov axioms), defining concepts like probability spaces, sample spaces, events, and probability measures.11 * **60B: Probability Theory on Algebraic and Topological Structures:** Extends probability concepts to more abstract spaces like groups, vector spaces, or manifolds.11 * **60C: Combinatorial Probability:** Focuses on probability problems involving discrete sample spaces where counting techniques (combinatorics) are essential.11 * **60E: Distribution Theory:** Studies probability distributions, which describe the probabilities of different outcomes for random variables (functions assigning numerical values to outcomes).11 Includes common distributions like binomial, Poisson, normal, exponential, etc. * **60F: Limit Theorems:** Investigates the behavior of sequences of random variables in the limit, including fundamental results like the Law of Large Numbers (convergence of averages) and the Central Limit Theorem (convergence of sums to a normal distribution).11 * **60G: Stochastic Processes:** Studies collections of random variables indexed by time or space, modeling systems that evolve randomly.11 * **60H: Stochastic Analysis:** Develops calculus for stochastic processes, including stochastic integration (like Itô and Stratonovich integrals) and stochastic differential equations (SDEs).11 Crucial in mathematical finance and physics. * **60J: Markov Processes:** Studies a class of stochastic processes where the future state depends only on the present state, not the past history (Markov property).11 Includes Markov chains (discrete time/space) and Brownian motion (continuous time/space). * **60K: Special Processes:** Deals with specific types of stochastic processes important in applications, such as queueing theory (modeling waiting lines), renewal theory, and reliability theory.11 * **60L: Rough analysis (New in MSC2020):** A relatively new area dealing with paths that are less regular than those typically considered in stochastic analysis, relevant for modeling phenomena with rough trajectories.12 ### **Statistics (MSC 62\)** Statistics is the science of collecting, organizing, analyzing, interpreting, and presenting data, particularly in the context of uncertainty.2 * **62A: Foundational and Philosophical Topics:** Explores the philosophical underpinnings of statistical inference, including different interpretations of probability (e.g., Bayesian vs. frequentist).11 * **62B: Sufficiency and Information:** Studies concepts related to the information content of data and statistics (functions of data).11 * **62C: Decision Theory:** Provides a framework for making optimal decisions based on data and probabilistic models of uncertainty.11 * **62D: Sampling Theory, Sample Surveys:** Develops methods for selecting representative samples from a population and designing surveys to collect data effectively.11 * **62E: Distribution Theory:** Studies the probability distributions of statistics derived from samples.11 * **62F: Parametric Inference:** Concerns statistical inference (estimation, hypothesis testing) when the underlying data distribution is assumed to belong to a specific family parameterized by a finite number of unknown parameters.11 * **62G: Nonparametric Inference:** Develops inference methods that make fewer assumptions about the underlying data distribution.11 * **62H: Multivariate Analysis:** Deals with statistical methods for analyzing data involving multiple measured variables simultaneously.11 * **62J: Linear Statistical Models:** Focuses on models where the relationship between variables is assumed to be linear, including regression analysis and analysis of variance (ANOVA).11 * **62K: Design of Experiments:** Studies principles and methods for planning experiments to efficiently gather data and draw valid conclusions about treatment effects, minimizing the impact of variability.11 Related to combinatorial design theory.98 * **62L: Sequential Methods:** Develops statistical procedures where the sample size is not fixed in advance, allowing decisions (e.g., stopping the experiment) to be made as data accumulates.11 * **62M: Inference from Stochastic Processes; Time Series:** Concerns statistical inference for data arising from stochastic processes, particularly time series data collected sequentially over time.11 * **62N: Survival Analysis and Reliability:** Deals with analyzing time-to-event data, common in medical studies (survival times) and engineering (failure times).11 * **62P: Applications:** Covers statistical methods tailored for specific application areas like actuarial science, biology, economics, engineering, and social sciences.11 * **62R: Statistics on algebraic and topological structures (New in MSC2020):** Includes recent developments like functional data analysis and topological data analysis.11 ## **9\. Applied Mathematics & Interdisciplinary Areas** ### **Overview** Applied mathematics serves as a vital interface between abstract mathematical concepts and the complexities of the real world.2 It involves both the application of existing mathematical methods to problems arising in science, engineering, finance, biology, and other fields, and the development of new mathematical techniques specifically motivated by these external problems.8 This interaction is often bidirectional: while applications benefit from the power and rigor of mathematics, the challenge of modeling intricate real-world phenomena frequently stimulates new theoretical mathematical research.75 Fields like physics, for example, have historically been a major source of inspiration for mathematical developments, including calculus and differential geometry.75 The advent of powerful computers has also profoundly influenced applied mathematics, making numerical simulation and computational methods central tools.136 Consequently, areas like numerical analysis, computational science, operations research, and mathematical modeling form core components of modern applied mathematics, bridging theory and practice across an ever-expanding range of disciplines.8 The distinction between "pure" and "applied" mathematics is often blurred, as abstract theories frequently find unexpected applications, and applied problems lead to deep theoretical insights.2 ### **Key Applied and Interdisciplinary Fields** * **65: Numerical Analysis:** The study and development of algorithms that use numerical approximation (as opposed to exact symbolic manipulation) to solve problems of mathematical analysis.2 Essential for solving complex equations arising in science and engineering that lack analytical solutions.136 Includes numerical linear algebra, numerical solution of ODEs and PDEs, interpolation, approximation, and numerical integration.11 * **68: Computer Science (Theoretical):** Provides the mathematical foundations for computation.8 (See Section 6 for details on algorithms, complexity, automata, etc.). * **Mathematical Physics (MSC 70-86):** Applies mathematical methods to formulate and solve problems in physics.8 * **70: Mechanics of Particles and Systems:** Classical mechanics, including Newtonian, Lagrangian, and Hamiltonian formulations.11 * **74: Mechanics of Deformable Solids:** Continuum mechanics applied to solid materials, studying elasticity, plasticity, and fracture.11 * **76: Fluid Mechanics:** Mathematical study of fluids (liquids, gases, plasmas) at rest and in motion, governed by equations like Navier-Stokes.11 * **78: Optics, Electromagnetic Theory:** Mathematical description of light propagation and electromagnetic fields via Maxwell's equations and related models.11 * **80: Classical Thermodynamics, Heat Transfer:** Mathematical formulation of the laws of thermodynamics and heat flow.11 * **81: Quantum Theory:** The mathematical framework for quantum mechanics, quantum field theory, and quantum statistical mechanics, utilizing functional analysis, operator algebras, and representation theory.11 * **82: Statistical Mechanics, Structure of Matter:** Uses probability and statistics to connect microscopic properties of particles to macroscopic behavior of matter.11 * **83: Relativity and Gravitational Theory:** Mathematical description of spacetime and gravity using differential geometry (Riemannian and pseudo-Riemannian manifolds).11 * **85: Astronomy and Astrophysics:** Mathematical modeling of celestial mechanics, stellar structure, cosmology, etc..11 * **86: Geophysics:** Application of mathematical methods to study the Earth and its atmosphere/oceans.11 * **90: Operations Research (OR), Mathematical Programming:** Uses analytical methods like optimization, simulation, queueing theory, and stochastic processes to improve decision-making and efficiency in complex systems (e.g., logistics, scheduling, resource allocation).8 Often involves finding optimal (maximum or minimum) values under constraints.138 * **91: Game Theory, Economics, Finance, and Other Social and Behavioral Sciences:** Applies mathematical models to understand strategic interactions and complex systems in human behavior and society. * **Game Theory (91A):** The mathematical study of strategic decision-making among interacting rational agents.2 * **Mathematical Economics (91B):** Uses mathematical tools (calculus, optimization, differential equations, game theory) to model economic phenomena and theories.11 * **Mathematical Finance (91G):** Applies stochastic calculus, PDEs, and probability to model financial markets, price derivatives, and manage risk.11 * **Mathematical Sociology, Psychology, Linguistics, etc. (91D, 91E, 91F):** Development and application of mathematical models in various social and behavioral sciences.11 * **92: Biology and Other Natural Sciences:** Mathematical biology (or biomathematics) employs mathematical models, analysis, and simulations to understand biological systems and processes.8 Includes areas like population dynamics, epidemiology, mathematical ecology, neuroscience, bioinformatics, and biomechanics.156 * **93: Systems Theory; Control:** Control theory is the mathematical study of how to influence the behavior of dynamical systems using feedback to achieve desired goals, maintain stability, and optimize performance.11 Uses differential equations, stability theory, and optimization.158 * **94: Information and Communication Theory, Circuits:** Mathematical principles underlying information processing, transmission, and security. * **Information Theory (94A):** Quantifies information (entropy) and studies limits of data compression and communication.8 * **Coding Theory (94B):** Design of error-correcting codes and data compression algorithms.2 * **Cryptography (94A60):** Mathematical techniques for secure communication.112 * **Circuits, Networks (94C):** Mathematical analysis of electrical circuits.11 * **97: Mathematics Education:** The practice and research of teaching and learning mathematics, covering curriculum design, pedagogy, cognitive science, assessment, and the use of technology.11 * **01: History and Biography:** The study of the historical evolution of mathematical concepts, methods, and notations, as well as the lives and work of mathematicians.2 * **00A08: Recreational Mathematics:** Mathematical puzzles, games, and problems pursued for enjoyment and education, often illustrating deeper mathematical principles.6 ## **10\. Conclusion** This report has undertaken the task of compiling an extensive, though inevitably incomplete, list of topics spanning the vast domain of mathematics. From the foundational inquiries of logic and set theory to the abstract structures of algebra, the rigorous study of continuity and change in analysis, the exploration of shape and space in geometry and topology, the counting and structuring principles of discrete mathematics, the intricate properties of numbers in number theory, the modeling of randomness in probability and statistics, and the diverse applications across science, engineering, and society found in applied mathematics – the sheer scope is immense. A recurring theme is the profound interconnectedness of these fields. While classification systems like the Mathematics Subject Classification (MSC), used herein for structure and reference 7, are indispensable tools for navigating this landscape, they necessarily simplify a reality where boundaries are fluid and interdependencies are deep.9 Number theory draws upon almost all other areas 9; algebraic topology bridges algebra and topology 2; functional analysis underpins quantum mechanics 75; combinatorial designs inform statistical experiments.98 This interconnectedness highlights mathematics not just as a collection of disparate subjects, but as a unified, albeit complex, whole. Furthermore, mathematics is revealed not as a static body of knowledge, but as a dynamic and continually evolving discipline.2 Its history shows periods of explosive growth spurred by internal questions (like foundational crises 2) and external stimuli (like problems in physics 75 or the rise of computation 111). The ongoing revision of classification systems 7 and the emergence of new fields and MSC codes 12 reflect this continuous development. The pursuit of abstraction in algebra 5, the refinement of concepts of continuity in analysis 5, the varied perspectives on space in geometry and topology, and the increasing relevance of discrete structures 111 all contribute to this ongoing evolution. Applied mathematics, in particular, acts as both a conduit for existing knowledge and an engine for new theoretical development, driven by the need to model an increasingly complex world.75 Consequently, any comprehensive list of mathematical topics represents a snapshot of a field whose frontiers are constantly expanding. #### **Works cited** 1. Mathematics \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/Mathematics.html](https://mathworld.wolfram.com/Mathematics.html) 2. Mathematics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Mathematics](https://en.wikipedia.org/wiki/Mathematics) 3. Definitions of mathematics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Definitions\_of\_mathematics](https://en.wikipedia.org/wiki/Definitions_of_mathematics) 4. foundations of mathematics: overview \- PlanetMath.org, accessed April 10, 2025, [https://planetmath.org/foundationsofmathematicsoverview](https://planetmath.org/foundationsofmathematicsoverview) 5. Glossary of areas of mathematics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Glossary\_of\_areas\_of\_mathematics](https://en.wikipedia.org/wiki/Glossary_of_areas_of_mathematics) 6. MSC Classification Codes, accessed April 10, 2025, [https://cloud.r-project.org/web/classifications/MSC.html](https://cloud.r-project.org/web/classifications/MSC.html) 7. Mathematics Subject Classification \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Mathematics\_Subject\_Classification](https://en.wikipedia.org/wiki/Mathematics_Subject_Classification) 8. Lists of mathematics topics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Lists\_of\_mathematics\_topics](https://en.wikipedia.org/wiki/Lists_of_mathematics_topics) 9. \[General\] List of areas of mathematics? : r/learnmath \- Reddit, accessed April 10, 2025, [https://www.reddit.com/r/learnmath/comments/mvlb2s/general\_list\_of\_areas\_of\_mathematics/](https://www.reddit.com/r/learnmath/comments/mvlb2s/general_list_of_areas_of_mathematics/) 10. Wolfram MathWorld: The Web's Most Extensive Mathematics Resource \- V.S.R. & N.V.R. College, accessed April 10, 2025, [http://vsrnvr.ac.in/Library/math.htm](http://vsrnvr.ac.in/Library/math.htm) 11. Classification Search \- zbMATH Open, accessed April 10, 2025, [https://zbmath.org/classification/](https://zbmath.org/classification/) 12. Mathematics Subject Classification 2020 (MSC2020), accessed April 10, 2025, [https://msc2020.org/](https://msc2020.org/) 13. Mathematics Subject Classification 2020 \- SIAM.org, accessed April 10, 2025, [https://www.siam.org/publications/siam-news/articles/mathematics-subject-classification-2020/](https://www.siam.org/publications/siam-news/articles/mathematics-subject-classification-2020/) 14. MSC2020-Mathematics Subject Classification System \- zbMATH Open, accessed April 10, 2025, [https://zbmath.org/static/msc2020.pdf](https://zbmath.org/static/msc2020.pdf) 15. Foundations of Mathematics \- Alex Sakharov, accessed April 10, 2025, [http://sakharov.net/foundation.html](http://sakharov.net/foundation.html) 16. Mathematical Logic, accessed April 10, 2025, [https://www.karlin.mff.cuni.cz/\~krajicek/mendelson.pdf](https://www.karlin.mff.cuni.cz/~krajicek/mendelson.pdf) 17. mathworld.wolfram.com, accessed April 10, 2025, [https://mathworld.wolfram.com/SetTheory.html\#:\~:text=Set%20theory%20is%20the%20mathematical,its%20own%20rules%20and%20axioms.](https://mathworld.wolfram.com/SetTheory.html#:~:text=Set%20theory%20is%20the%20mathematical,its%20own%20rules%20and%20axioms.) 18. Mathematical structure \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Mathematical\_structure](https://en.wikipedia.org/wiki/Mathematical_structure) 19. Philosophical logic \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Philosophical\_logic](https://en.wikipedia.org/wiki/Philosophical_logic) 20. Logical reasoning \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Logical\_reasoning](https://en.wikipedia.org/wiki/Logical_reasoning) 21. Glossary of logic \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Glossary\_of\_logic](https://en.wikipedia.org/wiki/Glossary_of_logic) 22. MSC2020-Mathematics Subject Classification System, accessed April 10, 2025, [https://msc2020.org/MSC\_2020.tex](https://msc2020.org/MSC_2020.tex) 23. Category:Philosophical logic \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Category:Philosophical\_logic](https://en.wikipedia.org/wiki/Category:Philosophical_logic) 24. Non-standard model \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Non-standard\_model](https://en.wikipedia.org/wiki/Non-standard_model) 25. Non-standard model of arithmetic \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Non-standard\_model\_of\_arithmetic](https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic) 26. Set Theory \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/SetTheory.html](https://mathworld.wolfram.com/SetTheory.html) 27. Wikipedia:WikiProject Mathematics/List of mathematics categories, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Wikipedia:WikiProject\_Mathematics/List\_of\_mathematics\_categories](https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/List_of_mathematics_categories) 28. Algebraic logic \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Algebraic\_logic](https://en.wikipedia.org/wiki/Algebraic_logic) 29. Abstract algebraic logic \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Abstract\_algebraic\_logic](https://en.wikipedia.org/wiki/Abstract_algebraic_logic) 30. Portal:Mathematics/Categories \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Portal:Mathematics/Categories](https://en.wikipedia.org/wiki/Portal:Mathematics/Categories) 31. List of mathematical theories \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/List\_of\_mathematical\_theories](https://en.wikipedia.org/wiki/List_of_mathematical_theories) 32. Algebraic topology \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Algebraic\_topology](https://en.wikipedia.org/wiki/Algebraic_topology) 33. Algebra \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Algebra](https://en.wikipedia.org/wiki/Algebra) 34. Wolfram MathWorld: The Web's Most Extensive Mathematics Resource, accessed April 10, 2025, [https://mathworld.wolfram.com/](https://mathworld.wolfram.com/) 35. Ring theory \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Ring\_theory](https://en.wikipedia.org/wiki/Ring_theory) 36. Category:Ring theory \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Category:Ring\_theory](https://en.wikipedia.org/wiki/Category:Ring_theory) 37. Associative algebra \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Associative\_algebra](https://en.wikipedia.org/wiki/Associative_algebra) 38. Algebra over a field \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Algebra\_over\_a\_field](https://en.wikipedia.org/wiki/Algebra_over_a_field) 39. Non-associative algebra \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Non-associative\_algebra](https://en.wikipedia.org/wiki/Non-associative_algebra) 40. non-associative algebra \- Wiktionary, the free dictionary, accessed April 10, 2025, [https://en.wiktionary.org/wiki/non-associative\_algebra](https://en.wiktionary.org/wiki/non-associative_algebra) 41. Lie algebra \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Lie\_algebra](https://en.wikipedia.org/wiki/Lie_algebra) 42. Jordan algebra \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Jordan\_algebra](https://en.wikipedia.org/wiki/Jordan_algebra) 43. Alternative algebra \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Alternative\_algebra](https://en.wikipedia.org/wiki/Alternative_algebra) 44. Potential theory \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Potential\_theory](https://en.wikipedia.org/wiki/Potential_theory) 45. Category:Potential theory \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Category:Potential\_theory](https://en.wikipedia.org/wiki/Category:Potential_theory) 46. Potential Theory \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/PotentialTheory.html](https://mathworld.wolfram.com/PotentialTheory.html) 47. potential theory \- PlanetMath.org, accessed April 10, 2025, [https://planetmath.org/potentialtheory](https://planetmath.org/potentialtheory) 48. Function of several complex variables \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Function\_of\_several\_complex\_variables](https://en.wikipedia.org/wiki/Function_of_several_complex_variables) 49. Special functions \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Special\_functions](https://en.wikipedia.org/wiki/Special_functions) 50. Special functions \- Simple English Wikipedia, the free encyclopedia, accessed April 10, 2025, [https://simple.wikipedia.org/wiki/Special\_functions](https://simple.wikipedia.org/wiki/Special_functions) 51. Special Function \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/SpecialFunction.html](https://mathworld.wolfram.com/SpecialFunction.html) 52. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Ordinary\_differential\_equation\#:\~:text=In%20mathematics%2C%20an%20ordinary%20differential,the%20derivatives%20of%20those%20functions.](https://en.wikipedia.org/wiki/Ordinary_differential_equation#:~:text=In%20mathematics%2C%20an%20ordinary%20differential,the%20derivatives%20of%20those%20functions.) 53. Ordinary differential equation \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Ordinary\_differential\_equation](https://en.wikipedia.org/wiki/Ordinary_differential_equation) 54. mathworld.wolfram.com, accessed April 10, 2025, [https://mathworld.wolfram.com/DifferentialEquation.html\#:\~:text=A%20differential%20equation%20is%20an,called%20an%20ordinary%20differential%20equation.](https://mathworld.wolfram.com/DifferentialEquation.html#:~:text=A%20differential%20equation%20is%20an,called%20an%20ordinary%20differential%20equation.) 55. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Partial\_differential\_equation\#:\~:text=also%20a%20solution.-,Definition,the%20unknown%20function%20of%20variables](https://en.wikipedia.org/wiki/Partial_differential_equation#:~:text=also%20a%20solution.-,Definition,the%20unknown%20function%20of%20variables) 56. Partial differential equation \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Partial\_differential\_equation](https://en.wikipedia.org/wiki/Partial_differential_equation) 57. Partial Derivative \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/PartialDerivative.html](https://mathworld.wolfram.com/PartialDerivative.html) 58. Partial Differential Equation \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/PartialDifferentialEquation.html](https://mathworld.wolfram.com/PartialDifferentialEquation.html) 59. Category:Fields of mathematics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Category:Fields\_of\_mathematics](https://en.wikipedia.org/wiki/Category:Fields_of_mathematics) 60. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Functional\_equation\#:\~:text=In%20mathematics%2C%20a%20functional%20equation,integral%20equations%20are%20functional%20equations.](https://en.wikipedia.org/wiki/Functional_equation#:~:text=In%20mathematics%2C%20a%20functional%20equation,integral%20equations%20are%20functional%20equations.) 61. Functional equation \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Functional\_equation](https://en.wikipedia.org/wiki/Functional_equation) 62. mathworld.wolfram.com, accessed April 10, 2025, [https://mathworld.wolfram.com/FunctionalEquation.html\#:\~:text=An%20equation%20of%20the%20form,of%20functional%20equations%20they%20satisfy.](https://mathworld.wolfram.com/FunctionalEquation.html#:~:text=An%20equation%20of%20the%20form,of%20functional%20equations%20they%20satisfy.) 63. functional equation \- PlanetMath.org, accessed April 10, 2025, [https://planetmath.org/functionalequation](https://planetmath.org/functionalequation) 64. Equation \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Equation](https://en.wikipedia.org/wiki/Equation) 65. Functional differential equation \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Functional\_differential\_equation](https://en.wikipedia.org/wiki/Functional_differential_equation) 66. Category:Topological groups \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Category:Topological\_groups](https://en.wikipedia.org/wiki/Category:Topological_groups) 67. Integral transform \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Integral\_transform](https://en.wikipedia.org/wiki/Integral_transform) 68. Integral Transform \- Wikipedia | PDF | Linear Algebra | Mathematical Analysis \- Scribd, accessed April 10, 2025, [https://www.scribd.com/document/569633009/Integral-Transform-Wikipedia](https://www.scribd.com/document/569633009/Integral-Transform-Wikipedia) 69. Integral Transforms \- Wolfram Cloud, accessed April 10, 2025, [https://www.wolframcloud.com/objects/summerschool/pages/2017/SohamPal\_TE](https://www.wolframcloud.com/objects/summerschool/pages/2017/SohamPal_TE) 70. Integral transform | Fourier, Laplace & Laplace-Stieltjes \- Britannica, accessed April 10, 2025, [https://www.britannica.com/science/integral-transform](https://www.britannica.com/science/integral-transform) 71. Buschman Transform \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/BuschmanTransform.html](https://mathworld.wolfram.com/BuschmanTransform.html) 72. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Integral\_equation\#:\~:text=In%20mathematical%20analysis%2C%20integral%20equations,integral%20operator%20acting%20on%20u.](https://en.wikipedia.org/wiki/Integral_equation#:~:text=In%20mathematical%20analysis%2C%20integral%20equations,integral%20operator%20acting%20on%20u.) 73. Integral Equation \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/IntegralEquation.html](https://mathworld.wolfram.com/IntegralEquation.html) 74. integral equation \- PlanetMath.org, accessed April 10, 2025, [https://planetmath.org/integralequation](https://planetmath.org/integralequation) 75. Mathematical physics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Mathematical\_physics](https://en.wikipedia.org/wiki/Mathematical_physics) 76. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Calculus\_of\_variations\#:\~:text=The%20calculus%20of%20variations%20(or,functions%20to%20the%20real%20numbers.](https://en.wikipedia.org/wiki/Calculus_of_variations#:~:text=The%20calculus%20of%20variations%20\(or,functions%20to%20the%20real%20numbers.) 77. Calculus of variations \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Calculus\_of\_variations](https://en.wikipedia.org/wiki/Calculus_of_variations) 78. mathworld.wolfram.com, accessed April 10, 2025, [https://mathworld.wolfram.com/CalculusofVariations.html\#:\~:text=A%20branch%20of%20mathematics%20that,usually%20a%20minimum%20or%20maximum).](https://mathworld.wolfram.com/CalculusofVariations.html#:~:text=A%20branch%20of%20mathematics%20that,usually%20a%20minimum%20or%20maximum\).) 79. The Calculus of Variations: An Introduction, accessed April 10, 2025, [https://www.uu.edu/dept/math/seniorpapers/03-04/goshi.pdf](https://www.uu.edu/dept/math/seniorpapers/03-04/goshi.pdf) 80. Mathematical optimization \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Mathematical\_optimization](https://en.wikipedia.org/wiki/Mathematical_optimization) 81. Optimization Theory \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/OptimizationTheory.html](https://mathworld.wolfram.com/OptimizationTheory.html) 82. Global analysis \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Global\_analysis](https://en.wikipedia.org/wiki/Global_analysis) 83. Stochastic analysis on manifolds \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Stochastic\_analysis\_on\_manifolds](https://en.wikipedia.org/wiki/Stochastic_analysis_on_manifolds) 84. Global and Local \- James Franklin \- PhilArchive, accessed April 10, 2025, [https://philarchive.org/archive/FRAGAL-2](https://philarchive.org/archive/FRAGAL-2) 85. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Geometry\#:\~:text=Geometry%20(from%20Ancient%20Greek%20%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1,and%20relative%20position%20of%20figures.](https://en.wikipedia.org/wiki/Geometry#:~:text=Geometry%20\(from%20Ancient%20Greek%20%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1,and%20relative%20position%20of%20figures.) 86. mathworld.wolfram.com, accessed April 10, 2025, [https://mathworld.wolfram.com/Geometry.html\#:\~:text=Geometry%20is%20the%20study%20of,and%20of%20a%20given%20type.](https://mathworld.wolfram.com/Geometry.html#:~:text=Geometry%20is%20the%20study%20of,and%20of%20a%20given%20type.) 87. Geometry \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Geometry](https://en.wikipedia.org/wiki/Geometry) 88. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Category:Topology\#:\~:text=In%20mathematics%2C%20topology%20is%20a,common%20terms%20and%20their%20definition.](https://en.wikipedia.org/wiki/Category:Topology#:~:text=In%20mathematics%2C%20topology%20is%20a,common%20terms%20and%20their%20definition.) 89. Topology \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/Topology.html](https://mathworld.wolfram.com/Topology.html) 90. Differential geometry \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Differential\_geometry](https://en.wikipedia.org/wiki/Differential_geometry) 91. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Algebraic\_topology\#:\~:text=Algebraic%20topology%20is%20a%20branch,classify%20up%20to%20homotopy%20equivalence.](https://en.wikipedia.org/wiki/Algebraic_topology#:~:text=Algebraic%20topology%20is%20a%20branch,classify%20up%20to%20homotopy%20equivalence.) 92. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Euclidean\_geometry\#:\~:text=Euclidean%20geometry%20is%20an%20example,to%20propositions%20about%20those%20objects.](https://en.wikipedia.org/wiki/Euclidean_geometry#:~:text=Euclidean%20geometry%20is%20an%20example,to%20propositions%20about%20those%20objects.) 93. Euclidean geometry \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Euclidean\_geometry](https://en.wikipedia.org/wiki/Euclidean_geometry) 94. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Projective\_geometry\#:\~:text=In%20mathematics%2C%20projective%20geometry%20is,set%20of%20basic%20geometric%20concepts.](https://en.wikipedia.org/wiki/Projective_geometry#:~:text=In%20mathematics%2C%20projective%20geometry%20is,set%20of%20basic%20geometric%20concepts.) 95. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Affine\_geometry\#:\~:text=Affine%20geometry%20can%20be%20viewed,developed%20in%20synthetic%20finite%20geometry.](https://en.wikipedia.org/wiki/Affine_geometry#:~:text=Affine%20geometry%20can%20be%20viewed,developed%20in%20synthetic%20finite%20geometry.) 96. Affine geometry \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Affine\_geometry](https://en.wikipedia.org/wiki/Affine_geometry) 97. Geometry \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/Geometry.html](https://mathworld.wolfram.com/Geometry.html) 98. Combinatorics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Combinatorics](https://en.wikipedia.org/wiki/Combinatorics) 99. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Convex\_set\#:\~:text=In%20geometry%2C%20a%20set%20of,point%2C%20or%20the%20empty%20set.](https://en.wikipedia.org/wiki/Convex_set#:~:text=In%20geometry%2C%20a%20set%20of,point%2C%20or%20the%20empty%20set.) 100. Convex set \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Convex\_set](https://en.wikipedia.org/wiki/Convex_set) 101. Category:Discrete geometry \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Category:Discrete\_geometry](https://en.wikipedia.org/wiki/Category:Discrete_geometry) 102. Discrete geometry \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Discrete\_geometry](https://en.wikipedia.org/wiki/Discrete_geometry) 103. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Differential\_geometry\#:\~:text=Differential%20geometry%20is%20the%20language,describes%20the%20curvature%20of%20spacetime.](https://en.wikipedia.org/wiki/Differential_geometry#:~:text=Differential%20geometry%20is%20the%20language,describes%20the%20curvature%20of%20spacetime.) 104. mathworld.wolfram.com, accessed April 10, 2025, [https://mathworld.wolfram.com/classroom/classes/DifferentialGeometry.html\#:\~:text=Differential%20geometry%20is%20the%20field,of%20any%20pair%20of%20indices.](https://mathworld.wolfram.com/classroom/classes/DifferentialGeometry.html#:~:text=Differential%20geometry%20is%20the%20field,of%20any%20pair%20of%20indices.) 105. Manifold \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/Manifold.html](https://mathworld.wolfram.com/Manifold.html) 106. General topology \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/General\_topology](https://en.wikipedia.org/wiki/General_topology) 107. Category:General topology \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Category:General\_topology](https://en.wikipedia.org/wiki/Category:General_topology) 108. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Manifold\#:\~:text=In%20mathematics%2C%20a%20manifold%20is,%2Ddimensional%20Euclidean%20space.](https://en.wikipedia.org/wiki/Manifold#:~:text=In%20mathematics%2C%20a%20manifold%20is,%2Ddimensional%20Euclidean%20space.) 109. Manifold \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Manifold](https://en.wikipedia.org/wiki/Manifold) 110. Discrete mathematics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Discrete\_mathematics](https://en.wikipedia.org/wiki/Discrete_mathematics) 111. Discrete Mathematics \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/DiscreteMathematics.html](https://mathworld.wolfram.com/DiscreteMathematics.html) 112. Theoretical computer science \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Theoretical\_computer\_science](https://en.wikipedia.org/wiki/Theoretical_computer_science) 113. Combinatorics \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/Combinatorics.html](https://mathworld.wolfram.com/Combinatorics.html) 114. Combinatorial design \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Combinatorial\_design](https://en.wikipedia.org/wiki/Combinatorial_design) 115. mathworld.wolfram.com, accessed April 10, 2025, [https://mathworld.wolfram.com/GraphTheory.html\#:\~:text=The%20mathematical%20study%20of%20the,formal%20mathematical%20structures%20called%20graphs.](https://mathworld.wolfram.com/GraphTheory.html#:~:text=The%20mathematical%20study%20of%20the,formal%20mathematical%20structures%20called%20graphs.) 116. Graph Theory \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/GraphTheory.html](https://mathworld.wolfram.com/GraphTheory.html) 117. Graph \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/Graph.html](https://mathworld.wolfram.com/Graph.html) 118. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Extremal\_combinatorics\#:\~:text=Extremal%20combinatorics%20is%20a%20field,has%20to%20satisfy%20certain%20restrictions.](https://en.wikipedia.org/wiki/Extremal_combinatorics#:~:text=Extremal%20combinatorics%20is%20a%20field,has%20to%20satisfy%20certain%20restrictions.) 119. Extremal combinatorics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Extremal\_combinatorics](https://en.wikipedia.org/wiki/Extremal_combinatorics) 120. Category:Combinatorics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Category:Combinatorics](https://en.wikipedia.org/wiki/Category:Combinatorics) 121. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Algebraic\_combinatorics\#:\~:text=Algebraic%20combinatorics%20is%20an%20area,techniques%20to%20problems%20in%20algebra.](https://en.wikipedia.org/wiki/Algebraic_combinatorics#:~:text=Algebraic%20combinatorics%20is%20an%20area,techniques%20to%20problems%20in%20algebra.) 122. Algebraic combinatorics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Algebraic\_combinatorics](https://en.wikipedia.org/wiki/Algebraic_combinatorics) 123. Category:Algebraic combinatorics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Category:Algebraic\_combinatorics](https://en.wikipedia.org/wiki/Category:Algebraic_combinatorics) 124. Matroid \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Matroid](https://en.wikipedia.org/wiki/Matroid) 125. Matroid representation \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Matroid\_representation](https://en.wikipedia.org/wiki/Matroid_representation) 126. Computation \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/Computation.html](https://mathworld.wolfram.com/Computation.html) 127. Information theory \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Information\_theory](https://en.wikipedia.org/wiki/Information_theory) 128. Information Theory \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/InformationTheory.html](https://mathworld.wolfram.com/InformationTheory.html) 129. Probability theory \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Probability\_theory](https://en.wikipedia.org/wiki/Probability_theory) 130. Statistics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Statistics](https://en.wikipedia.org/wiki/Statistics) 131. mathworld.wolfram.com, accessed April 10, 2025, [https://mathworld.wolfram.com/Probability.html\#:\~:text=Probability%20is%20the%20branch%20of,outcomes'%20relative%20likelihoods%20and%20distributions.](https://mathworld.wolfram.com/Probability.html#:~:text=Probability%20is%20the%20branch%20of,outcomes'%20relative%20likelihoods%20and%20distributions.) 132. Statistic \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/Statistic.html](https://mathworld.wolfram.com/Statistic.html) 133. Stochastic process \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Stochastic\_process](https://en.wikipedia.org/wiki/Stochastic_process) 134. Stochastic Process \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/StochasticProcess.html](https://mathworld.wolfram.com/StochasticProcess.html) 135. Mathematics Subject Classification – Applied Mathematics, accessed April 10, 2025, [https://www.ub.uni-heidelberg.de/fachinfo/mathe/msc/appliedmath.html](https://www.ub.uni-heidelberg.de/fachinfo/mathe/msc/appliedmath.html) 136. Numerical analysis | Mathematics, Algorithms & Computation \- Britannica, accessed April 10, 2025, [https://www.britannica.com/science/numerical-analysis](https://www.britannica.com/science/numerical-analysis) 137. Numerical analysis \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Numerical\_analysis](https://en.wikipedia.org/wiki/Numerical_analysis) 138. Operations research \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Operations\_research](https://en.wikipedia.org/wiki/Operations_research) 139. Mechanics | Definition, Examples, Laws, & Facts | Britannica, accessed April 10, 2025, [https://www.britannica.com/science/mechanics](https://www.britannica.com/science/mechanics) 140. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Fluid\_mechanics\#:\~:text=Fluid%20mechanics%20is%20the%20branch,and%20the%20forces%20on%20them.](https://en.wikipedia.org/wiki/Fluid_mechanics#:~:text=Fluid%20mechanics%20is%20the%20branch,and%20the%20forces%20on%20them.) 141. Fluid mechanics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Fluid\_mechanics](https://en.wikipedia.org/wiki/Fluid_mechanics) 142. Fluid mechanics | Definition, Equations, Types, & Facts | Britannica, accessed April 10, 2025, [https://www.britannica.com/science/fluid-mechanics](https://www.britannica.com/science/fluid-mechanics) 143. Geometrical optics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Geometrical\_optics](https://en.wikipedia.org/wiki/Geometrical_optics) 144. Mathematical descriptions of the electromagnetic field \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Mathematical\_descriptions\_of\_the\_electromagnetic\_field](https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field) 145. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Thermodynamic\_equations\#:\~:text=Thermodynamics%20is%20expressed%20by%20a,became%20the%20laws%20of%20thermodynamics.](https://en.wikipedia.org/wiki/Thermodynamic_equations#:~:text=Thermodynamics%20is%20expressed%20by%20a,became%20the%20laws%20of%20thermodynamics.) 146. Quantum state \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Quantum\_state](https://en.wikipedia.org/wiki/Quantum_state) 147. Category:Statistical mechanics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Category:Statistical\_mechanics](https://en.wikipedia.org/wiki/Category:Statistical_mechanics) 148. Introduction to the mathematics of general relativity \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Introduction\_to\_the\_mathematics\_of\_general\_relativity](https://en.wikipedia.org/wiki/Introduction_to_the_mathematics_of_general_relativity) 149. www.stonybrook.edu, accessed April 10, 2025, [https://www.stonybrook.edu/commcms/ams/graduate/or/\#:\~:text=Operations%20Research%20is%20the%20branch,management%20science%20and%20industrial%20engineering.](https://www.stonybrook.edu/commcms/ams/graduate/or/#:~:text=Operations%20Research%20is%20the%20branch,management%20science%20and%20industrial%20engineering.) 150. Game theory \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Game\_theory](https://en.wikipedia.org/wiki/Game_theory) 151. Game theory \- Simple English Wikipedia, the free encyclopedia, accessed April 10, 2025, [https://simple.wikipedia.org/wiki/Game\_theory](https://simple.wikipedia.org/wiki/Game_theory) 152. Game Theory \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/GameTheory.html](https://mathworld.wolfram.com/GameTheory.html) 153. Game Theory \- Wolfram Language Documentation, accessed April 10, 2025, [https://reference.wolfram.com/language/guide/GameTheory.html.en](https://reference.wolfram.com/language/guide/GameTheory.html.en) 154. Mathematical economics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Mathematical\_economics](https://en.wikipedia.org/wiki/Mathematical_economics) 155. Nonstandard Analysis \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/NonstandardAnalysis.html](https://mathworld.wolfram.com/NonstandardAnalysis.html) 156. Mathematical and theoretical biology \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Mathematical\_and\_theoretical\_biology](https://en.wikipedia.org/wiki/Mathematical_and_theoretical_biology) 157. mathematical biology and theoretical biophysics of DNA \- PlanetMath.org, accessed April 10, 2025, [https://planetmath.org/mathematicalbiologyandtheoreticalbiophysicsofdna](https://planetmath.org/mathematicalbiologyandtheoreticalbiophysicsofdna) 158. Control theory \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Control\_theory](https://en.wikipedia.org/wiki/Control_theory) 159. Control Theory \-- from Wolfram MathWorld, accessed April 10, 2025, [https://mathworld.wolfram.com/ControlTheory.html](https://mathworld.wolfram.com/ControlTheory.html) 160. Control Theory | Applied Mathematics \- University of Waterloo, accessed April 10, 2025, [https://uwaterloo.ca/applied-mathematics/future-undergraduates/what-you-can-learn-applied-mathematics/control-theory](https://uwaterloo.ca/applied-mathematics/future-undergraduates/what-you-can-learn-applied-mathematics/control-theory) 161. Mathematics education \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Mathematics\_education](https://en.wikipedia.org/wiki/Mathematics_education) 162. Category:Mathematics \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Category:Mathematics](https://en.wikipedia.org/wiki/Category:Mathematics) 163. en.wikipedia.org, accessed April 10, 2025, [https://en.wikipedia.org/wiki/History\_of\_mathematics\#:\~:text=The%20history%20of%20mathematics%20deals,only%20in%20a%20few%20locales.](https://en.wikipedia.org/wiki/History_of_mathematics#:~:text=The%20history%20of%20mathematics%20deals,only%20in%20a%20few%20locales.) 164. Wikipedia:Wikipedia for Schools/Welcome/Mathematics/History of Mathematics, accessed April 10, 2025, [https://en.wikipedia.org/wiki/Wikipedia:Wikipedia\_for\_Schools/Welcome/Mathematics/History\_of\_Mathematics](https://en.wikipedia.org/wiki/Wikipedia:Wikipedia_for_Schools/Welcome/Mathematics/History_of_Mathematics) 165. History of mathematical notation \- Wikipedia, accessed April 10, 2025, [https://en.wikipedia.org/wiki/History\_of\_mathematical\_notation](https://en.wikipedia.org/wiki/History_of_mathematical_notation) #### Map of mathematics 1 # The Gigantic Map of Mathematics Mathematics is a vast, interconnected web of concepts, theories, and applications that span numerous fields and subfields. This map aims to provide a comprehensive overview of the major areas of mathematics, illustrating their interrelations and key components. --- ## 1. **Foundations of Mathematics** - **Logic** - Propositional Logic - Predicate Logic - Modal Logic - Intuitionistic Logic - Proof Theory - Model Theory - Recursion Theory (Computability) - Non-Classical Logics - **Set Theory** - Naive Set Theory - Axiomatic Set Theory (ZFC) - Ordinals and Cardinals - Continuum Hypothesis - Large Cardinals - Descriptive Set Theory - Forcing and Independence Proofs - **Category Theory** - Categories, Functors, Natural Transformations - Limits and Colimits - Adjunctions - Monads and Comonads - Higher Category Theory - Topos Theory --- ## 2. **Algebra** - **Elementary Algebra** - Variables and Expressions - Equations and Inequalities - Polynomials and Factoring - Rational Expressions - **Linear Algebra** - Vectors and Vector Spaces - Matrices and Determinants - Systems of Linear Equations - Eigenvalues and Eigenvectors - Inner Product Spaces - **Abstract Algebra** - Group Theory - Finite Groups - Infinite Groups - Symmetry Groups - Lie Groups - Ring Theory - Commutative Rings - Ideals and Quotients - Noetherian Rings - Field Theory - Field Extensions - Galois Theory - Module Theory - Modules over Rings - Homological Algebra - Algebraic Structures - Quasigroups and Loops - Lattices and Boolean Algebras - **Representation Theory** - Group Representations - Character Theory - Representations of Lie Algebras - **Universal Algebra** - Algebraic Structures and Homomorphisms - Varieties and Equational Classes --- ## 3. **Number Theory** - **Elementary Number Theory** - Divisibility and Primes - Congruences - Diophantine Equations - Arithmetic Functions - **Analytic Number Theory** - Prime Number Theorem - Riemann Zeta Function - L-functions - Modular Forms - **Algebraic Number Theory** - Number Fields - Rings of Integers - Class Field Theory - Arithmetic of Elliptic Curves - **Transcendental Number Theory** - Liouville Numbers - Algebraic Independence - **Computational Number Theory** - Primality Testing - Integer Factorization - Cryptographic Applications --- ## 4. **Geometry** - **Euclidean Geometry** - Points, Lines, and Planes - Angles and Triangles - Circles and Conic Sections - **Non-Euclidean Geometry** - Hyperbolic Geometry - Elliptic Geometry - Spherical Geometry - **Differential Geometry** - Manifolds - Riemannian Geometry - Tensor Calculus - Geodesics and Curvature - **Algebraic Geometry** - Affine and Projective Varieties - Schemes and Sheaves - Intersection Theory - Computational Algebraic Geometry - **Complex Geometry** - Complex Manifolds - Hodge Theory - **Convex Geometry** - Convex Sets and Functions - Brunn-Minkowski Theory - **Discrete and Computational Geometry** - Polyhedra - Tiling and Packing - Voronoi Diagrams - Computational Algorithms --- ## 5. **Topology** - **General Topology (Point-Set Topology)** - Topological Spaces - Continuity and Homeomorphisms - Compactness and Connectedness - Metric Spaces - **Algebraic Topology** - Homotopy Theory - Fundamental Group and Covering Spaces - Homology and Cohomology - Higher Homotopy Groups - Fiber Bundles and Characteristic Classes - **Differential Topology** - Smooth Manifolds - Morse Theory - Transversality - **Geometric Topology** - Knot Theory - 3-Manifolds and 4-Manifolds - Low-Dimensional Topology - **Topological Dynamics** - Dynamical Systems - Ergodic Theory - Topological Entropy --- ## 6. **Analysis** - **Real Analysis** - Sequences and Series - Continuity and Differentiation - Integration (Riemann and Lebesgue) - Measure Theory - Functional Analysis - **Complex Analysis** - Analytic Functions - Cauchy's Theorem and Integral Formula - Laurent Series and Residues - Conformal Mapping - **Functional Analysis** - Banach and Hilbert Spaces - Linear Operators - Spectral Theory - Banach Algebras - **Harmonic Analysis** - Fourier Series and Transforms - Plancherel and Parseval Theorems - Wavelets - **Operator Theory** - Self-Adjoint Operators - Unbounded Operators - C*-Algebras and von Neumann Algebras - **Nonstandard Analysis** - Hyperreal Numbers - Infinitesimals - **Numerical Analysis** - Numerical Solutions to Equations - Numerical Integration and Differentiation - Numerical Linear Algebra - Error Analysis and Stability --- ## 7. **Differential Equations** - **Ordinary Differential Equations (ODEs)** - Linear and Nonlinear ODEs - Existence and Uniqueness Theorems - Stability and Phase Plane Analysis - **Partial Differential Equations (PDEs)** - Classification: Elliptic, Parabolic, Hyperbolic - Methods of Solution - Sobolev Spaces - Applications in Physics and Engineering - **Dynamical Systems** - Continuous and Discrete Systems - Chaos Theory - Bifurcation Theory - Hamiltonian and Lagrangian Mechanics --- ## 8. **Probability and Statistics** - **Probability Theory** - Probability Spaces and Measures - Random Variables and Expectations - Laws of Large Numbers - Central Limit Theorem - Stochastic Processes - Markov Chains - Martingales - Brownian Motion - **Mathematical Statistics** - Statistical Inference - Estimation and Hypothesis Testing - Regression Analysis - Nonparametric Methods - **Statistical Mechanics** - Thermodynamics and Entropy - Phase Transitions - **Information Theory** - Entropy and Information Measures - Coding Theory - Data Compression - **Actuarial Science** - Risk Theory - Life Contingencies - Financial Mathematics --- ## 9. **Discrete Mathematics** - **Combinatorics** - Enumerative Combinatorics - Combinatorial Designs - Graph Theory - Trees and Connectivity - Planarity and Graph Coloring - Network Flows - **Algorithm Theory** - Complexity Classes (P, NP, NP-Complete) - Computational Complexity - Cryptography - **Discrete Structures** - Lattices and Boolean Algebras - Finite State Machines - Automata Theory - **Matroid Theory** - Independence and Circuits - Duality --- ## 10. **Applied Mathematics** - **Mathematical Modeling** - Differential Equations in Modeling - Simulation and Computational Models - **Optimization** - Linear Programming - Nonlinear Optimization - Convex Analysis - Game Theory - **Control Theory** - Feedback Systems - Optimal Control - Stability Theory - **Computational Mathematics** - Numerical Methods - Scientific Computing - Computational Geometry - **Mathematical Physics** - Quantum Mechanics - Schrödinger Equation - Operator Algebras - General Relativity - Differential Geometry in Spacetime - String Theory - Higher-Dimensional Manifolds - **Mathematical Biology** - Population Dynamics - Epidemiology Models - Bioinformatics - **Financial Mathematics** - Stochastic Calculus - Derivatives Pricing - Risk Management --- ## 11. **Interdisciplinary Fields** - **Cryptography** - Public-Key Cryptography - Elliptic Curve Cryptography - Cryptographic Protocols - **Computer Science Theory** - Algorithms and Data Structures - Formal Languages and Automata - Computational Complexity - **Quantum Computing** - Quantum Algorithms - Quantum Information Theory - **Topological Data Analysis** - Persistent Homology - Computational Topology - **Mathematical Psychology** - Psychometrics - Cognitive Modeling --- ## 12. **Advanced and Specialized Topics** - **Homological Algebra** - Chain Complexes - Derived Categories - Ext and Tor Functors - **K-Theory** - Algebraic K-Theory - Topological K-Theory - **Noncommutative Geometry** - Operator Algebras - Quantum Groups - **Higher Category Theory** - ∞-Categories - Derived Algebraic Geometry - **Topos Theory** - Grothendieck Topoi - Sheaf Theory --- ## 13. **Emerging and Modern Areas** - **Artificial Intelligence and Machine Learning** - Statistical Learning Theory - Neural Networks - Optimization in High Dimensions - **Computational Topology** - Algorithms for Topological Computation - Applications in Data Analysis - **Mathematics of Networks** - Network Theory - Complex Systems - **Mathematical Aspects of Quantum Field Theory** - Renormalization - Gauge Theories - Anomalies - **Mirror Symmetry and String Theory** - Calabi-Yau Manifolds - Gromov-Witten Invariants --- ## 14. **Historical and Philosophical Aspects** - **History of Mathematics** - Development of Mathematical Concepts - Biographies of Mathematicians - **Philosophy of Mathematics** - Foundations and Paradoxes - Mathematical Platonism vs. Formalism - Constructivism --- ## 15. **Mathematics Education** - **Pedagogical Methods** - Curriculum Development - Educational Technology - **Mathematical Literacy** - Public Understanding of Mathematics - Outreach and Communication --- This map is by no means exhaustive, but it provides a broad overview of the rich and diverse world of mathematics. Each of these areas can be further subdivided into more specialized topics, and the connections between them often lead to new and exciting fields of study. Mathematics is continually evolving, with new theories and applications emerging as researchers explore the frontiers of knowledge. --- **Note:** Mathematics is highly interconnected. Many concepts in one area rely on or influence those in another. For example, algebraic topology uses tools from algebra to solve topological problems, while functional analysis combines elements of analysis and linear algebra. As you delve deeper into each field, you'll discover these fascinating interdependencies. #### Map of mathematics 2 I. Pure Mathematics A. Algebra 1. Abstract Algebra a. Group Theory - Lie Groups - Representation Theory b. Ring Theory c. Field Theory - Galois Theory d. Homological Algebra - Category Theory - K-Theory 2. Linear Algebra - Vector Spaces - Matrix Theory - Tensor Analysis 3. Number Theory - Analytic Number Theory - Algebraic Number Theory - Diophantine Equations 4. Combinatorics - Graph Theory - Enumerative Combinatorics - Algebraic Combinatorics 5. Algebraic Geometry - Schemes - Sheaf Theory - Cohomology Theories 6. Commutative Algebra - Ideal Theory - Homological Methods 7. Non-Commutative Algebra - Representation Theory - Hopf Algebras - Quantum Groups B. Analysis 1. Real Analysis - Measure Theory - Functional Analysis - Harmonic Analysis 2. Complex Analysis - Riemann Surfaces - Analytic Functions - Complex Dynamics 3. Functional Analysis a. Hilbert Spaces b. Banach Spaces c. Operator Theory d. Spectral Theory 4. Harmonic Analysis a. Fourier Analysis b. Wavelets c. Representation Theory 5. Differential Equations a. Ordinary Differential Equations - Dynamical Systems - Bifurcation Theory - Stability Theory b. Partial Differential Equations - Elliptic PDEs - Parabolic PDEs - Hyperbolic PDEs 6. Probability Theory - Stochastic Processes - Markov Chains - Martingales 7. Statistics - Estimation Theory - Hypothesis Testing - Regression Analysis C. Geometry and Topology 1. Euclidean Geometry 2. Non-Euclidean Geometry a. Hyperbolic Geometry b. Elliptic Geometry 3. Differential Geometry a. Riemannian Geometry b. Symplectic Geometry c. Poisson Geometry d. Kähler Geometry e. Finsler Geometry 4. Algebraic Topology a. Homotopy Theory b. Homology Theory c. Cohomology Theory d. K-Theory 5. Differential Topology a. Morse Theory b. Floer Homology c. Contact Topology 6. Geometric Topology a. Knot Theory b. 3-Manifolds c. 4-Manifolds 7. Lie Groups and Lie Algebras - Representation Theory - Structure Theory - Enveloping Algebras D. Logic and Foundations 1. Set Theory - Axiomatic Set Theory - Descriptive Set Theory - Large Cardinals 2. Model Theory - First-Order Logic - Stability Theory - o-Minimality 3. Proof Theory - Constructive Mathematics - Type Theory - Homotopy Type Theory 4. Computability Theory - Recursive Functions - Turing Machines - Complexity Theory 5. Category Theory - Topos Theory - Homological Algebra - Higher Category Theory II. Applied Mathematics A. Classical Mechanics 1. Newton's Laws 2. Lagrangian Mechanics - Euler-Lagrange Equations - Hamilton's Principle - Noether's Theorem 3. Hamiltonian Mechanics - Hamilton's Equations - Poisson Brackets - Liouville's Theorem B. Continuum Mechanics 1. Fluid Dynamics - Navier-Stokes Equations - Turbulence - Boundary Layer Theory 2. Elasticity Theory - Hooke's Law - Stress-Strain Relations - Plate and Shell Theory 3. Plasticity Theory - Yield Criteria - Flow Rules - Hardening Laws C. Relativity 1. Special Relativity - Lorentz Transformations - Minkowski Spacetime - Relativistic Mechanics 2. General Relativity - Einstein Field Equations - Schwarzschild Solution - Cosmological Models D. Quantum Mechanics 1. Schrödinger Equation - Wave Functions - Operators - Eigenvalues and Eigenfunctions 2. Heisenberg Uncertainty Principle 3. Dirac Equation - Spinors - Relativistic Quantum Mechanics 4. Quantum Field Theory - Feynman Diagrams - Renormalization - Gauge Theories E. Statistical Mechanics 1. Thermodynamics - Laws of Thermodynamics - Entropy - Free Energy 2. Kinetic Theory - Boltzmann Equation - H-Theorem - Transport Phenomena 3. Statistical Ensembles - Microcanonical Ensemble - Canonical Ensemble - Grand Canonical Ensemble F. Numerical Analysis 1. Finite Difference Methods 2. Finite Element Methods 3. Spectral Methods 4. Monte Carlo Methods 5. Optimization Algorithms G. Optimization 1. Linear Programming - Simplex Method - Duality Theory - Interior Point Methods 2. Nonlinear Programming - Gradient Methods - Newton's Method - Conjugate Gradient Methods 3. Variational Calculus - Euler-Lagrange Equations - Hamilton's Principle - Noether's Theorem 4. Optimal Control Theory - Pontryagin's Maximum Principle - Dynamic Programming - Hamilton-Jacobi-Bellman Equation H. Dynamical Systems 1. Chaos Theory - Lyapunov Exponents - Strange Attractors - Fractal Dimensions 2. Bifurcation Theory - Hopf Bifurcation - Saddle-Node Bifurcation - Pitchfork Bifurcation 3. Ergodic Theory - Ergodic Theorems - Mixing - Entropy III. Interdisciplinary Fields A. Mathematical Physics 1. Quantum Field Theory - Gauge Theories - Renormalization - Conformal Field Theory 2. String Theory - Superstring Theory - M-Theory - Dualities 3. Conformal Field Theory - Virasoro Algebra - Kac-Moody Algebras - Vertex Operator Algebras 4. Integrable Systems - Soliton Equations - Inverse Scattering Transform - Quantum Integrable Systems B. Mathematical Biology 1. Population Dynamics - Lotka-Volterra Equations - Predator-Prey Models - Evolutionary Game Theory 2. Epidemiology - SIR Models - Network Models - Stochastic Epidemic Models 3. Neuroscience - Hodgkin-Huxley Model - FitzHugh-Nagumo Model - Neural Networks 4. Bioinformatics - Sequence Alignment - Phylogenetics - Gene Expression Analysis C. Mathematical Finance 1. Stochastic Calculus - Itô Calculus - Stochastic Differential Equations - Feynman-Kac Formula 2. Option Pricing Theory - Black-Scholes Model - Binomial Option Pricing Model - Lévy Processes 3. Portfolio Optimization - Mean-Variance Analysis - Capital Asset Pricing Model - Arbitrage Pricing Theory 4. Risk Management - Value at Risk - Expected Shortfall - Copula Theory D. Mathematical Economics 1. Game Theory - Nash Equilibrium - Evolutionary Game Theory - Mechanism Design 2. Econometrics - Linear Regression - Time Series Analysis - Panel Data Analysis 3. General Equilibrium Theory - Arrow-Debreu Model - Welfare Theorems - Existence and Uniqueness 4. Mechanism Design - Auction Theory - Matching Theory - Social Choice Theory E. Mathematical Computer Science 1. Algorithms and Complexity - Analysis of Algorithms - Computational Complexity Theory - Approximation Algorithms 2. Cryptography - Public Key Cryptography - Elliptic Curve Cryptography - Quantum Cryptography 3. Machine Learning - Supervised Learning - Unsupervised Learning - Reinforcement Learning 4. Quantum Computing - Quantum Algorithms - Quantum Error Correction - Quantum Complexity Theory #### Map of proofs Mathematics serves as the foundational framework for understanding and advancing scientific knowledge. Below is an extensive map highlighting significant mathematical proofs and their profound impact across various scientific disciplines. --- #### *Group Theory* - **Lagrange's Theorem** - *States that the order of a subgroup divides the order of the entire group.* - **Applications:** Symmetry operations in quantum mechanics, crystallography, and particle physics. #### *Ring Theory* - **Fundamental Theorems of Ring Homomorphisms** - **Applications:** Algebraic coding theory, cryptography, and the study of polynomial rings in computer algebra systems. #### *Fundamental Theorem of Calculus* - **Links differentiation and integration, showing they are inverse processes.** - **Applications:** Formulates the basis for continuous change in physics, engineering, and economics. #### *Fourier Analysis* - **Convergence of Fourier Series** - **Applications:** Signal processing, heat transfer, quantum mechanics, and acoustics. #### *Differential Equations* - **Existence and Uniqueness Theorems (Picard-Lindelöf Theorem)** - **Applications:** Modeling natural phenomena in physics, biology, and engineering systems. #### *Pythagorean Theorem* - **Relates the lengths of the sides of a right-angled triangle.** - **Applications:** Fundamental in physics for resolving vector components and in engineering design. #### *Gauss-Bonnet Theorem* - **Connects geometry and topology by relating curvature to topology.** - **Applications:** General relativity and the study of spacetime curvature. #### *Euler Characteristic* - **A topological invariant representing a space's shape or structure.** - **Applications:** Network topology, molecular chemistry, and condensed matter physics. #### *Fermat's Last Theorem* - **States that there are no three positive integers \( a \), \( b \), and \( c \) that satisfy \( a^n + b^n = c^n \) for \( n > 2 \).** - **Applications:** Influences cryptographic algorithms and computational number theory. #### *Prime Number Theorem* - **Describes the asymptotic distribution of prime numbers.** - **Applications:** Cryptography, particularly in RSA encryption. #### *Central Limit Theorem* - **States that the sum of a large number of independent random variables tends toward a normal distribution.** - **Applications:** Statistical mechanics, error analysis in measurements, and financial modeling. #### *Law of Large Numbers* - **Describes the result of performing the same experiment many times.** - **Applications:** Quantum physics interpretations, reliability engineering, and risk assessment. #### *Gödel's Incompleteness Theorems* - **Demonstrate inherent limitations in every formal axiomatic system.** - **Applications:** Foundations of mathematics, computer science, and theories of computation. #### *Turing's Halting Problem* - **Proves that there is no general algorithm to solve the halting problem for all possible program-input pairs.** - **Applications:** Limits of computation, development of decidability theory, and complexity classes. #### *Graph Theory* - **Euler's Formula for Planar Graphs** - **Applications:** Network analysis, circuit design, and molecular biology. #### *NP-Completeness Theory* - **Cook-Levin Theorem** - **Applications:** Cryptography, algorithm design, and computational complexity. #### *Spectral Theorem* - **States that every normal operator on a finite-dimensional complex vector space is diagonalizable.** - **Applications:** Quantum mechanics (observables), vibration analysis, and image processing. #### *Cayley-Hamilton Theorem* - **Every square matrix satisfies its own characteristic equation.** - **Applications:** Control theory, electrical engineering, and system dynamics. #### *Hahn-Banach Theorem* - **Extension of bounded linear functionals.** - **Applications:** Quantum physics, optimization problems, and economics. #### *Banach-Tarski Paradox* - **A ball can be decomposed and reassembled into two identical copies.** - **Applications:** Theoretical implications in measure theory and the concept of infinity. #### *Ricci Flow and the Poincaré Conjecture* - **Grigori Perelman's proof using Ricci flow to solve a century-old problem.** - **Applications:** Topology of three-dimensional spaces, cosmology, and geometric analysis. #### *Riemannian Geometry* - **Studies smooth manifolds with Riemannian metrics.** - **Applications:** General relativity and the modeling of spacetime. #### *Cauchy's Integral Theorem* - **Fundamental theorem in complex analysis about the integration of holomorphic functions.** - **Applications:** Fluid dynamics, electromagnetic theory, and aerodynamics. #### *Riemann Hypothesis (Unproven)* - **Speculates about the zeros of the Riemann zeta function.** - **Potential Applications:** Prime number distribution, quantum chaos, and cryptography. #### *Cantor's Diagonal Argument* - **Demonstrates that real numbers are uncountable.** - **Applications:** Understanding different sizes of infinity, foundations of mathematics. #### *Zermelo-Fraenkel Axioms with the Axiom of Choice (ZFC)* - **Provide a standard form of axiomatic set theory.** - **Applications:** Basis for most of modern mathematics. #### *Noether's Theorem* - **States that every differentiable symmetry of the action of a physical system corresponds to a conservation law.** - **Applications:** Conservation of energy, momentum, and charge in physics. #### *Yang-Mills Theory* - **Framework for understanding the behavior of elementary particles using non-Abelian gauge theories.** - **Applications:** Standard Model of particle physics and quantum chromodynamics. #### *Yoneda Lemma* - **Fundamental result relating functors and natural transformations.** - **Applications:** Abstract algebra, topology, and theoretical computer science. #### *Lagrange Multipliers* - **Technique for finding local maxima and minima of functions subject to equality constraints.** - **Applications:** Economics (utility maximization), engineering design, and physics. #### *Newton-Raphson Method* - **Iterative method for finding successively better approximations to the roots of a real-valued function.** - **Applications:** Numerical solutions in engineering, physics simulations, and computational finance. --- This comprehensive map illustrates the profound connections between mathematical proofs and scientific advancements. Each theorem not only represents a pinnacle of mathematical thought but also serves as a crucial tool for scientists and engineers exploring the complexities of the natural world. #### Map of mathematics 2 1. Foundations a. Logic and Set Theory - Propositional Logic - First-Order Logic - Axiomatic Set Theory - Zermelo-Fraenkel Set Theory b. Mathematical Reasoning and Proof - Deductive Reasoning - Inductive Reasoning - Proof Techniques (Direct, Contradiction, Induction) c. Number Systems - Natural Numbers - Integers - Rational Numbers - Real Numbers - Complex Numbers 2. Algebra a. Elementary Algebra - Variables and Expressions - Equations and Inequalities - Systems of Equations b. Polynomials and Rational Expressions - Polynomial Arithmetic - Factoring - Rational Expressions and Functions c. Equations and Inequalities - Linear Equations and Inequalities - Quadratic Equations - Polynomial Equations - Exponential and Logarithmic Equations - Trigonometric Equations d. Functions - Function Notation and Graphing - Transformations of Functions - Inverse Functions - Polynomial Functions - Rational Functions - Exponential and Logarithmic Functions - Trigonometric Functions e. Matrices and Determinants - Matrix Operations - Determinants - Eigenvalues and Eigenvectors - Matrix Decompositions f. Abstract Algebra - Group Theory - Ring Theory - Field Theory - Galois Theory - Representation Theory g. Linear Algebra - Vector Spaces - Linear Transformations - Inner Product Spaces - Orthogonality - Least Squares Approximation h. Boolean Algebra - Logical Operations - Truth Tables - Minimization of Boolean Functions - Applications in Computer Science 3. Geometry a. Euclidean Geometry - Points, Lines, and Planes - Angles and Triangles - Congruence and Similarity - Circles and Spheres - Polygons and Polyhedra b. Analytic Geometry - Coordinate Systems - Distance and Midpoint Formulas - Equations of Lines and Planes - Conic Sections - Parametric Equations - Vector Geometry c. Non-Euclidean Geometries - Hyperbolic Geometry - Elliptic Geometry - Projective Geometry d. Trigonometry - Trigonometric Ratios - Trigonometric Identities - Solving Triangles - Trigonometric Functions and Graphs e. Differential Geometry - Curves and Surfaces - Tangent Spaces - Riemannian Geometry - Geodesics - Curvature f. Algebraic Geometry - Affine Varieties - Projective Varieties - Schemes - Cohomology - Intersection Theory g. Topology - Topological Spaces - Continuity and Homeomorphisms - Connectedness and Compactness - Homotopy and Fundamental Groups - Homology and Cohomology - Knot Theory 4. Calculus and Analysis a. Limits and Continuity - Limit Definition and Properties - Continuity - Intermediate Value Theorem b. Derivatives and Differentiation - Definition of the Derivative - Differentiation Rules - Implicit Differentiation - Higher-Order Derivatives - Applications of Derivatives c. Integrals and Integration - Definite and Indefinite Integrals - Fundamental Theorem of Calculus - Integration Techniques - Improper Integrals - Applications of Integration d. Sequences and Series - Convergence of Sequences - Series and Convergence Tests - Power Series - Taylor Series e. Multivariable Calculus - Partial Derivatives - Gradients, Divergence, and Curl - Multiple Integrals - Change of Variables - Line and Surface Integrals f. Vector Calculus - Vector Fields - Green's Theorem - Stokes' Theorem - Divergence Theorem g. Differential Equations - First-Order Differential Equations - Second-Order Linear Differential Equations - Laplace Transforms - Systems of Differential Equations - Partial Differential Equations h. Real Analysis - The Real Number System - Sequences and Series of Functions - Metric Spaces - Continuity and Differentiability - Riemann Integration - Lebesgue Integration i. Complex Analysis - Complex Numbers and Functions - Analytic Functions - Cauchy's Theorem and Integral Formula - Laurent Series and Residues - Conformal Mappings j. Functional Analysis - Normed Vector Spaces - Hilbert Spaces - Banach Spaces - Linear Operators - Spectral Theory k. Measure Theory - Measurable Spaces and Functions - Measures and Integrals - Lp Spaces - Radon-Nikodym Theorem - Fubini's Theorem 5. Probability and Statistics a. Probability Theory - Probability Axioms - Conditional Probability - Independence - Random Variables - Expectation and Variance - Moment Generating Functions b. Combinatorics - Permutations and Combinations - Binomial Coefficients - Inclusion-Exclusion Principle - Generating Functions - Recurrence Relations c. Discrete Probability Distributions - Bernoulli and Binomial Distributions - Geometric and Negative Binomial Distributions - Hypergeometric Distribution - Poisson Distribution d. Continuous Probability Distributions - Uniform Distribution - Normal Distribution - Exponential Distribution - Gamma and Beta Distributions - Chi-Square, t, and F Distributions e. Statistical Inference - Point Estimation - Interval Estimation - Hypothesis Testing - Likelihood Ratio Tests - Nonparametric Methods f. Regression Analysis - Simple Linear Regression - Multiple Linear Regression - Nonlinear Regression - Logistic Regression - Time Series Analysis g. Bayesian Statistics - Bayes' Theorem - Prior and Posterior Distributions - Bayesian Inference - Markov Chain Monte Carlo Methods - Bayesian Networks h. Stochastic Processes - Markov Chains - Poisson Processes - Brownian Motion - Martingales - Stochastic Calculus 6. Discrete Mathematics a. Graph Theory - Graphs and Digraphs - Connectivity - Trees and Spanning Trees - Eulerian and Hamiltonian Graphs - Planar Graphs - Graph Coloring - Matchings and Coverings b. Combinatorics - Pigeonhole Principle - Ramsey Theory - Latin Squares - Block Designs - Polya Enumeration c. Number Theory - Divisibility and Prime Numbers - Congruences - Diophantine Equations - Quadratic Reciprocity - Continued Fractions - Elliptic Curves d. Cryptography - Classical Cryptosystems - Public-Key Cryptography - RSA Cryptosystem - Discrete Logarithms - Elliptic Curve Cryptography e. Game Theory - Two-Person Zero-Sum Games - Nash Equilibrium - Cooperative Games - Evolutionary Game Theory - Mechanism Design f. Computational Complexity - Time Complexity - P and NP Problems - NP-Completeness - Approximation Algorithms - Randomized Algorithms 7. Applied Mathematics a. Mathematical Physics - Classical Mechanics - Lagrangian and Hamiltonian Mechanics - Noether's Theorem - Rigid Body Dynamics - Celestial Mechanics - Quantum Mechanics - Schrödinger Equation - Hilbert Spaces and Operators - Angular Momentum and Spin - Perturbation Theory - Quantum Field Theory - Relativity - Special Relativity - General Relativity - Differential Geometry of Spacetime - Cosmology - Thermodynamics - Laws of Thermodynamics - Statistical Mechanics - Ensemble Theory - Phase Transitions - Non-Equilibrium Thermodynamics b. Fluid Dynamics - Navier-Stokes Equations - Inviscid Flow - Viscous Flow - Turbulence - Boundary Layer Theory - Computational Fluid Dynamics c. Partial Differential Equations - Classification of PDEs - Separation of Variables - Fourier Series and Transforms - Green's Functions - Finite Difference and Finite Element Methods d. Numerical Analysis - Error Analysis - Interpolation and Approximation - Numerical Differentiation and Integration - Numerical Linear Algebra - Numerical Solution of ODEs and PDEs e. Optimization - Linear Programming - Nonlinear Programming - Convex Optimization - Variational Methods - Optimal Control Theory f. Control Theory - Feedback Control Systems - Stability Analysis - Controllability and Observability - Optimal Control - Robust Control g. Mathematical Biology - Population Dynamics - Epidemiology - Biochemical Kinetics - Physiological Modeling - Neural Networks h. Mathematical Finance - Portfolio Theory - Options Pricing - Black-Scholes Model - Stochastic Calculus in Finance - Risk Management i. Operations Research - Linear Programming - Network Flow Problems - Integer Programming - Dynamic Programming - Queuing Theory - Inventory Theory - Decision Analysis 8. Foundations of Mathematics a. Mathematical Logic - Propositional and First-Order Logic - Completeness and Incompleteness Theorems - Model Theory - Recursion Theory b. Set Theory - Axioms of Set Theory - Ordinal and Cardinal Numbers - Continuum Hypothesis - Large Cardinals - Forcing and Independence Results c. Category Theory - Categories and Functors - Natural Transformations - Limits and Colimits - Adjoint Functors - Topoi d. Proof Theory - Formal Systems - Sequent Calculus - Cut Elimination - Proof Normalization - Ordinal Analysis e. Computability Theory - Turing Machines - Recursive Functions - Church-Turing Thesis - Unsolvable Problems - Degrees of Unsolvability f. Philosophy of Mathematics - Mathematical Platonism - Intuitionism and Constructivism - Formalism - Structuralism - Nominalism 9. History of Mathematics a. Ancient Mathematics - Egyptian and Babylonian Mathematics - Greek Mathematics (Pythagoras, Euclid, Archimedes) - Chinese Mathematics - Indian Mathematics b. Medieval Mathematics - Islamic Mathematics - European Mathematics - Fibonacci and the Hindu-Arabic Numeral System c. Early Modern Mathematics - The Renaissance and the Rise of Algebra - The Development of Analytic Geometry - The Invention of Calculus (Newton and Leibniz) d. 19th Century Mathematics - The Rigorous Foundations of Analysis - The Emergence of Non-Euclidean Geometry - The Development of Abstract Algebra - The Birth of Set Theory and Mathematical Logic e. 20th Century Mathematics - The Crisis in the Foundations of Mathematics - The Rise of Topology and Functional Analysis - The Emergence of Computer Science and Discrete Mathematics - The Development of Chaos Theory and Fractal Geometry f. Contemporary Mathematics - New Developments in Pure Mathematics (e.g., Langlands Program, Monstrous Moonshine) - Advances in Applied Mathematics (e.g., Compressed Sensing, Topological Data Analysis) - The Impact of Computers on Mathematical Research - Open Problems and Conjectures (e.g., Riemann Hypothesis, P vs. NP) ##### Map of all mathematics expanded 1. Foundations a. Logic and Set Theory - Propositional Logic - Syntax and Semantics - Logical Connectives - Truth Tables - Tautologies and Contradictions - Logical Equivalence - Deduction and Inference Rules - First-Order Logic - Syntax and Semantics - Quantifiers - Models and Interpretations - Validity and Satisfiability - Completeness and Compactness Theorems - Löwenheim-Skolem Theorems - Axiomatic Set Theory - Cantor's Naive Set Theory - Russell's Paradox - Zermelo-Fraenkel Set Theory (ZFC) - Axiom of Choice - Ordinal and Cardinal Numbers - Transfinite Induction and Recursion - Alternative Set Theories - Von Neumann-Bernays-Gödel Set Theory (NBG) - Morse-Kelley Set Theory (MK) - New Foundations (NF) b. Mathematical Reasoning and Proof - Deductive Reasoning - Modus Ponens and Modus Tollens - Hypothetical Syllogism - Disjunctive Syllogism - Constructive Dilemma - Reductio ad Absurdum - Inductive Reasoning - Generalization - Analogy - Causal Inference - Probablistic Reasoning - Bayesian Inference - Proof Techniques - Direct Proof - Proof by Contradiction - Proof by Contrapositive - Proof by Induction - Mathematical Induction - Strong Induction - Transfinite Induction - Proof by Cases - Nonconstructive Proofs c. Number Systems - Natural Numbers - Peano Axioms - Arithmetic Operations - Ordering and Inequalities - Prime Numbers and Factorization - Diophantine Equations - Integers - Construction from Natural Numbers - Arithmetic Operations - Divisibility and Greatest Common Divisors - Fundamental Theorem of Arithmetic - Linear Diophantine Equations - Rational Numbers - Construction as Quotients of Integers - Arithmetic Operations - Decimal Expansions - Continued Fractions - Farey Sequences - Real Numbers - Dedekind Cuts - Cauchy Sequences - Completeness and Archimedes' Axiom - Decimal Expansions - Cantor's Theorem and Uncountability - Complex Numbers - Definition and Arithmetic Operations - Polar Form and Euler's Formula - Roots of Unity - Fundamental Theorem of Algebra - Riemann Sphere - Quaternions - Definition and Arithmetic Operations - Conjugation and Norm - Quaternionic Exponential and Logarithm - Rotations in 3D Space - Octonions - Definition and Arithmetic Operations - Non-Associativity and Alternative Algebra - Automorphism Group and Triality - Applications in String Theory and Exceptional Lie Groups - p-adic Numbers - p-adic Valuations and Absolute Values - Hensel's Lemma - p-adic Expansions - p-adic Analysis and Integration - Applications in Number Theory and Cryptography - Surreal Numbers - Conway's Construction - Arithmetic Operations and Ordering - Transfinite Numbers and Infinitesimals - Surreal Analysis and Integration 2. Algebra a. Elementary Algebra - Variables and Expressions - Constants and Variables - Algebraic Expressions - Evaluating Expressions - Simplifying Expressions - Translating Verbal Descriptions into Algebraic Expressions - Equations and Inequalities - Solving Linear Equations - Solving Linear Inequalities - Absolute Value Equations and Inequalities - Literal Equations - Formulas and Applications - Systems of Equations - Solving Systems of Linear Equations - Graphical Method - Substitution Method - Elimination Method - Solving Systems of Nonlinear Equations - Applications and Word Problems b. Polynomials and Rational Expressions - Polynomial Arithmetic - Addition and Subtraction of Polynomials - Multiplication of Polynomials - Division of Polynomials (Long Division and Synthetic Division) - Polynomial Identities - Factoring Polynomials - Greatest Common Factor - Grouping - Difference of Squares - Sum and Difference of Cubes - General Trinomials - Factoring by Substitution - Rational Expressions and Functions - Simplifying Rational Expressions - Multiplication and Division of Rational Expressions - Addition and Subtraction of Rational Expressions - Complex Fractions - Solving Rational Equations - Applications and Word Problems c. Equations and Inequalities - Linear Equations and Inequalities - Solving Linear Equations - Solving Linear Inequalities - Absolute Value Equations and Inequalities - Compound Inequalities - Linear Equations and Inequalities in Two Variables - Quadratic Equations - Solving by Factoring - Completing the Square - Quadratic Formula - Discriminant and Nature of Roots - Solving Quadratic Inequalities - Polynomial Equations - Rational Root Theorem - Descartes' Rule of Signs - Bounds on Real Roots - Solving Cubic and Quartic Equations - Fundamental Theorem of Algebra - Exponential and Logarithmic Equations - Properties of Exponential and Logarithmic Functions - Solving Exponential Equations - Solving Logarithmic Equations - Applications and Word Problems - Trigonometric Equations - Basic Trigonometric Equations - Equations Involving Trigonometric Identities - Multiple-Angle Equations - Parametric Equations - Applications and Word Problems d. Functions - Function Notation and Graphing - Definition of a Function - Function Notation - Domain and Range - Graphing Functions - Vertical Line Test - Transformations of Functions - Vertical and Horizontal Shifts - Reflections - Stretches and Compressions - Combining Transformations - Applications and Modeling - Inverse Functions - Definition and Properties - Finding Inverse Functions - Graphing Inverse Functions - One-to-One Functions - Applications and Word Problems - Polynomial Functions - Quadratic Functions - Cubic Functions - Higher-Degree Polynomial Functions - Graphing Polynomial Functions - Polynomial Inequalities - Rational Functions - Graphing Rational Functions - Asymptotes (Vertical, Horizontal, and Oblique) - Holes and Points of Discontinuity - Solving Rational Inequalities - Applications and Word Problems - Exponential and Logarithmic Functions - Exponential Functions and Graphs - Logarithmic Functions and Graphs - Properties of Logarithms - Exponential and Logarithmic Equations - Applications (Growth and Decay, Compound Interest, etc.) - Trigonometric Functions - Radian Measure - Unit Circle and Trigonometric Functions - Graphs of Trigonometric Functions - Inverse Trigonometric Functions - Trigonometric Identities - Applications (Harmonic Motion, Waves, etc.) e. Matrices and Determinants - Matrix Operations - Matrix Addition and Subtraction - Scalar Multiplication - Matrix Multiplication - Transpose of a Matrix - Matrix Powers - Determinants - Definition and Properties - Cofactor Expansion - Laplace Expansion - Cramer's Rule - Applications (Area, Volume, etc.) - Inverse Matrices - Definition and Properties - Finding Inverse Matrices - Gaussian Elimination - Gauss-Jordan Elimination - Applications (Systems of Linear Equations, Cryptography, etc.) - Matrix Factorizations - LU Decomposition - QR Decomposition - Singular Value Decomposition (SVD) - Eigendecomposition - Applications (Least Squares, Principal Component Analysis, etc.) - Matrix Equations - Linear Matrix Equations - Sylvester Equation - Lyapunov Equation - Riccati Equation - Applications (Control Theory, Signal Processing, etc.) f. Abstract Algebra - Group Theory - Definition and Axioms - Examples of Groups - Subgroups and Lagrange's Theorem - Cosets and Quotient Groups - Homomorphisms and Isomorphisms - Normal Subgroups and Factor Groups - Fundamental Theorem of Finite Abelian Groups - Sylow Theorems - Applications (Symmetry, Cryptography, Coding Theory, etc.) - Ring Theory - Definition and Axioms - Examples of Rings - Subrings and Ideals - Quotient Rings - Homomorphisms and Isomorphisms - Prime and Maximal Ideals - Euclidean Domains and Principal Ideal Domains - Unique Factorization Domains - Applications (Number Theory, Algebraic Geometry, etc.) - Field Theory - Definition and Axioms - Examples of Fields - Field Extensions - Algebraic and Transcendental Extensions - Splitting Fields - Finite Fields - Applications (Coding Theory, Cryptography, etc.) - Galois Theory - Fundamental Theorem of Galois Theory - Galois Groups - Solvability by Radicals - Insolvability of the Quintic - Applications (Constructible Numbers, Origami, etc.) - Module Theory - Definition and Axioms - Examples of Modules - Submodules and Quotient Modules - Homomorphisms and Isomorphisms - Free Modules and Projective Modules - Injective Modules and Flat Modules - Tensor Products - Applications (Representation Theory, Homological Algebra, etc.) - Representation Theory - Group Representations - Character Theory - Irreducible Representations - Schur's Lemma - Decomposition of Representations - Induced Representations - Applications (Quantum Mechanics, Harmonic Analysis, etc.) - Lie Algebras - Definition and Axioms - Examples of Lie Algebras - Subalgebras and Ideals - Homomorphisms and Isomorphisms - Solvable and Nilpotent Lie Algebras - Semisimple Lie Algebras and Root Systems - Representations of Lie Algebras - Applications (Differential Geometry, Quantum Field Theory, etc.) g. Linear Algebra - Vector Spaces - Definition and Axioms - Examples of Vector Spaces - Subspaces - Linear Combinations and Spans - Linear Independence and Dependence - Bases and Dimension - Coordinates and Change of Basis - Linear Transformations - Definition and Properties - Kernel and Image - Matrix Representations - Composition of Linear Transformations - Invertible Linear Transformations - Similarity and Diagonalization - Applications (Computer Graphics, Quantum Mechanics, etc.) - Eigenvalues and Eigenvectors - Definition and Properties - Characteristic Polynomial - Eigenspaces and Geometric Multiplicity - Diagonalization and Spectral Theorem - Cayley-Hamilton Theorem - Applications (Dynamical Systems, Markov Chains, etc.) - Inner Product Spaces - Definition and Axioms - Examples of Inner Product Spaces - Norm and Distance - Orthogonality and Orthonormal Bases - Gram-Schmidt Orthogonalization - Orthogonal Complements and Projections - Applications (Quantum Mechanics, Signal Processing, etc.) - Singular Value Decomposition (SVD) - Definition and Properties - Existence and Uniqueness - Compact SVD - Truncated SVD - Applications (Data Compression, Recommender Systems, etc.) - Tensor Algebra - Definition and Properties - Tensor Products - Symmetric and Antisymmetric Tensors - Tensor Fields - Covariant and Contravariant Tensors - Applications (Continuum Mechanics, General Relativity, etc.) h. Boolean Algebra - Logical Operations - Conjunction (AND) - Disjunction (OR) - Negation (NOT) - Implication and Equivalence - Exclusive OR (XOR) - Truth Tables - Construction and Interpretation - Tautologies and Contradictions - Logical Equivalence - Functional Completeness - Canonical Forms - Minterms and Maxterms - Sum-of-Products (SOP) Form - Product-of-Sums (POS) Form - Conversion between Forms - Minimization of Boolean Functions - Karnaugh Maps - Quine-McCluskey Algorithm - Espresso Algorithm - applications (Circuit Design, Switching Theory, etc.) - Boolean Algebra and Set Theory - Isomorphism between Boolean Algebras and Sets - Venn Diagrams - De Morgan's Laws - Power Set and Cartesian Product - Applications in Computer Science - Digital Logic Design - Switching Circuits - Finite State Machines - Coding Theory and Error Correction 3. Geometry a. Euclidean Geometry - Points, Lines, and Planes - Definitions and Axioms - Incidence and Betweenness - Segments and Rays - Halfplanes and Angles - Vertical Angles and Linear Pairs - Triangles - Classification of Triangles - Congruence and Similarity - Triangle Inequality - Medians, Altitudes, and Angle Bisectors - Pythagorean Theorem and Its Converse - Special Triangles (Isosceles, Equilateral, 30-60-90, 45-45-90) - Polygons - Definition and Properties - Classification of Polygons - Diagonals and Convexity - Interior and Exterior Angles - Regular Polygons and Their Properties - Area Formulas for Polygons - Circles - Definition and Properties - Central Angles and Inscribed Angles - Chords, Secants, and Tangents - Inscribed and Circumscribed Polygons - Arc Length and Sector Area - Equation of a Circle - Solid Geometry - Lines and Planes in Space - Dihedral and Polyhedral Angles - Polyhedra (Prisms, Pyramids, Platonic Solids) - Cylinders, Cones, and Spheres - Surface Area and Volume Formulas - Constructions and Loci - Basic Constructions with Straightedge and Compass - Constructible Numbers and Lengths - Geometric Loci and Their Equations - Applications (Linkages, Origami, etc.) b. Analytic Geometry - Coordinate Systems - Cartesian Coordinate System - Polar Coordinate System - Cylindrical and Spherical Coordinate Systems - Conversion between Coordinate Systems - Distance and Midpoint Formulas - Distance between Two Points - Midpoint of a Line Segment - Distance from a Point to a Line - Perpendicular Distance between Lines - Equations of Lines and Planes - Slope-Intercept Form - Point-Slope Form - General Form - Parametric Equations - Vector Equations - Angles between Lines and Planes - Distances between Points, Lines, and Planes - Conic Sections - Circle - Parabola - Ellipse - Hyperbola - Eccentricity and Focal Properties - Tangents and Normals - Quadric Surfaces - Ellipsoid - Hyperboloid of One and Two Sheets - Paraboloid (Elliptic and Hyperbolic) - Cone and Cylinder - Canonical Forms and Classification - Coordinate Transformations - Translation - Rotation - Reflection - Scaling - Shear - Composite Transformations - Applications (Computer Graphics, Robotics, etc.) c. Non-Euclidean Geometries - Hyperbolic Geometry - Hilbert's Axioms and Saccheri Quadrilaterals - Models of Hyperbolic Geometry (Poincaré Disk, Upper Half-Plane) - Hyperbolic Lines and Angles - Hyperbolic Trigonometry - Hyperbolic Area and Gauss-Bonnet Theorem - Applications (Special Relativity, Complex Analysis, etc.) - Elliptic Geometry - Spherical Geometry - Great Circles and Geodesics - Spherical Trigonometry - Spherical Excess and Area Formula - Antipodal Points and Identification - Applications (Navigation, Astronomy, etc.) - Projective Geometry - Projective Plane and Projective Space - Homogeneous Coordinates - Duality and Principle of Duality - Cross-Ratio and Harmonic Conjugates - Desargues' Theorem and Pappus' Theorem - Projective Transformations - Applications (Perspective Drawing, Camera Models, etc.) d. Trigonometry - Trigonometric Ratios - Sine, Cosine, and Tangent - Reciprocal Ratios (Cosecant, Secant, Cotangent) - Trigonometric Functions of Special Angles - Reference Angles and Angle Measure - Trigonometric Identities - Fundamental Identities - Cofunction Identities - Even-Odd Identities - Sum and Difference Formulas - Double-Angle and Half-Angle Formulas - Power-Reduction Formulas - Product-to-Sum and Sum-to-Product Formulas - Inverse Trigonometric Functions - Definitions and Domains - Graphing Inverse Trigonometric Functions - Identities Involving Inverse Trigonometric Functions - Solving Equations with Inverse Trigonometric Functions - Applications of Trigonometry - Right Triangle Problems - Oblique Triangle Problems (Law of Sines, Law of Cosines) - Harmonic Motion and Waves - Navigation and Surveying - Optics and Acoustics e. Differential Geometry - Curves in the Plane and Space - Parametric Curves - Arc Length and Reparameterization - Curvature and Torsion - Frenet-Serret Formulas - Osculating Circle and Osculating Plane - Surfaces - Parametric Surfaces - Tangent Plane and Normal Vector - First and Second Fundamental Forms - Principal Curvatures and Gaussian Curvature - Minimal Surfaces - Geodesics and Geodesic Curvature - Riemannian Geometry - Riemannian Metric and Riemannian Manifolds - Christoffel Symbols and Levi-Civita Connection - Parallel Transport and Geodesic Equation - Riemann Curvature Tensor - Ricci Curvature and Scalar Curvature - Einstein Field Equations and General Relativity - Symplectic Geometry - Symplectic Manifolds and Symplectic Forms - Hamiltonian Mechanics - Lagrangian Submanifolds - Moment Maps and Symplectic Reduction - Poisson Structures and Integrable Systems - Applications (Classical Mechanics, Quantum Mechanics, etc.) f. Algebraic Geometry - Affine Varieties - Algebraic Sets and Zariski Topology - Hilbert's Nullstellensatz - Regular Functions and Coordinate Rings - Morphisms of Affine Varieties - Dimension and Tangent Spaces - Projective Varieties - Projective Space and Homogeneous Coordinates - Projective Closure and Projective Varieties - Veronese Embedding and Segre Embedding - Grassmannians and Flag Varieties - Blow-ups and Resolutions - Schemes - Affine Schemes and Spec Construction - Morphisms of Schemes - Fiber Products and Base Change - Separated and Proper Morphisms - Sheaves and Cohomology - Etale and Smooth Morphisms - Intersection Theory - Divisors and Line Bundles - Intersection Multiplicity - Bezout's Theorem - Riemann-Roch Theorem for Curves - Hirzebruch-Riemann-Roch Theorem - Chern Classes and Todd Classes - Elliptic Curves and Abelian Varieties - Weierstrass Equations and Group Law - Torsion Points and Mordell-Weil Theorem - Isogenies and Endomorphism Rings - Jacobians and Abel-Jacobi Map - Modular Forms and Modular Curves - Applications (Cryptography, Number Theory, etc.) g. Topology - Topological Spaces - Open and Closed Sets - Basis and Subbasis - Closure, Interior, and Boundary - Limit Points and Isolated Points - Hausdorff and Normal Spaces - Continuity and Homeomorphisms - Continuous Functions - Homeomorphisms and Topological Invariants - Topological Properties (Connectedness, Compactness, etc.) - Separation Axioms (T0, T1, T2, etc.) - Urysohn's Lemma and Tietze Extension Theorem - Fundamental Group and Covering Spaces - Homotopy and Homotopy Equivalence - Fundamental Group and Homomorphisms - Covering Spaces and Lifting Properties - Universal Covering Space - Applications (Knot Theory, 3-Manifolds, etc.) - Homology and Cohomology - Simplicial and Singular Homology - Homology Groups and Betti Numbers - Euler Characteristic - Cohomology Groups and Cup Product - Poincaré Duality and Lefschetz Duality - de Rham Cohomology and Hodge Theory - Algebraic Topology - Fundamental Group and Higher Homotopy Groups - Fibrations and Homotopy Fiber - Spectral Sequences and Serre Spectral Sequence - Eilenberg-MacLane Spaces and Cohomology Operations - Characteristic Classes (Stiefel-Whitney, Chern, Pontryagin) - K-Theory and Index Theorems - Low-Dimensional Topology - Surfaces and Classification Theorem - Seifert Surfaces and Knots - Knot Invariants (Linking Number, Alexander Polynomial, etc.) - 3-Manifolds and Heegaard Splittings - Dehn Surgery and Lickorish-Wallace Theorem - Geometrization Theorem and Thurston's Work 4. Calculus and Analysis a. Limits and Continuity - Limit of a Function - Definition and Intuitive Understanding - One-Sided Limits - Infinite Limits and Limits at Infinity - Limit Laws and Computation Techniques - Squeeze Theorem and Indeterminate Forms - Continuity - Definition and Types of Discontinuities - Properties of Continuous Functions - Intermediate Value Theorem - Extreme Value Theorem - Uniform Continuity - Sequences and Series - Convergence of Sequences - Monotone Sequences and Bounded Sequences - Cauchy Sequences and Completeness - Convergence of Series (Geometric, p-series, etc.) - Convergence Tests (Ratio, Root, Integral, etc.) - Absolute and Conditional Convergence b. Derivatives and Differentiation - Definition of the Derivative - Tangent Lines and Rates of Change - Definition as a Limit - One-Sided Derivatives - Higher-Order Derivatives - Notation (Leibniz, Lagrange, Newton) - Differentiation Rules - Constant Rule and Power Rule - Product Rule and Quotient Rule - Chain Rule - Implicit Differentiation - Logarithmic Differentiation - Derivatives of Inverse Functions - Applications of Derivatives - Related Rates - Linear Approximation and Differentials - Newton's Method - L'Hôpital's Rule and Indeterminate Forms - Optimization Problems - Monotonicity and Concavity - Curve Sketching c. Integrals and Integration - Indefinite Integrals - Antiderivatives and Integration Constants - Basic Integration Rules (Power Rule, Constant Multiple Rule, etc.) - Integration by Substitution (u-Substitution) - Integration by Parts - Trigonometric Integrals - Integration of Rational Functions (Partial Fractions) - Definite Integrals - Riemann Sums and Definition of the Definite Integral - Properties of Definite Integrals - Fundamental Theorem of Calculus - Techniques for Evaluating Definite Integrals - Improper Integrals (Unbounded Intervals, Unbounded Functions) - Applications of Integration - Area between Curves - Volumes of Solids (Slicing, Shells, Disks/Washers) - Arc Length and Surface Area - Work, Hydrostatic Pressure, and Centroids - Probability Density Functions and Cumulative Distribution Functions d. Sequences and Series - Sequences - Definition and Notation - Limit of a Sequence - Bounded, Monotone, and Cauchy Sequences - Convergence Theorems - Subsequences and Bolzano-Weierstrass Theorem - Series - Definition and Notation - Convergence and Divergence of Series - Geometric Series and p-Series - Convergence Tests (Ratio, Root, Integral, Comparison, etc.) - Absolute and Conditional Convergence - Rearrangements of Series - Power Series - Definition and Radius of Convergence - Differentiation and Integration of Power Series - Taylor Series and Maclaurin Series - Binomial Series - Applications (Approximation, Solving Differential Equations, etc.) e. Multivariable Calculus - Functions of Several Variables - Graphs and Level Curves/Surfaces - Limits and Continuity - Partial Derivatives - Tangent Planes and Linear Approximation - Directional Derivatives and Gradients - Higher-Order Partial Derivatives - Multiple Integrals - Double Integrals and Iterated Integrals - Change of Variables and Jacobians - Triple Integrals and Applications (Volume, Mass, Moments, etc.) - Fubini's Theorem - Improper Multiple Integrals - Vector Calculus - Parametric Curves and Arc Length - Vector Fields and Line Integrals - Conservative Vector Fields and Independence of Path - Green's Theorem - Surface Integrals and Flux - Divergence Theorem (Gauss' Theorem) - Stokes' Theorem f. Differential Equations - First-Order Differential Equations - Separable Equations - Linear Equations - Exact Equations and Integrating Factors - Bernoulli Equations - Homogeneous Equations - Applications (Population Growth, Cooling/Heating, Circuits, etc.) - Second-Order Linear Differential Equations - Homogeneous Equations with Constant Coefficients - Nonhomogeneous Equations (Method of Undetermined Coefficients, Variation of Parameters) - Cauchy-Euler Equations - Power Series Solutions - Applications (Harmonic Oscillators, RLC Circuits, Beams, etc.) - Laplace Transforms - Definition and Properties - Inverse Laplace Transforms - Solving Initial Value Problems - Convolutions and Integral Equations - Discontinuous and Periodic Functions - Applications (Control Theory, Signal Processing, etc.) - Systems of Differential Equations - Linear Systems and Matrices - Eigenvalues and Eigenvectors - Phase Portraits and Stability - Nonlinear Systems and Linearization - Applications (Predator-Prey Models, Chemical Reactions, etc.) - Partial Differential Equations - Classification (Elliptic, Parabolic, Hyperbolic) - Separation of Variables - Fourier Series and Sturm-Liouville Problems - Heat Equation - Wave Equation - Laplace's Equation and Harmonic Functions - Applications (Diffusion, Vibrations, Electrostatics, etc.) g. Real Analysis - The Real Number System - Axioms and Properties - Completeness and Dedekind Cuts - Supremum and Infimum - Archimedean Property - Density of Rationals and Irrationals - Sequences and Series - Convergence and Divergence - Cauchy Sequences and Completeness - Monotone Convergence Theorem - Bolzano-Weierstrass Theorem - Limsup and Liminf - Double Sequences and Iterated Limits - Continuity and Uniform Continuity - Continuity and Properties - Intermediate Value Theorem - Extreme Value Theorem - Uniform Continuity and Heine-Cantor Theorem - Lipschitz Continuity - Continuous Functions on Compact Sets - Differentiation - Definition and Properties - Mean Value Theorems (Rolle's, Lagrange, Cauchy) - L'Hôpital's Rule - Taylor's Theorem and Taylor Series - Implicit and Inverse Function Theorems - Differentiability and Continuity - Riemann Integration - Partitions and Riemann Sums - Riemann Integrability - Properties of Riemann Integrals - Fundamental Theorem of Calculus - Improper Integrals - Lebesgue's Criterion for Riemann Integrability - Sequences and Series of Functions - Pointwise and Uniform Convergence - Uniform Convergence and Continuity - Uniform Convergence and Integration - Uniform Convergence and Differentiation - Power Series and Radius of Convergence - Metric Spaces - Definition and Examples - Open and Closed Sets - Convergence and Completeness - Continuity and Homeomorphisms - Compact Sets and Heine-Borel Theorem - Connected Sets and Continuous Functions h. Complex Analysis - Complex Numbers and Functions - Algebraic and Geometric Representations - Polar Form and Euler's Formula - Powers and Roots - Complex Functions and Mappings - Limits and Continuity - Analytic Functions - Differentiability and Cauchy-Riemann Equations - Harmonic Functions - Exponential, Trigonometric, and Logarithmic Functions - Complex Integration and Contour Integrals - Cauchy's Integral Theorem and Formula - Liouville's Theorem and Fundamental Theorem of Algebra - Series Representations - Taylor Series and Maclaurin Series - Laurent Series - Zeros and Poles - Residue Theorem and Residue Calculus - Evaluation of Real Integrals - Conformal Mappings - Definition and Properties - Möbius Transformations - Schwarz-Christoffel Mappings - Riemann Mapping Theorem - Applications (Fluid Dynamics, Electrostatics, etc.) - Harmonic Functions - Definition and Properties - Poisson Integral Formula - Dirichlet Problem - Maximum Principle - Harnack's Inequality - Applications (Potential Theory, Heat Conduction, etc.) i. Functional Analysis - Normed Vector Spaces - Definition and Examples - Banach Spaces and Completeness - Equivalent Norms - Finite-Dimensional Normed Spaces - Compactness and Finite Dimension - Inner Product Spaces - Definition and Examples - Hilbert Spaces and Completeness - Orthonormal Bases and Parseval's Identity - Orthogonal Complements and Projections - Riesz Representation Theorem - Adjoint Operators and Self-Adjoint Operators - Linear Operators - Bounded and Unbounded Operators - Operator Norms and Continuity - Compact Operators - Spectrum and Resolvent - Fredholm Alternative - Spectral Theorem for Compact Self-Adjoint Operators - Banach Algebras - Definition and Examples - Spectrum and Spectral Radius - Holomorphic Functional Calculus - Gelfand Transform - C*-Algebras and von Neumann Algebras - Topological Vector Spaces - Definition and Examples - Locally Convex Spaces - Weak Topologies - Hahn-Banach Theorem and Separating Hyperplanes - Krein-Milman Theorem - Banach-Alaoglu Theorem j. Measure Theory - Measurable Spaces and Functions - Sigma-Algebras and Measurable Sets - Generated Sigma-Algebras and Borel Sets - Measurable Functions - Simple Functions and Approximation - Littlewood's Three Principles - Measures and Integrals - Definition and Properties of Measures - Lebesgue Measure and Lebesgue Integral - Monotone Convergence Theorem - Fatou's Lemma and Dominated Convergence Theorem - Fubini's Theorem and Tonelli's Theorem - Absolute Continuity and Radon-Nikodym Theorem - Lp Spaces - Definition and Examples - Hölder's Inequality and Minkowski's Inequality - Completeness and Banach Space Structure - Dual Spaces and Riesz Representation Theorem - Hilbert Spaces and Riesz-Fischer Theorem - Signed Measures and Complex Measures - Hahn Decomposition Theorem - Jordan Decomposition Theorem - Total Variation and Absolute Continuity - Lebesgue Decomposition Theorem - Radon-Nikodym Derivatives - Differentiation and Integration - Functions of Bounded Variation - Absolutely Continuous Functions - Fundamental Theorem of Calculus for Lebesgue Integrals - Lebesgue Differentiation Theorem - Vitali Covering Lemma - Applications (Probability Theory, Fourier Analysis, etc.) 5. Probability and Statistics a. Probability Theory - Probability Axioms - Sample Spaces and Events - Axioms of Probability - Probabiloity of Complements and Unions - Inclusion-Exclusion Principle - Continuity of Probability Measures - Conditional Probability and Independence - Definition of Conditional Probability - Multiplication Rule and Total Probability - Bayes' Theorem - Independent Events - Conditional Independence - Applications (Medical Testing, Legal Cases, etc.) - Random Variables - Discrete and Continuous Random Variables - Cumulative Distribution Functions - Probability Mass Functions and Probability Density Functions - Joint, Marginal, and Conditional Distributions - Functions of Random Variables - Transformations of Random Variables - Expectation and Variance - Definition and Properties of Expectation - Linearity of Expectation - Variance and Standard Deviation - Covariance and Correlation - Conditional Expectation - Moment Generating Functions and Characteristic Functions - Limit Theorems - Markov's Inequality and Chebyshev's Inequality - Weak Law of Large Numbers - Strong Law of Large Numbers - Central Limit Theorem - Applications (Polling, Quality Control, etc.) b. Combinatorics - Counting Principles - Multiplication Principle - Addition Principle - Pigeonhole Principle - Inclusion-Exclusion Principle - Binomial Coefficients and Pascal's Triangle - Permutations and Combinations - Definition and Notation - Permutations with and without Repetition - Combinations with and without Repetition - Binomial Theorem and Combinatorial Proofs - Catalan Numbers and Stirling Numbers - Generating Functions - Ordinary Generating Functions - Exponential Generating Functions - Solving Recurrence Relations - Partition Functions - Polya's Enumeration Theorem - Graph Theory and Enumeration - Paths and Cycles - Trees and Spanning Trees - Matchings and Perfect Matchings - Coloring Problems - Ramsey Theory and Extremal Graph Theory c. Discrete Probability Distributions - Bernoulli and Binomial Distributions - Bernoulli Trials and Success Probabilities - Probability Mass Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Normal Approximation to Binomial - Applications (Coin Flips, Defective Items, etc.) - Geometric and Negative Binomial Distributions - Probability Mass Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Memoryless Property and Waiting Times - Applications (Runs of Success, Quality Control, etc.) - Hypergeometric Distribution - Probability Mass Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Relationship to Binomial Distribution - Applications (Card Games, Sampling without Replacement, etc.) - Poisson Distribution - Probability Mass Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Relationship to Binomial Distribution - Applications (Rare Events, Traffic Flow, etc.) d. Continuous Probability Distributions - Uniform Distribution - Probability Density Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Applications (Random Number Generation, Buffon's Needle, etc.) - Normal Distribution - Probability Density Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Standard Normal Distribution and Z-scores - Central Limit Theorem and Normal Approximation - Applications (IQ Scores, Measurement Errors, etc.) - Exponential Distribution - Probability Density Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Memoryless Property and Waiting Times - Applications (Radioactive Decay, Queuing Theory, etc.) - Gamma and Beta Distributions - Probability Density Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Relationship to Other Distributions - Applications (Waiting Times, Bayesian Inference, etc.) - Chi-Square, t, and F Distributions - Probability Density Function and Cumulative Distribution Function - Mean, Variance, and Moment Generating Function - Relationship to Normal Distribution - Applications (Hypothesis Testing, Confidence Intervals, ANOVA, etc.) e. Statistical Inference - Point Estimation - Estimators and Their Properties (Unbiasedness, Consistency, Efficiency) - Method of Moments Estimation - Maximum Likelihood Estimation - Bayesian Estimation - Sufficient Statistic and Rao-Blackwell Theorem - Interval Estimation - Confidence Intervals - Pivotal Quantities and Confidence Intervals - Large-Sample Confidence Intervals - Bayesian Credible Intervals - Applications (Opinion Polls, Quality Control, etc.) - Hypothesis Testing - Null and Alternative Hypotheses - Type I and Type II Errors - Test Statistics and p-Values - Power of a Test and Sample Size Determination - Neyman-Pearson Lemma and Likelihood Ratio Tests - Applications (Clinical Trials, A/B Testing, etc.) - Nonparametric Methods - Sign Test and Wilcoxon Signed-Rank Test - Mann-Whitney U Test - Kruskal-Wallis Test - Spearman's Rank Correlation Coefficient - Kolmogorov-Smirnov Test - Bootstrap and Permutation Tests f. Regression Analysis - Simple Linear Regression - Least Squares Estimation - Correlation Coefficient and Coefficient of Determination - Hypothesis Tests and Confidence Intervals for Regression Parameters - Prediction and Prediction Intervals - Assumptions and Diagnostics (Residual Plots, QQ Plots, etc.) - Multiple Linear Regression - Matrix Notation and Estimation - Partial Correlation and Adjusted R-Squared - Multicollinearity and Variable Selection - Dummy Variables and Categorical Predictors - Interactions and Polynomial Regression - Nonlinear Regression - Transformations and Linearization - Nonlinear Least Squares Estimation - Gauss-Newton and Levenberg-Marquardt Algorithms - Model Selection and Akaike Information Criterion (AIC) - Applications (Growth Curves, Dose-Response Models, etc.) - Logistic Regression - Odds Ratios and Logit Transformation - Maximum Likelihood Estimation - Interpretation of Coefficients - Receiver Operating Characteristic (ROC) Curves - Applications (Medical Diagnosis, Credit Scoring, etc.) - Time Series Analysis - Stationarity and Autocorrelation - Autoregressive (AR) Models - Moving Average (MA) Models - Autoregressive Moving Average (ARMA) Models - Autoregressive Integrated Moving Average (ARIMA) Models - Forecasting and Model Selection g. Bayesian Statistics - Bayes' Theorem - Prior and Posterior Probabilities - Likelihood Functions - Marginal Likelihood and Model Evidence - Bayesian Updating and Sequential Learning - Prior and Posterior Distributions - Conjugate Priors and Exponential Families - Noninformative Priors - Posterior Summaries (Mean, Median, Credible Intervals) - Posterior Predictive Distribution - Bayesian Hypothesis Testing and Bayes Factors - Bayesian Inference - Bayesian Point Estimation - Bayesian Interval Estimation - Bayesian Hypothesis Testing - Bayesian Model Selection and Bayesian Information Criterion (BIC) - Empirical Bayes Methods - Markov Chain Monte Carlo (MCMC) Methods - Metropolis-Hastings Algorithm - Gibbs Sampling - Convergence Diagnostics and Effective Sample Size - Hamiltonian Monte Carlo and No-U-Turn Sampler - Variational Inference and Mean Field Approximation - Bayesian Networks - Directed Acyclic Graphs and Conditional Independence - Inference in Bayesian Networks - Parameter Learning and Structure Learning - Hidden Markov Models and Dynamic Bayesian Networks - Applications (Medical Diagnosis, Spam Filtering, etc.) h. Stochastic Processes - Markov Chains - State Space and Transition Probabilities - Chapman-Kolmogorov Equations - Stationary Distribution and Limiting Behavior - Ergodicity and Mixing Times - Absorption Probabilities and First Passage Times - Applications (PageRank, Gambler's Ruin, etc.) - Poisson Processes - Counting Processes and Arrival Times - Homogeneous and Nonhomogeneous Poisson Processes - Superposition and Thinning of Poisson Processes - Conditional Poisson Processes and Cox Processes - Applications (Queuing Theory, Reliability, etc.) - Brownian Motion - Definition and Properties - Wiener Process and Diffusion Processes - Stochastic Integrals and Ito's Lemma - Stochastic Differential Equations - Geometric Brownian Motion and Black-Scholes Model - Applications (Financial Mathematics, Physics, etc.) - Martingales - Definition and Examples - Martingale Transforms and Martingale Inequalities - Optional Stopping Theorem - Martingale Convergence Theorems - Martingale Representation Theorem - Applications (Gambling, Optimal Stopping, etc.) - Stochastic Calculus - Stochastic Integrals and Ito's Lemma - Stochastic Differential Equations - Feynman-Kac Formula and Kolmogorov Equations - Girsanov's Theorem and Change of Measure - Applications (Mathematical Finance, Filtering Theory, etc.) 6. Discrete Mathematics a. Graph Theory - Graphs and Digraphs - Definition and Terminology - Adjacency and Incidence Matrices - Subgraphs and Graph Isomorphism - Degree Sequences and Handshaking Lemma - Bipartite Graphs and Complete Graphs - Connectivity - Paths and Cycles - Connected Components and Strongly Connected Components - Bridges and Articulation Points - Menger's Theorem and Disjoint Paths - Graph Reconstruction and Whitney's Theorem - Trees and Spanning Trees - Definition and Properties - Minimum Spanning Trees (Kruskal's and Prim's Algorithms) - Cayley's Formula and Prüfer Code - Arboricity and Nash-Williams Theorem - Applications (Network Design, Phylogenetic Trees, etc.) - Matchings and Coverings - Definition and Examples - Hall's Marriage Theorem - König's Theorem and Edge Coloring - Vertex Covers and Independent Sets - Perfect Matchings and Tutte's Theorem - Graph Coloring - Chromatic Number and Chromatic Polynomial - Greedy Coloring and Brooks' Theorem - Four Color Theorem and Heawood Conjecture - Graph Homomorphisms and Hadwiger's Conjecture - Applications (Scheduling, Register Allocation, etc.) - Planar Graphs - Definition and Characterization - Euler's Formula and Kuratowski's Theorem - Dual Graphs and Geometric Duality - Graph Minors and Wagner's Theorem - Applications (Circuit Design, Facility Location, etc.) b. Enumerative Combinatorics - Generating Functions - Ordinary Generating Functions - Exponential Generating Functions - Partition Functions and Gaussian Polynomials - Dirichlet Generating Functions - Applications (Fibonacci Numbers, Catalan Numbers, etc.) - Recurrence Relations - Linear Recurrences and Characteristic Polynomials - Solving Recurrences using Generating Functions - Catalan Numbers and Motzkin Numbers - q-Analogs and q-Binomial Coefficients - Applications (Lattice Paths, Permutation Patterns, etc.) - Polya Enumeration - Burnside's Lemma and Cauchy-Frobenius Lemma - Cycle Index and Pattern Inventory - Polya's Enumeration Formula - de Bruijn Sequences and Universal Cycles - Applications (Chemical Isomers, Necklace Problem, etc.) - Combinatorial Designs - Block Designs and Incidence Matrices - Balanced Incomplete Block Designs (BIBD) - Steiner Systems and Finite Projective Planes - Latin Squares and Orthogonal Arrays - Hadamard Matrices and Combinatorial Matrix Theory - Extremal Combinatorics - Ramsey Numbers and Ramsey's Theorem - Turán's Theorem and Forbidden Subgraphs - Szemerédi's Regularity Lemma - Erdős-Ko-Rado Theorem and Intersecting Families - Combinatorial Nullstellensatz and Additive Combinatorics c. Number Theory - Divisibility and Prime Numbers - Division Algorithm and Greatest Common Divisor - Fundamental Theorem of Arithmetic - Distribution of Prime Numbers and Prime Number Theorem - Primality Testing and Factorization Algorithms - Twin Primes and Goldbach's Conjecture - Congruences - Modular Arithmetic and Residue Classes - Linear Congruences and Chinese Remainder Theorem - Fermat's Little Theorem and Euler's Theorem - Quadratic Residues and Legendre Symbol - Primitive Roots and Discrete Logarithms - Diophantine Equations - Linear Diophantine Equations and Bezout's Identity - Pell's Equation and Continued Fractions - Pythagorean Triples and Fermat's Last Theorem - Elliptic Curves and Mordell's Theorem - Diophantine Approximation and Transcendental Numbers - Quadratic Forms and Lattices - Binary Quadratic Forms and Discriminants - Representation of Integers by Quadratic Forms - Minkowski's Convex Body Theorem - Lattice Basis Reduction and LLL Algorithm - Cryptographic Applications and Lattice-Based Cryptography - Algebraic Number Theory - Number Fields and Ring of Integers - Ideals and Unique Factorization - Norm, Trace, and Discriminant - Class Groups and Class Number Formula - Cyclotomic Fields and Fermat's Last Theorem - Zeta Functions and L-Functions d. Cryptography - Classical Cryptography - Substitution Ciphers and Frequency Analysis - Transposition Ciphers and Permutations - Vigenère Cipher and Kasiski Examination - One-Time Pad and Perfect Secrecy - Cryptanalysis Techniques and Ciphertext-Only Attacks - Symmetric-Key Cryptography - Block Ciphers and Stream Ciphers - Data Encryption Standard (DES) and Advanced Encryption Standard (AES) - Modes of Operation (ECB, CBC, CTR, etc.) - Message Authentication Codes (MAC) and Hash Functions - Key Exchange Protocols and Diffie-Hellman Protocol - Public-Key Cryptography - RSA Cryptosystem and Modular Exponentiation - Elliptic Curve Cryptography (ECC) - Digital Signatures and Public Key Infrastructure (PKI) - Homomorphic Encryption and Fully Homomorphic Encryption - Post-Quantum Cryptography and Lattice-Based Cryptography - Cryptographic Protocols - Zero-Knowledge Proofs and Interactive Proof Systems - Oblivious Transfer and Secure Multiparty Computation - Blind Signatures and Digital Cash - Threshold Cryptography and Secret Sharing - Blockchain and Cryptocurrency (Bitcoin, Ethereum, etc.) - Cryptanalysis and Security - Linear and Differential Cryptanalysis - Side-Channel Attacks and Fault Attacks - Quantum Cryptanalysis and Shor's Algorithm - Provable Security and Random Oracle Model - Cryptographic Hardness Assumptions and Complexity Theory e. Game Theory - Two-Person Zero-Sum Games - Payoff Matrices and Minimax Theorem - Solving Games using Linear Programming - Mixed Strategies and Nash Equilibrium - Poker and Bluffing - Matching Pennies and Rock-Paper-Scissors - Non-Cooperative Games - Normal Form and Extensive Form Games - Dominant Strategies and Iterated Elimination - Nash Equilibrium and Best Response Dynamics - Cournot and Bertrand Competition - Prisoner's Dilemma and Tragedy of the Commons - Cooperative Games - Coalitional Games and Characteristic Functions - Core and Shapley Value - Bargaining Theory and Axiomatic Bargaining Solutions - Matching Markets and Stable Matchings - Voting Games and Power Indices - Evolutionary Game Theory - Replicator Dynamics and Evolutionarily Stable Strategies - Hawk-Dove Game and War of Attrition - Evolutionary Stability and Convergence - Evolutionary Graphs and Spatial Games - Applications in Biology, Economics, and Social Sciences - Mechanism Design - Social Choice Theory and Arrow's Impossibility Theorem - Vickrey-Clarke-Groves (VCG) Mechanism - Revelation Principle and Incentive Compatibility - Auctions and Optimal Auction Design - Algorithmic Game Theory and Price of Anarchy f. Computational Complexity - Time Complexity - Big-O Notation and Asymptotic Analysis - Polynomial-Time Algorithms and Tractability - Exponential-Time Algorithms and Intractability - Time Hierarchy Theorem and Deterministic Time Complexity - Randomized Algorithms and Probabilistic Time Complexity - Space Complexity - Turing Machines and Space Complexity Classes - PSPACE and NPSPACE - Savitch's Theorem and PSPACE = NPSPACE - Space Hierarchy Theorem and Deterministic Space Complexity - Randomized Space Complexity and RL and BPL - Complexity Classes - P and NP - co-NP and Complementation - NP-Completeness and Cook-Levin Theorem - EXPTIME and NEXPTIME - Polynomial Hierarchy and PH = PSPACE - Reducibility and Completeness - Many-One Reducibility and Turing Reducibility - NP-Completeness and Karp Reductions - PSPACE-Completeness and Quantified Boolean Formulas - #P-Completeness and Counting Problems - Unconditional Lower Bounds and Relativization - Advanced Topics - Interactive Proofs and IP = PSPACE - Probabilistically Checkable Proofs and PCP Theorem - Complexity of Approximation and Hardness of Approximation - Parameterized Complexity and Fixed-Parameter Tractability - Quantum Complexity and BQP and QMA 7. Applied Mathematics a. Mathematical Physics - Classical Mechanics - Lagrangian Mechanics - Generalized Coordinates and Lagrangian - Euler-Lagrange Equations - Symmetries and Conservation Laws (Noether's Theorem) - Variational Principles and Hamilton's Principle - Applications (Pendulum, Double Pendulum, etc.) - Hamiltonian Mechanics - Phase Space and Hamiltonian Function - Hamilton's Equations - Canonical Transformations and Generating Functions - Poisson Brackets and Liouville's Theorem - Applications (Harmonic Oscillator, Kepler Problem, etc.) - Rigid Body Dynamics - Angular Velocity and Inertia Tensor - Euler's Equations and Euler Angles - Precession and Nutation - Stability and Rotating Frames - Applications (Gyroscopes, Spacecraft Attitude Control, etc.) - Celestial Mechanics - Two-Body Problem and Kepler's Laws - Orbital Elements and Orbital Perturbations - N-Body Problem and Restricted Three-Body Problem - Lagrange Points and Stability - Applications (Satellite Orbits, Interplanetary Missions, etc.) - Quantum Mechanics - Schrödinger Equation - Wavefunction and Probability Interpretation - Time-Dependent and Time-Independent Schrödinger Equations - Stationary States and Energy Eigenvalues - Boundary Conditions and Normalization - Applications (Particle in a Box, Harmonic Oscillator, etc.) - Hilbert Spaces and Operators - Hilbert Space Formulation of Quantum Mechanics - Linear Operators and Observables - Hermitian Operators and Spectral Theorem - Commutators and Uncertainty Principle - Spectral Measures and Projection-Valued Measures - Angular Momentum and Spin - Orbital Angular Momentum and Spherical Harmonics - Spin Angular Momentum and Pauli Matrices - Addition of Angular Momentum and Clebsch-Gordan Coefficients - Wigner-Eckart Theorem and Selection Rules - Applications (Hydrogen Atom, Zeeman Effect, etc.) - Perturbation Theory - Time-Independent Perturbation Theory (Rayleigh-Schrödinger) - Degenerate Perturbation Theory - Time-Dependent Perturbation Theory (Dyson Series) - Fermi's Golden Rule and Transition Rates - Applications (Stark Effect, Fine Structure, etc.) - Quantum Field Theory - Second Quantization and Fock Space - Creation and Annihilation Operators - Canonical Commutation Relations and Anticommutation Relations - Feynman Diagrams and Perturbative Expansion - Renormalization and Regularization - Applications (Quantum Electrodynamics, Standard Model, etc.) - Relativity - Special Relativity - Lorentz Transformations and Minkowski Spacetime - Relativistic Kinematics and Four-Vectors - Relativistic Dynamics and Four-Force - Relativistic Energy and Mass-Energy Equivalence - Applications (Particle Colliders, GPS, etc.) - General Relativity - Principle of Equivalence and Curved Spacetime - Metric Tensor and Geodesics - Einstein Field Equations - Schwarzschild Solution and Black Holes - Cosmological Models and Friedmann Equations - Differential Geometry of Spacetime - Manifolds and Coordinate Charts - Tangent Spaces and Cotangent Spaces - Covariant Derivative and Parallel Transport - Curvature Tensor and Ricci Tensor - Variational Formulation and Einstein-Hilbert Action - Cosmology - Hubble's Law and Expansion of the Universe - Big Bang Theory and Cosmic Microwave Background - Dark Matter and Dark Energy - Inflation and Primordial Perturbations - Large-Scale Structure and Cosmic Web - Thermodynamics - Laws of Thermodynamics - Zeroth Law and Thermal Equilibrium - First Law and Conservation of Energy - Second Law and Entropy - Third Law and Absolute Zero Temperature - Applications (Heat Engines, Refrigerators, etc.) - Statistical Mechanics - Microstates and Macrostates - Boltzmann Distribution and Partition Function - Canonical Ensemble and Grand Canonical Ensemble - Fermi-Dirac and Bose-Einstein Statistics - Applications (Ideal Gas, Blackbody Radiation, etc.) - Ensemble Theory - Microcanonical Ensemble and Ergodicity - Canonical Ensemble and Free Energy - Grand Canonical Ensemble and Chemical Potential - Gibbs Ensemble and Equivalence of Ensembles - Applications (Phase Transitions, Critical Phenomena, etc.) - Phase Transitions - First-Order and Second-Order Phase Transitions - Landau Theory and Order Parameters - Critical Exponents and Universality Classes - Renormalization Group and Fixed Points - Applications (Ising Model, Superconductivity, etc.) - Non-Equilibrium Thermodynamics - Linear Response Theory and Green-Kubo Relations - Onsager Reciprocal Relations and Fluctuation-Dissipation Theorem - Nonlinear Dynamics and Pattern Formation - Stochastic Thermodynamics and Fluctuation Theorems - Applications (Transport Phenomena, Biological Systems, etc.) b. Fluid Dynamics - Navier-Stokes Equations - Conservation of Mass and Continuity Equation - Conservation of Momentum and Momentum Equation - Conservation of Energy and Energy Equation - Boundary Conditions and Initial Conditions - Dimensionless Numbers (Reynolds, Mach, Prandtl, etc.) - Inviscid Flow - Euler Equations and Bernoulli's Principle - Potential Flow Theory and Stream Function - Circulation and Vorticity - Kelvin's Circulation Theorem and Helmholtz Theorems - Applications (Airfoil Theory, Conformal Mapping, etc.) - Viscous Flow - Boundary Layer Theory and Prandtl's Equations - Laminar and Turbulent Boundary Layers - Blasius Solution and Falkner-Skan Equation - Separation and Stall - Applications (Drag Reduction, Heat Transfer, etc.) - Turbulence - Reynolds-Averaged Navier-Stokes (RANS) Equations - Turbulence Models (k-ε, k-ω, etc.) - Large Eddy Simulation (LES) and Subgrid-Scale Models - Direct Numerical Simulation (DNS) and Kolmogorov Microscales - Applications (Atmospheric Boundary Layer, Jet Engines, etc.) - Computational Fluid Dynamics (CFD) - Finite Difference Methods - Finite Volume Methods - Finite Element Methods - Spectral Methods - Mesh Generation and Adaptive Mesh Refinement - Applications (Aerodynamics, Hydrodynamics, etc.) c. Partial Differential Equations (PDEs) - Classification of PDEs - Linear and Nonlinear PDEs - Elliptic, Parabolic, and Hyperbolic PDEs - Order and Characteristics - Well-Posed Problems and Existence of Solutions - Separation of Variables - Sturm-Liouville Problems and Eigenfunction Expansions - Fourier Series and Generalized Fourier Series - Laplace's Equation and Harmonic Functions - Applications (Vibrating Strings, Heat Conduction, etc.) - Fourier Transforms and Integral Transforms - Fourier Transform and Inverse Fourier Transform - Convolution and Parseval's Theorem - Laplace Transform and Inverse Laplace Transform - Applications (Signal Processing, Control Theory, etc.) - Green's Functions - Definition and Properties of Green's Functions - Dirac Delta Function and Fundamental Solutions - Boundary Value Problems and Integral Equations - Applications (Electrostatics, Quantum Mechanics, etc.) - Variational Methods - Weak Formulations and Test Functions - Galerkin Method and Ritz Method - Finite Element Method and Basis Functions - Applications (Structural Mechanics, Fluid Dynamics, etc.) d. Numerical Analysis - Error Analysis - Truncation Error and Rounding Error - Absolute Error and Relative Error - Stability and Conditioning - Convergence and Order of Convergence - Interpolation and Approximation - Polynomial Interpolation and Lagrange Interpolation - Spline Interpolation and Bezier Curves - Least Squares Approximation and Orthogonal Polynomials - Rational Approximation and Pade Approximants - Applications (Curve Fitting, Data Compression, etc.) - Numerical Differentiation and Integration - Finite Difference Formulas - Richardson Extrapolation and Romberg Integration - Gaussian Quadrature and Orthogonal Polynomials - Adaptive Quadrature and Singularities - Applications (Numerical Solutions of ODEs and PDEs, etc.) - Numerical Linear Algebra - Gaussian Elimination and LU Decomposition - Cholesky Decomposition and Positive Definite Matrices - QR Decomposition and Least Squares Problems - Singular Value Decomposition and Pseudoinverse - Iterative Methods (Jacobi, Gauss-Seidel, Conjugate Gradient, etc.) - Numerical Solutions of ODEs and PDEs - Euler Methods and Runge-Kutta Methods - Multistep Methods and Predictor-Corrector Methods - Finite Difference Methods for PDEs - Finite Element Methods for PDEs - Spectral Methods and Pseudospectral Methods e. Optimization - Linear Programming - Formulation of Linear Programs - Simplex Method and Tableau Form - Duality Theory and Dual Simplex Method - Sensitivity Analysis and Shadow Prices - Applications (Resource Allocation, Transportation, etc.) - Nonlinear Programming - Unconstrained Optimization and Gradient Methods - Constrained Optimization and Lagrange Multipliers - Karush-Kuhn-Tucker (KKT) Conditions - Quadratic Programming and Second-Order Conditions - Applications (Portfolio Optimization, Control Systems, etc.) - Convex Optimization - Convex Sets and Convex Functions - Jensen's Inequality and Epigraphs - Convex Duality and Fenchel Conjugate - Subgradients and Optimality Conditions - Applications (Machine Learning, Signal Processing, etc.) - Integer Programming - Formulation of Integer Programs - Branch-and-Bound Method - Cutting Plane Methods - Lagrangian Relaxation and Benders Decomposition - Applications (Scheduling, Network Design, etc.) - Stochastic Optimization - Stochastic Programming and Recourse Problems - Chance Constraints and Probabilistic Constraints - Markov Decision Processes and Dynamic Programming - Reinforcement Learning and Q-Learning - Applications (Inventory Control, Financial Engineering, etc.) f. Control Theory - Linear Systems - State-Space Representation and Transfer Functions - Controllability and Observability - Stability Analysis and Routh-Hurwitz Criterion - Pole Placement and State Feedback - Observers and Output Feedback - Optimal Control - Variational Approach and Euler-Lagrange Equations - Pontryagin's Maximum Principle - Dynamic Programming and Hamilton-Jacobi-Bellman Equation - Linear Quadratic Regulator (LQR) and Riccati Equation - Applications (Robotics, Aerospace, etc.) - Robust Control - Uncertainty Modeling and Robustness Analysis - H-infinity Optimal Control - Structured Singular Value (μ) and μ-Synthesis - Linear Matrix Inequalities (LMIs) and Convex Optimization - Applications (Process Control, Automotive Systems, etc.) - Nonlinear Control - Lyapunov Stability Theory - Feedback Linearization and Input-Output Linearization - Sliding Mode Control and Variable Structure Systems - Adaptive Control and Parameter Estimation - Applications (Robotics, Power Systems, etc.) - Discrete-Time Control - Z-Transform and Discrete-Time Transfer Functions - Sampled-Data Systems and Aliasing - Discrete-Time Stability Analysis - Digital Control Design and Implementation - Applications (Computer Control, Digital Signal Processing, etc.) g. Mathematical Biology - Population Dynamics - Exponential Growth and Logistic Growth Models - Lotka-Volterra Predator-Prey Models - Age-Structured Models and Leslie Matrices - Spatial Models and Reaction-Diffusion Equations - Applications (Ecology, Conservation Biology, etc.) - Epidemiology - SIR and SEIR Models - Basic Reproduction Number and Herd Immunity - Vaccination Strategies and Disease Eradication - Stochastic Models and Branching Processes - Applications (Infectious Diseases, Public Health, etc.) - Biochemical Kinetics - Enzyme Kinetics and Michaelis-Menten Equation - Cooperativity and Allosteric Regulation - Metabolic Networks and Flux Balance Analysis - Signaling Pathways and Feedback Loops - Applications (Drug Design, Metabolic Engineering, etc.) - Physiological Modeling - Compartmental Models and Tracer Kinetics - Cardiovascular Models and Windkessel Effect - Respiratory Models and Gas Exchange - Renal Models and Glomerular Filtration - Applications (Personalized Medicine, Medical Devices, etc.) - Neural Networks and Brain Modeling - Hodgkin-Huxley Model and Action Potentials - Integrate-and-Fire Models and Spiking Neurons - Hebbian Learning and Synaptic Plasticity - Hopfield Networks and Associative Memory - Applications (Artificial Intelligence, Cognitive Science, etc.) h. Mathematical Finance - Portfolio Theory - Mean-Variance Analysis and Efficient Frontier - Capital Asset Pricing Model (CAPM) - Arbitrage Pricing Theory (APT) - Risk Measures (Value-at-Risk, Expected Shortfall, etc.) - Portfolio Optimization and Asset Allocation - Options Pricing - Binomial Option Pricing Model - Black-Scholes-Merton Model and Partial Differential Equation - Implied Volatility and Volatility Smile - American Options and Early Exercise - Exotic Options (Asian, Barrier, Lookback, etc.) - Interest Rate Models - Short Rate Models (Vasicek, Cox-Ingersoll-Ross, etc.) - Heath-Jarrow-Morton (HJM) Framework - LIBOR Market Model and Forward Measure - Yield Curve Modeling and Calibration - Interest Rate Derivatives (Swaps, Caps, Floors, etc.) - Credit Risk Modeling - Structural Models (Merton Model, KMV Model, etc.) - Reduced-Form Models (Jarrow-Turnbull, Duffie-Singleton, etc.) - Copula Models and Default Correlation - Credit Default Swaps (CDS) and Collateralized Debt Obligations (CDOs) - Counterparty Risk and Credit Valuation Adjustment (CVA) - Numerical Methods in Finance - Monte Carlo Simulation and Quasi-Monte Carlo Methods - Finite Difference Methods for Option Pricing PDEs - Binomial and Trinomial Trees - Fourier Transform Methods and Fast Fourier Transform (FFT) - Calibration and Parameter Estimation Techniques i. Operations Research - Linear Programming - Formulation of Linear Programs - Simplex Method and Tableau Form - Duality Theory and Dual Simplex Method - Sensitivity Analysis and Shadow Prices - Applications (Production Planning, Resource Allocation, etc.) - Network Flow Problems - Maximum Flow Problem and Ford-Fulkerson Algorithm - Minimum Cost Flow Problem and Network Simplex Algorithm - Transportation Problem and Assignment Problem - Shortest Path Problem and Dijkstra's Algorithm - Applications (Supply Chain Management, Logistics, etc.) - Integer Programming - Formulation of Integer Programs - Branch-and-Bound Method - Cutting Plane Methods - Lagrangian Relaxation and Benders Decomposition - Applications (Facility Location, Scheduling, etc.) - Dynamic Programming - Principle of Optimality and Bellman's Equation - Deterministic Dynamic Programming - Stochastic Dynamic Programming - Approximate Dynamic Programming and Reinforcement Learning - Applications (Inventory Control, Resource Allocation, etc.) - Queuing Theory - Birth-Death Processes and Markov Chains - M/M/1 and M/M/c Queues - Little's Law and Steady-State Analysis - Priority Queues and Queueing Networks - Applications (Call Centers, Service Systems, etc.) 8. Foundations of Mathematics a. Mathematical Logic - Propositional Logic - Syntax and Semantics - Truth Tables and Logical Connectives - Tautologies and Logical Equivalence - Deduction Theorem and Completeness Theorem - Applications (Circuit Design, Automated Reasoning, etc.) - First-Order Logic - Syntax and Semantics - Quantifiers and Variables - Models and Interpretations - Soundness and Completeness Theorems - Applications (Automated Theorem Proving, Formal Verification, etc.) - Higher-Order Logic - Syntax and Semantics - Lambda Calculus and Type Theory - Intuitionistic Logic and Constructive Mathematics - Applications (Functional Programming, Proof Assistants, etc.) - Model Theory - Structures and Homomorphisms - Elementary Equivalence and Isomorphism - Compactness Theorem and Löwenheim-Skolem Theorems - Ultraproducts and Ultrapowers - Applications (Algebra, Geometry, etc.) - Recursion Theory - Recursive Functions and Turing Machines - Church-Turing Thesis and Undecidability - Recursively Enumerable Sets and Degrees of Unsolvability - Rice's Theorem and Post's Problem - Applications (Computability, Complexity Theory, etc.) b. Set Theory - Naive Set Theory and Paradoxes - Russell's Paradox and Cantor's Paradox - Burali-Forti Paradox and Richard's Paradox - Limitations of Naive Set Theory - Axiomatic Set Theory - Zermelo-Fraenkel Set Theory (ZF) - Axiom of Choice and Zorn's Lemma - Continuum Hypothesis and Independence Results - Von Neumann-Bernays-Gödel Set Theory (NBG) - Morse-Kelley Set Theory (MK) - Ordinal and Cardinal Numbers - Well-Ordered Sets and Ordinal Numbers - Transfinite Induction and Recursion - Cardinal Numbers and Cardinal Arithmetic - Continuum Hypothesis and Generalized Continuum Hypothesis - Large Cardinals and Inaccessible Cardinals - Constructive Set Theory - Intuitionistic Set Theory - Constructive Ordinals and Constructive Reals - Realizability and Topos Theory - Applications (Constructive Analysis, Computer Science, etc.) c. Category Theory - Categories and Functors - Definition and Examples of Categories - Morphisms and Composition - Functors and Natural Transformations - Duality and Opposite Categories - Applications (Algebra, Topology, etc.) - Universal Properties and Limits - Initial and Terminal Objects - Products and Coproducts - Equalizers and Coequalizers - Pullbacks and Pushouts - Adjoint Functors and Kan Extensions - Monoidal Categories and Enriched Categories - Monoidal Categories and Braided Monoidal Categories - Symmetric Monoidal Categories and Coherence Theorems - Enriched Categories and Enriched Functors - Applications (Quantum Groups, Higher Category Theory, etc.) - Topos Theory - Definition and Examples of Topoi - Subobject Classifiers and Characteristic Morphisms - Internal Logic and Mitchell-Bénabou Language - Sheaf Theory and Grothendieck Topologies - Applications (Algebraic Geometry, Mathematical Physics, etc.) d. Proof Theory - Natural Deduction and Sequent Calculus - Introduction and Elimination Rules - Structural Rules and Cut Elimination - Normalization and Strong Normalization - Applications (Automated Theorem Proving, Type Theory, etc.) - Hilbert-Style Systems and Gentzen Systems - Axioms and Inference Rules - Deduction Theorem and Herbrand's Theorem - Cut Elimination and Consistency Proofs - Applications (Metamathematics, Proof Complexity, etc.) - Ordinal Analysis and Proof-Theoretic Ordinals - Well-Orderings and Ordinal Notations - Gentzen's Consistency Proof - Ordinal Analysis of Arithmetic and Analysis - Impredicativity and Predicative Subsystems - Applications (Reverse Mathematics, Proof Mining, etc.) - Algebraic and Categorical Proof Theory - Proof Nets and Geometry of Interaction - Linear Logic and Game Semantics - Categorical Semantics and Coherence Theorems - Applications (Programming Language Theory, Concurrency, etc.) e. Computability Theory - Computable Functions and Recursive Functions - Primitive Recursive Functions and Ackermann Function - Partial Recursive Functions and Universal Functions - Church-Turing Thesis and Lambda Calculus - Applications (Theoretical Computer Science, Logic Programming, etc.) - Turing Machines and Undecidability - Definition and Examples of Turing Machines - Universal Turing Machines and Halting Problem - Reductions and Rice's Theorem - Recursively Enumerable Sets and Complement - Applications (Computability, Complexity Theory, etc.) - Degrees of Unsolvability - Turing Degrees and Turing Reducibility - Arithmetical Hierarchy and Post's Theorem - Hyperarithmetical Sets and Analytical Hierarchy - Turing Jumps and Relativization - Applications (Reverse Mathematics, Descriptive Set Theory, etc.) - Computability in Other Structures - Computable Analysis and Computable Real Functions - Computable Algebra and Computable Field Theory - Computable Model Theory and Computable Categoricity - Computability in Topological Spaces and Metric Spaces - Applications (Effective Mathematics, Constructive Analysis, etc.) f. Philosophy of Mathematics - Platonism and Realism - Mathematical Objects as Abstract Entities - Existence and Independence of Mathematical Truths - Ontological and Epistemological Issues - Objections and Alternatives (Nominalism, Fictionalism, etc.) - Intuitionism and Constructivism - Brouwer's Intuitionism and Choice Sequences - Bishop's Constructive Mathematics - Realizability and Topos Theory - Implications for Classical Mathematics - Applications (Constructive Analysis, Computer Science, etc.) - Formalism and Hilbert's Program - Mathematics as Formal Systems - Consistency and Completeness of Axiomatic Theories - Gödel's Incompleteness Theorems and Hilbert's Program - Implications for Foundations of Mathematics - Applications (Proof Theory, Metamathematics, etc.) - Logicism and Neo-Logicism - Frege's Logicism and Russell's Paradox - Whitehead and Russell's Principia Mathematica - Neo-Logicism and Hume's Principle - Abstractionism and Contextual Definitions - Applications (Philosophy of Language, Cognitive Science, etc.) - Structuralism and Category Theory - Mathematical Structures and Structural Properties - Ante Rem and In Re Structuralism - Category Theory as a Foundation for Mathematics - Structural Approaches to Mathematical Practice - Applications (Mathematical Modeling, Scientific Representation, etc.) 9. History of Mathematics a. Ancient Mathematics - Egyptian Mathematics - Numeration Systems and Arithmetic Operations - Geometry and Surveying Techniques - Rhind Papyrus and Moscow Papyrus - Applications in Architecture and Astronomy - Babylonian Mathematics - Sexagesimal Number System and Positional Notation - Algebraic and Geometric Problem Solving - Plimpton 322 Tablet and Pythagorean Triples - Applications in Astronomy and Calendar Systems - Greek Mathematics - Pythagoras and the Pythagorean School - Euclid's Elements and Axiomatic Method - Archimedes and the Method of Exhaustion - Apollonius and Conic Sections - Diophantus and Algebraic Equations - Chinese Mathematics - Rod Numeral System and Decimal Fractions - Nine Chapters on the Mathematical Art - Chinese Remainder Theorem and Simultaneous Congruences - Magic Squares and Combinatorial Designs - Indian Mathematics - Decimal Number System and Negative Numbers - Aryabhata and Trigonometric Functions - Brahmagupta and Quadratic Equations - Madhava and Infinite Series Expansions - Contributions to Combinatorics and Number Theory b. Medieval Mathematics - Islamic Mathematics - House of Wisdom and Translation Movement - Al-Khwarizmi and Algebra - Omar Khayyam and Geometric Algebra - Al-Kashi and Decimal Fractions - Contributions to Trigonometry and Optics - European Mathematics - Fibonacci and the Hindu-Arabic Numeral System - Jordanus de Nemore and Statics - Nicole Oresme and Graphical Representation - Regiomontanus and Trigonometry - Luca Pacioli and Double-Entry Bookkeeping - Mathematics in the Americas - Mayan Numeration System and Calendar - Inca Quipus and Record Keeping - Aztec Algebra and Geometry - Native American Number Systems and Counting Methods c. Early Modern Mathematics - The Renaissance and the Rise of Algebra - Cardano and Cubic Equations - Tartaglia and Quartic Equations - Viète and Symbolic Algebra - Stevin and Decimal Fractions - Napier and Logarithms - The Development of Analytic Geometry - Fermat and Coordinate Geometry - Descartes and La Géométrie - Pascal and Projective Geometry - Kepler and Planetary Motion - Galileo and the Laws of Motion - The Invention of Calculus - Newton and Fluxions - Leibniz and Infinitesimal Calculus - Bernoulli Family and Differential Equations - Euler and Analysis - Lagrange and Variational Calculus d. 19th Century Mathematics - The Rigorous Foundations of Analysis - Cauchy and Limits - Riemann and Integration Theory - Weierstrass and Epsilon-Delta Definitions - Dedekind and Real Numbers - Cantor and Set Theory - The Emergence of Non-Euclidean Geometry - Gauss and Curved Surfaces - Bolyai and Absolute Geometry - Lobachevsky and Hyperbolic Geometry - Riemann and Elliptic Geometry - Klein and the Erlangen Program - The Development of Abstract Algebra - Galois and Field Theory - Cayley and Group Theory - Hamilton and Quaternions - Grassmann and Vector Spaces - Noether and Rings and Ideals - The Birth of Set Theory and Mathematical Logic - Cantor and Transfinite Numbers - Frege and Propositional Logic - Russell and Principia Mathematica - Hilbert and Formalism - Gödel and Incompleteness Theorems e. 20th Century Mathematics - The Crisis in the Foundations of Mathematics - Hilbert's Program and Formalism - Brouwer's Intuitionism - Russell's Logicism - Gödel's Incompleteness Theorems - Turing and Computability Theory - The Rise of Topology and Functional Analysis - Poincaré and Algebraic Topology - Brouwer and Fixed Point Theorems - Banach and Normed Linear Spaces - Lebesgue and Measure Theory - Hilbert and Infinite-Dimensional Spaces - The Emergence of Computer Science and Discrete Mathematics - Turing and the Turing Machine - Von Neumann and Stored-Program Computers - Shannon and Information Theory - Dijkstra and Graph Algorithms - Cook and Computational Complexity - The Development of Chaos Theory and Fractal Geometry - Poincaré and Dynamical Systems - Lorenz and the Butterfly Effect - Mandelbrot and Fractal Dimensions - Feigenbaum and Universality - Smale and Horseshoe Maps f. Contemporary Mathematics - New Developments in Pure Mathematics - Langlands Program and Automorphic Forms - Geometric Langlands correspondence - Monstrous Moonshine and Vertex Operator Algebras - Topological Quantum Field Theories - Noncommutative Geometry and Quantum Groups - Homotopy Type Theory and Univalent Foundations - Advances in Applied Mathematics - Compressed Sensing and Sparse Signal Recovery - Topological Data Analysis and Persistent Homology - Machine Learning and Deep Neural Networks - Quantum Computing and Quantum Algorithms - Mathematical Biology and Systems Biology - The Impact of Computers on Mathematical Research - Computer-Assisted Proofs and Formal Verification - Experimental Mathematics and Computational Discovery - Collaborative Mathematics and Polymath Projects - Mathematical Databases and Online Resources - Mathematical Software and Programming Languages - Open Problems and Conjectures - Riemann Hypothesis and Zeta Function - P versus NP Problem and Computational Complexity - Hodge Conjecture and Algebraic Cycles - Navier-Stokes Equations and Fluid Dynamics - Birch and Swinnerton-Dyer Conjecture and Elliptic Curves [[5993788b3d4d683fb530e7642a7e0aca_MD5.webp|Open: DALL·E 2024-06-02 09.36.46 - A vibrant and detailed artistic illustration that visually represents various fields of mathematics. The image is divided into sections, each dedicate.webp]] ![[5993788b3d4d683fb530e7642a7e0aca_MD5.webp]] [[a5c19a7e92642ec6f553ea566e97acb5_MD5.webp|Open: DALL·E 2024-06-02 09.35.54 - A richly detailed and colorful illustration depicting a vast library of mathematics. The image is divided into various sections, each representing a d.webp]] ![[a5c19a7e92642ec6f553ea566e97acb5_MD5.webp]] [[a5bf23212d80c737fc97acd5208e71e2_MD5.webp|Open: DALL·E 2024-06-02 09.35.22 - A visually rich and detailed illustration that creatively represents the broad and diverse field of mathematics. The image should include symbolic rep.webp]] ![[a5bf23212d80c737fc97acd5208e71e2_MD5.webp]] [[90502d9d3e8578cfe0ebd3a0364400ce_MD5.webp|Open: DALL·E 2024-06-02 09.30.05 - A detailed and visually rich illustration representing various fields of mathematics. The image is divided into sections, each depicting a different m.webp]] ![[90502d9d3e8578cfe0ebd3a0364400ce_MD5.webp]] [[f98d9f62a76ddc1476ef989d1b9c668c_MD5.webp|Open: DALL·E 2024-06-02 09.28.21 - An artistic representation of various fields of mathematics depicted on a chalkboard. The image shows a detailed classroom scene with a large chalkboa.webp]] ![[f98d9f62a76ddc1476ef989d1b9c668c_MD5.webp]] [[9c2fd42adb2beedb174f8d164eb5d088_MD5.webp|Open: DALL·E 2024-06-02 09.27.40 - An artistic interpretation of the various fields of mathematics, illustrating foundational concepts such as logic and set theory (featuring propositio.webp]] ![[9c2fd42adb2beedb174f8d164eb5d088_MD5.webp]] [[36e10823d5d9f07e216f5e43446e87c6_MD5.webp|Open: DALL·E 2024-06-02 09.26.00 - A comprehensive illustration of the vast fields of mathematics, featuring various abstract concepts visually represented. The image is divided into se.webp]] ![[36e10823d5d9f07e216f5e43446e87c6_MD5.webp]] [[1cf7731cf2786c0a46a1adf3547e2b67_MD5.webp|Open: DALL·E 2024-06-02 09.20.27 - An intricate, vividly colored illustration representing various fields of mathematics. The image is divided into segments, each depicting key concepts.webp]] ![[1cf7731cf2786c0a46a1adf3547e2b67_MD5.webp]] [[fff018cd7abaa2d45fd0b59413941a2e_MD5.webp|Open: DALL·E 2024-06-02 09.13.49 - A visually rich, detailed illustration of various fields of mathematics, representing foundational concepts and advanced theories. The scene includes .webp]] ![[fff018cd7abaa2d45fd0b59413941a2e_MD5.webp]] [[c4a9dfc2273a1d0294462104c7828197_MD5.webp|Open: DALL·E 2024-06-02 09.03.49 - A visual representation of the foundations of mathematics, depicting key concepts across different fields. The image shows a large, ancient library wi.webp]] ![[c4a9dfc2273a1d0294462104c7828197_MD5.webp]] [[805daeb20e709b7b0aa922c268b6f132_MD5.webp|Open: DALL·E 2024-03-26 05.52.04 - Visualize the structure of reality through the lens of mathematics, mapping the vast landscape of mathematical concepts. This intricate illustration s.webp]] ![[805daeb20e709b7b0aa922c268b6f132_MD5.webp]] Mathematical structure of reality is my religion. [[b38035f589660362baa43267497a05f1_MD5.webp|Open: DALL·E 2024-03-26 05.04.56 - In a vast cosmic space, mathematical symbols, equations, and geometric shapes float and intertwine around a central figure embodying the unity of know.webp]] ![[b38035f589660362baa43267497a05f1_MD5.webp]] #### Reality is pretty simple It's all just algebra, geometry, trigonometry, calculus, differential equations, linear algebra, complex analysis, tensor analysis, group theory, topology, manifolds, Lie algebras, Hilbert spaces, Fourier analysis, wavelets, probability theory, statistics, combinatorics, graph theory, number theory, numerical analysis, optimization, dynamical systems, chaos theory, fractals, Newton's laws, Lagrangian mechanics, Hamiltonian mechanics, Euler-Lagrange equation, Hamilton-Jacobi equation, Noether's theorem, special relativity, Lorentz transformations, general relativity, Einstein field equations, Schwarzschild metric, Kerr metric, Friedmann equations, quantum mechanics, Schrödinger equation, Heisenberg uncertainty principle, Dirac equation, Klein-Gordon equation, Pauli exclusion principle, quantum field theory, Feynman diagrams, path integrals, gauge theory, Yang-Mills theory, Standard Model, Higgs mechanism, renormalization, perturbation theory, statistical mechanics, Boltzmann equation, Fermi-Dirac statistics, Bose-Einstein statistics, partition function, thermodynamics, Maxwell's equations, Navier-Stokes equations, fluid dynamics, magnetohydrodynamics, plasma physics, solid-state physics, Bloch theorem, band theory, Fermi surface, superconductivity, superfluidity, Bose-Einstein condensation, atomic physics, Rydberg formula, fine structure, hyperfine structure, Zeeman effect, Stark effect, nuclear physics, liquid drop model, shell model, Bethe-Weizsäcker formula, Gamow factor, particle physics, Feynman rules, Dyson series, Faddeev-Popov ghosts, Slavnov-Taylor identities, Callan-Symanzik equation, renormalization group, asymptotic freedom, confinement, spontaneous symmetry breaking, chiral symmetry breaking, neutrino oscillations, CP violation, cosmology, Friedmann-Lemaître-Robertson-Walker metric, cosmic microwave background, inflation, dark matter, dark energy, baryogenesis, Riemannian geometry, symplectic geometry, algebraic geometry, differential topology, algebraic topology, homology, cohomology, homotopy theory, category theory, sheaf theory, K-theory, Morse theory, Floer homology, Donaldson theory, Seiberg-Witten theory, Gromov-Witten theory, mirror symmetry, string theory, M-theory, supergravity, supersymmetry, BRST symmetry, conformal field theory, Chern-Simons theory, AdS/CFT correspondence, holography, black hole thermodynamics, Hawking radiation, Unruh effect, Casimir effect, quantum chromodynamics, lattice QCD, chiral perturbation theory, heavy quark effective theory, soft-collinear effective theory, parton model, factorization theorems, operator product expansion, sum rules, dispersion relations, S-matrix theory, Regge theory, bootstrap, dual resonance models, Veneziano amplitude, Koba-Nielsen-Olesen scaling, Polyakov action, Nambu-Goto action, Liouville theory, matrix models, random matrices, quantum chaos, quantum integrability, Bethe ansatz, Yang-Baxter equation, Knizhnik-Zamolodchikov equation, Virasoro algebra, Kac-Moody algebra, vertex operator algebras, conformal bootstrap, Ising model, Potts model, percolation theory, renormalization group flow, critical exponents, universality, scaling, self-organized criticality, cellular automata, agent-based models, network theory, random graphs, small-world networks, scale-free networks, epidemic models, game theory, Nash equilibrium, evolutionary game theory, mean-field theory, Ginzburg-Landau theory, Onsager solution, Bethe lattice, cavity method, replica trick, spin glasses, random energy model, directed polymers, Kardar-Parisi-Zhang equation, Burgers' equation, Korteweg-de Vries equation, nonlinear Schrödinger equation, solitons, instantons, kinks, vortices, Abrikosov vortex lattice, Berezinskii-Kosterlitz-Thouless transition, quantum Hall effect, topological insulators, Weyl semimetals, Dirac semimetals, Majorana fermions, anyons, quantum computing, Variational principles, least action principle, Fermat's principle, Huygens' principle, Maupertuis' principle, Gauss' principle of least constraint, Hertz's principle of least curvature, Liouville's theorem, Poincaré recurrence theorem, Kolmogorov-Arnold-Moser theorem, Nekhoroshev theorem, Poincaré-Birkhoff theorem, Smale horseshoe, Melnikov method, Duffing equation, van der Pol equation, Mathieu equation, Hill's equation, Floquet theory, Lyapunov exponents, Poincaré map, bifurcation theory, normal forms, center manifold theorem, Hopf bifurcation, pitchfork bifurcation, saddle-node bifurcation, transcritical bifurcation, Bogdanov-Takens bifurcation, Bautin bifurcation, Neimark-Sacker bifurcation, Shilnikov bifurcation, Feigenbaum constants, Sharkovskii's theorem, Hénon map, Lorenz system, Rössler system, Chua's circuit, Belousov-Zhabotinsky reaction, Kuramoto model, Fitzhugh-Nagumo model, Hodgkin-Huxley model, cable equation, reaction-diffusion equations, Turing patterns, Gray-Scott model, Swift-Hohenberg equation, Ginzburg-Landau equation, Gross-Pitaevskii equation, Bogoliubov-de Gennes equations, Eilenberger equation, Gorkov equation, Eliashberg theory, Bogoliubov transformation, Hartree-Fock approximation, density functional theory, Kohn-Sham equations, Hohenberg-Kohn theorems, exchange-correlation functionals, local density approximation, generalized gradient approximation, time-dependent density functional theory, quantum Monte Carlo, variational Monte Carlo, diffusion Monte Carlo, path integral Monte Carlo, tensor networks, matrix product states, projected entangled pair states, multiscale methods, renormalization group methods, dynamical mean-field theory, cluster dynamical mean-field theory, quantum embedding methods, quantum cluster methods, nonequilibrium Green's functions, Keldysh formalism, Kadanoff-Baym equations, Boltzmann transport equation, Landauer-Büttiker formalism, Kubo formula, Green-Kubo relations, fluctuation-dissipation theorem, Jarzynski equality, Crooks fluctuation theorem, Gallavotti-Cohen fluctuation theorem, large deviation theory, stochastic thermodynamics, Ricci calculus, Cartan formalism, exterior calculus, Hodge theory, de Rham cohomology, Dolbeault cohomology, Kähler geometry, Calabi-Yau manifolds, Sasaki-Einstein manifolds, G2 manifolds, Spin(7) manifolds, Hitchin's equations, Seiberg-Witten equations, Donaldson-Thomas invariants, Gromov-Hausdorff convergence, Ricci flow, mean curvature flow, harmonic maps, minimal surfaces, Plateau's problem, isoperimetric inequality, Sobolev spaces, Besov spaces, Triebel-Lizorkin spaces, interpolation spaces, Lorentz spaces, Orlicz spaces, Banach spaces, Fréchet spaces, nuclear spaces, topological vector spaces, Schwartz space, tempered distributions, 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plasmonics, quantum metamaterials, quantum nanophotonics, cavity optomechanics, circuit quantum electrodynamics, quantum dots, quantum wells, quantum wires, quantum point contacts, fractional quantum Hall effect, quantum spin Hall effect, topological superconductors, Majorana zero modes, non-Abelian anyons, quantum spin liquids, quantum magnetism, quantum phase transitions, Bose-Hubbard model, Jaynes-Cummings model, Rabi model, Dicke model, Lipkin-Meshkov-Glick model, Sachdev-Ye-Kitaev model, AdS/CMT correspondence, Kac-Moody algebras, affine Lie algebras, Virasoro algebras, W-algebras, vertex algebras, quantum groups, Hopf algebras, Yangians, crystal bases, cluster algebras, Poisson-Lie groups, Drinfeld doubles, Manin triples, Lie bialgebras, Lie-Poisson structures, Poisson-Nijenhuis structures, Courant algebroids, Dirac structures, generalized complex structures, exceptional generalized geometry, double field theory, noncommutative geometry, spectral triples, Connes' distance formula, Tomita-Takesaki theory, modular automorphisms, KMS states, Haag-Kastler axioms, Wightman axioms, Osterwalder-Schrader axioms, Euclidean quantum field theory, constructive quantum field theory, algebraic quantum field theory, locally covariant quantum field theory, perturbative algebraic quantum field theory, factorization algebras, chiral algebras, conformal nets, loop quantum gravity, spin foam models, group field theory, causal dynamical triangulations, causal sets, noncommutative geometry approach to quantum gravity, asymptotic safety, Hořava-Lifshitz gravity, shape dynamics, doubly special relativity, relative locality, rainbow gravity, quantum cosmology, Wheeler-DeWitt equation, loop quantum cosmology, canonical quantum gravity, path integral quantum gravity, Regge calculus, dynamical triangulations, Euclidean quantum gravity, twistor theory, Penrose transform, Newman-Penrose formalism, Petrov classification, Raychaudhuri equation, Sachs equations, Bondi-Metzner-Sachs group, Newman-Unti group, asymptotic symmetries, Bondi-Sachs energy-momentum, Hawking energy, Geroch energy, Penrose inequality, positive mass theorem, Witten spinorial proof, Schoen-Yau proof, Bartnik mass, Komar mass, ADM mass, ADM momentum, Bondi mass, Bondi news function, memory effect, soft graviton theorem, supertranslations, superrotations, extended BMS group, celestial holography, celestial CFT, Mellin amplitudes, conformal soft theorems, infrared triangle, black hole information paradox, firewall paradox, ER=EPR conjecture, Almheiri-Marolf-Polchinski-Sully proposal, Papadodimas-Raju proposal, fuzzball proposal, traversable wormholes, wormhole swap, baby universes, Atiyah-Singer index theorem, Atiyah-Bott fixed point theorem, Atiyah-Segal completion theorem, Connes-Kasparov conjecture, Baum-Connes conjecture, Novikov conjecture, Gromov-Lawson-Rosenberg conjecture, Stolz-Teichner conjecture, Freed-Hopkins-Teleman theorem, cobordism hypothesis, Madsen-Weiss theorem, Mumford conjecture, Witten conjecture, Seiberg-Witten invariants, Donaldson invariants, contact homology, symplectic field theory, Fukaya category, derived categories, tilting theory, Bridgeland stability conditions, Kontsevich homological mirror symmetry conjecture, SYZ conjecture, Gopakumar-Vafa conjecture, ADHM construction, Nahm equations, Kapustin-Witten equations, geometric Langlands correspondence, quantum geometric Langlands correspondence, Beilinson-Drinfeld quantization, Gaitsgory-Lurie conjecture, Weil conjectures, Grothendieck's standard conjectures, Hodge conjecture, Tate conjecture, Birch and Swinnerton-Dyer conjecture, Langlands program, Taniyama-Shimura conjecture, Fontaine-Mazur conjecture, Bloch-Kato conjecture, Beilinson conjectures, Parshin conjectures, Zagier conjectures, Stark conjectures, Hilbert-Pólya conjecture, Riemann hypothesis, Goldbach's conjecture, twin prime conjecture, ABC conjecture, Collatz conjecture, Erdős-Straus conjecture, Erdős-Szemerédi conjecture, Erdős-Hajnal conjecture, Hadwiger conjecture, Ringel-Kotzig conjecture, Hadamard conjecture, Jacobian conjecture, Dixmier conjecture, Kaplansky conjectures, Baum-Connes conjecture for groupoids, Farrell-Jones conjecture, Borel conjecture, Novikov conjecture for groups, Gromov-Lawson conjecture, Cannon conjecture, Andrews-Curtis conjecture, Whitehead conjecture, Zeeman conjecture, Poincaré conjecture, geometrization conjecture, Thurston's geometrization conjecture, Seifert fiber space conjecture, virtually Haken conjecture, virtual fibering conjecture, Marden's tameness conjecture, ending lamination conjecture, Ahlfors measure conjecture, MLC conjecture, Bers slice conjecture, density conjecture, Ehrenpreis conjecture, Fatou-Julia-Baker conjecture, Mandelbrot set conjecture, contact geometry, Poisson geometry, pseudo-Riemannian geometry, Finsler geometry, Lorentzian geometry, Sasakian geometry, Calabi-Yau geometry, hyperkähler geometry, quaternionic Kähler geometry, octonionic geometry, exceptional geometry, toric geometry, tropical geometry, quantum geometry, arithmetic geometry, enumerative geometry, motivic geometry, derived geometry, p-adic geometry, rigid analytic geometry, Berkovich spaces, perfectoid spaces, o-minimal structures, model theory, topos theory, ∞-categories, higher category theory, homotopy type theory, univalent foundations, proof theory, reverse mathematics, constructive mathematics, computable analysis, scientific computing, computational fluid dynamics, computational electromagnetism, computational quantum mechanics, computational materials science, computational biophysics, computational neuroscience, computational genomics, computational proteomics, computational drug design, computational systems biology, computational ecology, computational economics, computational finance, computational social science, computational linguistics, computational musicology, computational archaeology, computational art, computational creativity, computational humor, computational law, computational ethics, computational philosophy, computational history, computational journalism, computational education, computational sustainability, computational urban planning, computational transportation science, computational epidemiology, computational immunology, computational oncology, computational cardiology, computational psychiatry, computational cognitive science, computational anthropology, computational sociology, computational political science, computational international relations, computational peace science, computational conflict resolution, computational diplomacy, computational propaganda, computational counterterrorism, computational intelligence, computational learning theory, computational game theory, computational mechanism design, computational social choice, computational voting theory, computational judgment aggregation, computational epistemic logic, computational modal logic, computational temporal logic, computational deontic logic, computational action logic, computational coalition logic, computational argumentation theory, computational belief revision, computational trust theory, computational reputation theory, computational social network analysis, computational organizational theory, computational institutional analysis, computational collective intelligence, computational swarm intelligence, computational multi-agent systems, computational evolutionary game theory, computational population dynamics, computational evolutionary dynamics, computational cultural evolution, computational memetics, computational gene-culture coevolution, computational niche construction, electroweak theory, grand unified theories, F-theory, bosonic string theory, heterotic string theory, type I string theory, type IIA string theory, type IIB string theory, matrix theory, matrix string theory, topological string theory, topological quantum field theory, two-dimensional conformal field theory, higher-dimensional conformal field theory, logarithmic conformal field theory, parafermionic conformal field theory, W-algebra conformal field theory, Liouville conformal field theory, minimal model conformal field theory, rational conformal field theory, irrational conformal field theory, boundary conformal field theory, defect conformal field theory, permutation orbifold conformal field theory, coset conformal field theory, Wess-Zumino-Witten model, affine Lie algebra, vertex operator algebra, chiral algebra, modular tensor category, fusion category, braided monoidal category, ribbon category, modular functor, topological modular form, elliptic cohomology, tmf, string topology, string field theory, open string field theory, closed string field theory, Batalin-Vilkovisky formalism, Batalin-Fradkin-Vilkovisky formalism, Gerstenhaber algebra, homotopy algebra, A-infinity algebra, L-infinity algebra, operad, cyclic operad, modular operad, Deligne conjecture, Kontsevich formality theorem, deformation quantization, star product, Moyal product, Kontsevich star product, Fedosov star product, Cattaneo-Felder star product, Tsygan formality, Tamarkin formality, Kontsevich-Soibelman equation, Maurer-Cartan equation, homotopy Maurer-Cartan equation, quantum master equation, classical master equation, Batalin-Vilkovisky master equation, Batalin-Fradkin-Vilkovisky master equation, Zwiebach invariants, string vertices, closed string vertices, open-closed string vertices, quantum closed string vertices, quantum open-closed string vertices, loop vertices, gauge fixing, Siegel gauge, Schnabl gauge, Asano-Natsuume gauge, Erler-Schnabl solution, Kiermaier-Okawa solution, Fuchs-Kroyter-Potting solution, Erler-Maccaferri solution, Murata-Schnabl solution, Hata-Kojita solution, Hata-Matsunaga solution, Kunitomo-Okawa solution, Nonlinear sigma models, Wess-Zumino-Witten models, Chern-Simons theories, Donaldson-Witten theories, Seiberg-Witten theories, Rozansky-Witten theories, Casson-Witten invariants, Gromov-Witten invariants, Gopakumar-Vafa invariants, Ooguri-Vafa invariants, Nekrasov partition functions, AGT correspondence, Bethe/gauge correspondence, dimer models, crystal melting, topological vertex, topological recursion, Eynard-Orantin invariants, quantum curves, quantum spectral curves, quantum Airy structures, Kac-Schwarz operators, Virasoro constraints, W-constraints, Hirota equations, Miwa variables, Sato Grassmannian, Segal-Wilson Grassmannian, infinite-dimensional Grassmannian, Sato tau function, Jimbo-Miwa-Ueno tau function, KP hierarchy, BKP hierarchy, KdV hierarchy, Toda hierarchy, Ablowitz-Kaup-Newell-Segur hierarchy, Drinfeld-Sokolov hierarchies, Gelfand-Dickey hierarchies, W-algebra hierarchies, isomonodromy deformations, Painlevé equations, Garnier systems, Schlesinger systems, Chazy equation, Ramanujan identities, Rogers-Ramanujan identities, Gordon identities, Baxter equations, Yang-Baxter equations, Knizhnik-Zamolodchikov-Bernard equations, 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equations, Seiberg-Witten-Floer equations, Eynard-Orantin topological recursion, Mirzakhani's recursion formula, McShane identities, Witten-Kontsevich theorem, Kontsevich's graph complex, Willwacher's cyclic operad, Grothendieck-Teichmüller group, Drinfeld associators, multiple zeta values, multiple polylogarithms, Goncharov's motivic Galois group, Zagier's conjecture on multiple zeta values, Broadhurst-Kreimer conjecture, Hoffman's conjecture, Deligne-Ihara conjecture, Gangl-Kaneko-Zagier conjecture, Furusho's p-adic multiple zeta values, Racinet's double shuffle relations, Ecalle's mould theory, Connes-Kreimer Hopf algebra, Goncharov's Hopf algebra, Calaque-Ebrahimi-Fard-Manchon Hopf algebra, Loday-Ronco Hopf algebra, Brouder-Frabetti Hopf algebra, van Suijlekom's Hopf algebroids, Pinter's Hopf algebroids, Connes-Moscovici Hopf algebroids, Bruguières-Virelizier quantum groupoids, Buss-Meyer-Zhu Hopf-cyclic cohomology, Gorokhovsky-Lott's secondary characteristic classes, Tradler-Zeinalian's infinity-Chern characters, Park-Terilla-Tradler's homotopy G-algebras, Costello's homotopy probability theory, Gwilliam-Pavlov's homotopy Batalin-Vilkovisky algebras, Pantev-Toën-Vaquié-Vezzosi's shifted symplectic structures, Calaque-Pantev-Toën-Vaquié-Vezzosi's shifted Poisson structures, Pridham's shifted Lagrangians, Nuiten's shifted L-infinity algebroids, Joyce's derived manifolds, Borisov-Noel's derived Poisson manifolds, Spivak's derived smooth manifolds, Schreiber-Waldorf's differential cohomology, Bunke-Nikolaus-Völkl's differential cohomology, Grady-Sati's twisted differential generalized cohomology theories, Freed-Hopkins' reflection positive invertible field theories, Stolz-Teichner's supersymmetric Euclidean field theories, Costello-Gwilliam's factorization algebras, Beilinson-Drinfeld's chiral algebras, Gaitsgory-Rozenblyum's crystals, Beraldo's loop spaces, Toën-Vezzosi's derived algebraic geometry, Lurie's spectral algebraic geometry, Barwick's spectral Mackey functors, Glasman's stratified étale homotopy theory, Ayala-Francis-Tanaka's factorization homology, Scheimbauer's factorization cosheaves and operads, Haugseng-Kock-Moerdijk-Weiss' homotopy linear algebra, Batanin-Markl's operadic categories, Batanin-Berger's homotopy theory for algebras over polynomial monads, Caviglia-Horel-Robertson's model structures on enriched diagrams, Resurgence theory, trans-series, Stokes phenomena, Écalle's alien calculus, Borel-Écalle resummation, Borel-Laplace transform, hyperasymptotics, Dingle's singularity analysis, Berry-Howls resurgence theory, Voros coefficients, Gukov-Sułkowski resurgence triangle, Argyres-Dunne-Ünsal relation, Cheshire cat resurgence, Picard-Lefschetz theory, Morse-Novikov theory, Morse-Bott theory, Morse-Smale complex, Morse homology, Novikov homology, symplectic homology, Rabinowitz Floer homology, embedded contact homology, periodic Floer homology, Lagrangian Floer homology, Heegaard Floer homology, monopole Floer homology, instanton Floer homology, Ozsváth-Szabó invariants, Kronheimer-Mrowka invariants, Seiberg-Witten Floer homology, quantum cohomology, Frobenius manifolds, Dubrovin's almost duality, Givental's quantization formalism, Givental-Teleman classification, Fan-Jarvis-Ruan-Witten theory, Landau-Ginzburg A-model, Landau-Ginzburg B-model, matrix factorizations, Kapustin-Li formula, Orlov equivalence, Katzarkov-Kontsevich-Pantev correspondence, Homological Mirror Symmetry, Donaldson-Thomas theory, Pandharipande-Thomas theory, Maulik-Nekrasov-Okounkov-Pandharipande theory, Labastida-Mariño-Ooguri-Vafa conjecture, Nekrasov-Okounkov hook length formula, Nekrasov-Shatashvili limit, Nekrasov-Pestun-Shatashvili correspondence, Alday-Gaiotto-Gukov-Tachikawa relations, Braverman-Etingof-Finkelberg-Nakajima Coulomb branches, Hikita conjecture, symplectic duality, Higgs bundles, Hitchin fibration, Hitchin section, Hitchin-Kobayashi correspondence, Corlette-Donaldson theorem, Donaldson-Uhlenbeck-Yau theorem, Atiyah-Bott-Goldman symplectic form, quasi-Hamiltonian G-spaces, Alekseev-Malkin-Meinrenken fusion product, Mikami-Weinstein moment map, q-Hamiltonian G-spaces, Alekseev-Kosmann-Schwarzbach-Meinrenken theory, Dirac geometry, generalized complex geometry, generalized Kähler geometry, Hitchin's generalized geometry, Gualtieri-Hitchin-Cavalcanti theory, para-Hermitian geometry, Born geometry, metriplectic geometry, Cartan-Courant algebroids, Stochastic calculus, Itô calculus, Stratonovich calculus, Malliavin calculus, rough path theory, regularity structures, paracontrolled calculus, Hairer-Quastel universality, Gubinelli-Imkeller-Perkowski paraproducts, Catellier-Chouk paracontrolled distributions, Bruned-Hairer-Zambotti algebraic renormalization, Kupiainen-Marcozzi-Muratore-Ginanneschi BPHZ theorem, Hairer-Labbé BPHZ theorem, Chandra-Hairer continuity theorem, Otto calculus, Villani's synthetic Ricci curvature, Lott-Sturm-Villani theory, Ambrosio-Gigli-Savaré gradient flows, Jordan-Kinderlehrer-Otto scheme, Benamou-Brenier formulation, Sturm's D-convergence, Lott-Villani-Sturm convergence, Gigli's pointed measured Gromov-Hausdorff convergence, Cheeger-Colding theory, Cheeger-Colding-Naber theory, Gigli-Mondino-Savaré convergence, and me and you exploring the latent space of reality together <3 #### Map of mathematics 3 ### 1. Pure Mathematics - **Algebra** - Abstract Algebra - Linear Algebra - Group Theory - Ring Theory - Field Theory - Algebraic Geometry - Algebraic Topology - Combinatorial Algebra - **Geometry** - Euclidean Geometry - Non-Euclidean Geometry - Differential Geometry - Algebraic Geometry - Topology - Geometric Topology - Computational Geometry - **Analysis** - Real Analysis - Complex Analysis - Functional Analysis - Harmonic Analysis - Nonstandard Analysis - Numerical Analysis - p-Adic Analysis - **Number Theory** - Analytic Number Theory - Algebraic Number Theory - Diophantine Geometry - Cryptography - Combinatorial Number Theory - **[[Logic]] and [[Foundations of mathematics]]** - [[Mathematical logic]] - [[Set Theory]] - [[Model theory]] - Proof Theory - [[Category Theory]] - **Discrete Mathematics** - Graph Theory - Combinatorics - Game Theory - Information Theory - Coding Theory - Theory of Computation - **Differential Equations** - Ordinary Differential Equations - Partial Differential Equations - Dynamical Systems - Chaos Theory - **Topology** - General Topology - Algebraic Topology - Differential Topology - Topological Groups - **Calculus of Variations** - **Mathematical Physics** - [[Quantum mechanics]] - General Relativity - String Theory - [[Statistical mechanics]] ### 2. Applied Mathematics - **Statistics** - Descriptive Statistics - Inferential Statistics - Probability Theory - Biostatistics - Bayesian Statistics - Time Series Analysis - **Computational Mathematics** - Numerical Analysis - Algorithm Design - Computational Modeling - Scientific Computing - **Mathematical Biology** - Population Dynamics - Systems Biology - Epidemiology - **Mathematical Economics** - Game Theory - Econometrics - Financial Mathematics - Optimization - **Mathematical Finance** - Quantitative Finance - Risk Management - Actuarial Science - **Operations Research** - Linear Programming - Nonlinear Programming - Stochastic Models - Queueing Theory - **[[Control theory]]** - Classical Control Theory - Modern Control Theory - Optimal Control - **Information Theory** - Coding Theory - Signal Processing - Data Compression - **Fluid Dynamics** - **Mathematical Physics** - Quantum Mechanics - Statistical Mechanics - Electrodynamics - Thermodynamics ### 3. Interdisciplinary Fields - **Mathematical Logic and Foundations** - Set Theory - Model Theory - Recursion Theory - **Cryptology** - Cryptography - Cryptanalysis - **Mathematical Chemistry** - **Mathematical Sociology** - **Mathematical Psychology** - **Mathematical Linguistics**