Mathematics [x.com](https://twitter.com/burny_tech/status/1771884959369748634) Absolutely! Let's dive even deeper into the vast ocean of mathematics, exploring the intricate connections and the myriad of subfields that make up this fascinating discipline. Prepare for a long and comprehensive journey through the mathematical iceberg. Above the surface (commonly known): - Basic arithmetic (addition, subtraction, multiplication, division, exponents, roots, order of operations) - Fractions, decimals, percentages, ratios, and proportions - Basic geometry (shapes, angles, area, volume, surface area, Pythagorean theorem, similarity, congruence) - Basic algebra (solving simple equations, linear functions, inequalities, absolute value, polynomials) - Basic probability (coin flips, dice rolls, simple events, basic combinatorics, probability trees) - Basic statistics (mean, median, mode, range, standard deviation, variance, z-scores, normal distribution) - Consumer mathematics (interest, taxes, budgeting, loans, annuities, depreciation, inflation) - Measurement and unit conversions (length, area, volume, mass, time, temperature) - Basic trigonometry (sine, cosine, tangent, basic identities, right triangle trigonometry) - Coordinate geometry (distance formula, midpoint formula, slope, equations of lines) Just below the surface (high school level): - Advanced trigonometry (law of sines, law of cosines, trigonometric equations, inverse trigonometric functions, trigonometric form of complex numbers) - Precalculus (function composition, transformations, inverse functions, rational functions, parametric equations, polar coordinates, vectors) - Probability distributions (binomial, normal, Poisson, geometric, hypergeometric, exponential, chi-square, t-distribution, F-distribution) - Inferential statistics (hypothesis testing, confidence intervals, t-tests, chi-square tests, ANOVA, regression, correlation) - Matrices, determinants, and systems of linear equations (Gaussian elimination, Cramer's rule, matrix operations, eigenvalues, eigenvectors) - Complex numbers and polar form (arithmetic, De Moivre's theorem, roots of unity, geometric interpretation) - Calculus (limits, continuity, derivatives, integrals, fundamental theorem, applications, Taylor series, power series) - Vectors, vector spaces, and linear transformations (dot product, cross product, span, linear independence, basis, matrix representation) - Sequences, series, and convergence tests (arithmetic, geometric, harmonic, telescoping, ratio test, root test, integral test, alternating series) - Conic sections (ellipses, hyperbolas, parabolas, eccentricity, foci, directrices, polar equations) - Combinatorics (permutations, combinations, binomial theorem, Pascal's triangle, inclusion-exclusion principle, pigeonhole principle) - Introductory proof techniques (direct, contradiction, contraposition, induction, case analysis, counterexamples) - Elementary number theory (divisibility rules, prime factorization, greatest common divisor, least common multiple, Diophantine equations) - Graph theory basics (vertices, edges, degree, paths, cycles, trees, Euler and Hamilton paths/cycles, planar graphs) - Introductory set theory (sets, subsets, unions, intersections, complements, Venn diagrams, cardinality, power sets) - Mathematical logic basics (propositions, truth tables, logical connectives, quantifiers, validity, satisfiability) Deeper (undergraduate level): - Multivariable calculus (partial derivatives, gradients, directional derivatives, multiple integrals, change of variables, vector calculus, Green's, Stokes', and Divergence theorems) - Differential equations (first-order, second-order, higher-order, systems, Laplace transforms, series solutions, numerical methods, phase plane analysis, stability) - Linear algebra (vector spaces, linear independence, bases, dimension, linear transformations, matrices, determinants, eigenvalues, eigenvectors, diagonalization, inner product spaces, orthogonality, least squares, singular value decomposition) - Abstract algebra (groups, subgroups, cosets, Lagrange's theorem, normal subgroups, quotient groups, homomorphisms, isomorphisms, rings, ideals, integral domains, fields, polynomial rings, Galois theory, Sylow theorems, finitely generated abelian groups) - Real analysis (completeness, uniform continuity, differentiation, Riemann integration, sequences and series of functions, uniform convergence, Lebesgue measure, Lebesgue integration, function spaces, Banach spaces, Hilbert spaces, Fourier series, Fourier transforms) - Complex analysis (analytic functions, Cauchy-Riemann equations, harmonic functions, contour integrals, Cauchy's theorem, Cauchy's integral formula, residues, Laurent series, conformal mappings, Riemann mapping theorem, Riemann surfaces) - Number theory (divisibility, congruences, prime numbers, Diophantine equations, quadratic reciprocity, primitive roots, continued fractions, Pell's equation, p-adic numbers, algebraic number theory, analytic number theory, elliptic curves, modular forms) - Topology (metric spaces, topological spaces, continuity, homeomorphisms, compactness, connectedness, separation axioms, product spaces, quotient spaces, Tychonoff's theorem, fundamental group, covering spaces, knots, surfaces, manifolds) - Combinatorics (recurrence relations, generating functions, Ramsey theory, Polya counting, partially ordered sets, lattices, Möbius inversion, Catalan numbers, Young tableaux, symmetric functions, graph enumeration) - Graph theory (connectivity, Eulerian and Hamiltonian paths/cycles, coloring, trees, planarity, network flows, matching, independent sets, dominating sets, Ramsey numbers, random graphs, spectral graph theory, expanders) - Numerical analysis (error analysis, interpolation, approximation theory, numerical integration, finite differences, numerical linear algebra, optimization, iterative methods, finite element methods, wavelets, compressed sensing) - Probability theory (random variables, expectation, conditional probability, independence, central limit theorem, laws of large numbers, martingales, Markov chains, Poisson processes, Brownian motion, stochastic processes, ergodic theory) - Mathematical statistics (point estimation, sufficient statistics, maximum likelihood, Bayesian inference, decision theory, hypothesis testing, confidence intervals, linear models, multivariate analysis, nonparametric statistics, bootstrap, EM algorithm) - Differential geometry (manifolds, tangent spaces, Riemannian metrics, connections, curvature, geodesics, tensors, Lie groups, homogeneous spaces, symmetric spaces, Kähler manifolds, Hodge theory, Chern classes, characteristic classes) - Fourier analysis (Fourier series, Fourier transforms, discrete Fourier transforms, wavelets, signal processing, time-frequency analysis, pseudodifferential operators, microlocal analysis, Gabor analysis, Littlewood-Paley theory) - Partial differential equations (classification, characteristics, separation of variables, Fourier methods, Green's functions, Sobolev spaces, weak solutions, regularity, maximum principles, variational methods, nonlinear PDEs, conservation laws, shock waves) - Optimization (linear programming, convexity, duality, Lagrange multipliers, Karush-Kuhn-Tucker conditions, interior point methods, semidefinite programming, stochastic optimization, optimal control, calculus of variations, Hamilton-Jacobi equations) - Financial mathematics (stochastic calculus, Black-Scholes model, options pricing, risk management, portfolio optimization, interest rate models, credit risk, Monte Carlo methods, machine learning in finance, algorithmic trading) - Dynamical systems (discrete and continuous dynamical systems, fixed points, stability, bifurcations, chaos, attractors, Lyapunov exponents, ergodic theory, topological dynamics, symbolic dynamics, complex dynamics, Hamiltonian systems) - Coding theory and cryptography (error-correcting codes, linear codes, cyclic codes, BCH codes, Reed-Solomon codes, convolutional codes, turbo codes, LDPC codes, public-key cryptography, RSA, elliptic curve cryptography, lattice-based cryptography, quantum cryptography) - Mathematical biology (population dynamics, epidemiology, ecology, genetics, neuroscience, pattern formation, morphogenesis, biomechanics, bioinformatics, systems biology, biochemical networks, evolutionary game theory) Even deeper (graduate level and beyond): - Measure theory (σ-algebras, measures, Lebesgue integration, Radon-Nikodym theorem, Fubini's theorem, Lp spaces, Banach spaces, Hilbert spaces, spectral theory, ergodic theory, probability measures, stochastic processes) - Functional analysis (Banach spaces, Hilbert spaces, bounded linear operators, spectral theory, Hahn-Banach theorem, open mapping theorem, closed graph theorem, Banach algebras, C*-algebras, von Neumann algebras, noncommutative geometry) - Algebraic geometry (affine and projective varieties, schemes, sheaves, cohomology, intersection theory, moduli spaces, abelian varieties, algebraic curves, surfaces, Hodge theory, étale cohomology, p-adic cohomology, motives, Grothendieck's theory) - Algebraic topology (fundamental groups, covering spaces, homology, cohomology, homotopy groups, spectral sequences, K-theory, characteristic classes, cobordism, surgery theory, higher categories, infinity-groupoids, homotopy type theory) - Differential topology (smooth manifolds, transversality, Morse theory, cobordism, foliations, surgery theory, h-cobordism theorem, pseudoisotopy, Cerf theory, Kirby calculus, contact topology, symplectic topology) - Representation theory (linear representations, character theory, Lie groups, Lie algebras, Weyl groups, Schur-Weyl duality, Clifford algebras, spinors, Hecke algebras, quantum groups, crystal bases, categorification, geometric representation theory) - Commutative algebra (Noetherian rings, primary decomposition, regular sequences, depth, Cohen-Macaulay rings, Gorenstein rings, homological dimensions, local cohomology, Hilbert functions, Gröbner bases, toric varieties, Stanley-Reisner theory) - Homological algebra (complexes, homology, cohomology, derived categories, spectral sequences, Ext and Tor functors, Koszul duality, Hochschild homology, cyclic homology, A-infinity algebras, L-infinity algebras, operads, homotopical algebra) - Category theory (monoidal categories, Abelian categories, Yoneda lemma, adjoint functors, Kan extensions, topos theory, Grothendieck topoi, derived categories, triangulated categories, model categories, higher categories, ∞-categories) - K-theory (vector bundles, Chern classes, Adams operations, Bott periodicity, index theorems, motivic homotopy theory, algebraic K-theory, topological K-theory, Waldhausen K-theory, noncommutative motives, K-homology, KK-theory) - Moduli spaces (moduli of curves, abelian varieties, vector bundles, representations, Hilbert schemes, Gromov-Witten invariants, Donaldson-Thomas invariants, quantum cohomology, mirror symmetry, Calabi-Yau manifolds, Bridgeland stability) - Topos theory (Grothendieck topoi, sheaf cohomology, classifying topoi, higher topoi, homotopy type theory, univalent foundations, synthetic differential geometry, synthetic topology, modal homotopy type theory, cohesive homotopy type theory) - Homotopy theory (model categories, simplicial sets, spectra, stable homotopy theory, chromatic homotopy theory, ∞-categories, A-infinity algebras, E-infinity algebras, operads, Goodwillie calculus, Waldhausen K-theory, Lurie's higher topos theory) - Symplectic geometry (Hamiltonian mechanics, Poisson manifolds, moment maps, quantization, Floer homology, mirror symmetry, Fukaya categories, Lagrangian submanifolds, Gromov-Witten theory, Symplectic field theory, contact homology) - Noncommutative geometry (C*-algebras, von Neumann algebras, cyclic cohomology, quantum groups, Hopf algebras, Connes' theory, spectral triples, Dixmier traces, Baum-Connes conjecture, noncommutative tori, quantum tori, quantum spheres) - Arithmetic geometry (elliptic curves, modular forms, Shimura varieties, L-functions, Galois representations, Iwasawa theory, p-adic Hodge theory, Fontaine's rings, Weil conjectures, Langlands program, automorphic forms, Hecke algebras) - Geometric group theory (hyperbolic groups, CAT(0) spaces, automatic groups, amenability, Kazhdan's property (T), Gromov's theory, Tits alternative, growth of groups, Dehn functions, asymptotic cones, boundaries, quasi-isometries) - Harmonic analysis (maximal functions, singular integrals, Littlewood-Paley theory, Bochner-Riesz means, restriction problems, Kakeya conjecture, Fourier restriction theory, oscillatory integrals, Carleson measures, BMO, H^1, Hardy spaces) - Algebraic number theory (number fields, class groups, units, ramification, local fields, adeles, ideles, class field theory, Brauer groups, Tate cohomology, Galois cohomology, étale cohomology, L-functions, Birch and Swinnerton-Dyer conjecture) - Geometric topology (low-dimensional topology, knot theory, 3-manifolds, 4-manifolds, Heegaard Floer homology, contact topology, Seiberg-Witten theory, Donaldson invariants, Khovanov homology, Khovanov-Rozansky homology, Reshetikhin-Turaev invariants) - Quantum topology (quantum invariants, Jones polynomial, Khovanov homology, topological quantum field theories, conformal field theory, vertex operator algebras, quantum groups, Reshetikhin-Turaev invariants, Witten-Reshetikhin-Turaev invariants) - Stochastic analysis (Brownian motion, martingales, Itô calculus, stochastic differential equations, Malliavin calculus, rough paths, stochastic partial differential equations, stochastic control, backward stochastic differential equations, mean field games) - Infinite-dimensional analysis (Banach manifolds, Fréchet spaces, inverse problems, calculus of variations, optimal transport, Monge-Ampère equations, Wasserstein spaces, Γ-convergence, gradient flows, Ricci flow, mean curvature flow) - Descriptive set theory (Borel sets, analytic sets, projective sets, determinacy, large cardinals, forcing, inner model theory, Wadge hierarchy, Martin's axiom, Solovay model, Woodin cardinals, Π^1_1 sets, Σ^1_2 sets) - Computability theory (recursive functions, Turing machines, degrees of unsolvability, constructive mathematics, reverse mathematics, hyperarithmetical hierarchy, Π^1_1-completeness, Π^1_2-completeness, higher recursion theory, α-recursion theory) - Mathematical logic (first-order logic, model theory, proof theory, set theory, independence results, nonstandard analysis, o-minimality, stability theory, simplicity theory, NIP theories, Shelah's classification theory, Hrushovski constructions) - Theoretical computer science (complexity theory, algorithms, data structures, cryptography, machine learning, quantum computing, approximation algorithms, parameterized complexity, descriptive complexity, circuit complexity, communication complexity) - Quantum mathematics (quantum probability, quantum information, quantum algorithms, quantum error correction, quantum cryptography, quantum Shannon theory, quantum Markov chains, quantum groups, quantum topology, quantum field theory) - Tropical geometry (tropical curves, tropical surfaces, tropical varieties, tropical Grassmannians, tropical convexity, tropical linear algebra, tropical combinatorics, tropical Hodge theory, tropical mirror symmetry, connections with toric geometry) - Higher category theory (∞-categories, (∞,1)-categories, (∞,n)-categories, quasi-categories, complete Segal spaces, Segal categories, Rezk categories, model categories, homotopy coherent nerve, Grothendieck construction, Lurie's theory of ∞-topoi) - Derived algebraic geometry (derived schemes, derived stacks, derived categories of sheaves, cotangent complexes, derived intersection theory, derived moduli spaces, derived Azumaya algebras, derived Brauer groups, derived Picard groups) - Geometric representation theory (perverse sheaves, D-modules, microlocal sheaf theory, Hodge modules, Kazhdan-Lusztig theory, geometric Langlands program, Beilinson-Bernstein localization, Kac-Moody algebras, crystal bases) - Topological data analysis (persistent homology, barcodes, Mapper algorithm, Reeb graphs, Morse theory, Morse-Smale complexes, zigzag persistence, multidimensional persistence, topological machine learning, topological signal processing) - Algebraic statistics (toric models, Markov bases, Gröbner bases, contingency tables, graphical models, phylogenetic trees, algebraic exponential families, maximum likelihood estimation, Bayesian networks, causal inference) - Nonlinear dispersive equations (Korteweg-de Vries equation, nonlinear Schrödinger equation, wave maps, Schrödinger maps, Gross-Pitaevskii equation, Hartree equation, Zakharov system, Boussinesq equation, Camassa-Holm equation) - Integrable systems (solitons, inverse scattering transform, Lax pairs, Hamiltonian structures, Painlevé equations, Toda lattices, Calogero-Moser systems, Hitchin systems, Seiberg-Witten theory, cluster algebras, Y-systems) - Algebraic combinatorics (symmetric functions, Schur functions, Macdonald polynomials, Schubert polynomials, Kazhdan-Lusztig polynomials, cluster algebras, quiver representations, Littlewood-Richardson coefficients, Kostka numbers) - Extremal combinatorics (Szemerédi's theorem, Green-Tao theorem, Roth's theorem, Sárközy's theorem, Erdős-Ko-Rado theorem, Sperner's theorem, Turán's theorem, Ramsey theory, Szemerédi regularity lemma, graph limits) - Additive combinatorics (Freiman's theorem, Plünnecke-Ruzsa inequalities, Balog-Szemerédi-Gowers theorem, sum-product estimates, Bourgain-Katz-Tao theorem, Erdős-Volkmann theorem, Bohr sets, Roth's theorem in the primes) - Geometric measure theory (Hausdorff measures, rectifiable sets, Lipschitz maps, area and coarea formulas, currents, varifolds, minimal surfaces, Plateau's problem, isoperimetric inequalities, Brunn-Minkowski theory) - Metric geometry (length spaces, Alexandrov spaces, Gromov-Hausdorff convergence, Gromov hyperbolicity, asymptotic cones, coarse geometry, Lipschitz extensions, Assouad dimension, Nagata dimension, Markov convexity) - Fractal geometry (Hausdorff dimension, box dimension, packing dimension, self-similarity, iterated function systems, Julia sets, Mandelbrot set, Sierpiński triangle, Cantor set, Moran constructions, multifractal analysis) - Random matrix theory (Wigner matrices, Wishart matrices, Tracy-Widom distribution, Airy point process, determinantal point processes, Gaussian unitary ensemble, Gaussian orthogonal ensemble, Gaussian symplectic ensemble, free probability) - Compressed sensing (sparse recovery, basis pursuit, matching pursuit, restricted isometry property, null space property, coherence, matrix completion, phase retrieval, dictionary learning, low-rank matrix recovery) - Topological quantum field theory (Atiyah-Segal axioms, Witten-Reshetikhin-Turaev invariants, Chern-Simons theory, Jones polynomial, Khovanov homology, Heegaard Floer homology, Ozsvath-Szabo invariants, Crane-Yetter invariants) - Quantum algebra (quantum groups, Hopf algebras, Drinfeld doubles, quasitriangular Hopf algebras, ribbon Hopf algebras, Yetter-Drinfeld modules, Nichols algebras, pointed Hopf algebras, quantum cluster algebras) - Noncommutative algebraic geometry (noncommutative projective geometry, noncommutative schemes, noncommutative resolutions, Calabi-Yau algebras, Sklyanin algebras, Artin-Schelter regular algebras, twisted homogeneous coordinate rings) - Quantum chaos (quantum ergodicity, quantum unique ergodicity, Hecke quantum unique ergodicity, quantum graphs, quantum maps, quantum billiards, Gutzwiller trace formula, Selberg trace formula, Weyl law, scarring) - Quantum topology and knot theory (Jones polynomial, HOMFLY-PT polynomial, Kauffman polynomial, Reshetikhin-Turaev invariants, Witten-Reshetikhin-Turaev invariants, Khovanov homology, Khovanov-Rozansky homology, knot Floer homology)