Structure of reality [x.com](https://twitter.com/burny_tech/status/1772487227592884402) 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