- General relativity: Friedmann equations, cosmic microwave background, gravitational lensing - Fluid dynamics: stellar structure, accretion disks, shock waves - Nuclear physics: nucleosynthesis, stellar evolution, supernovae - Particle physics: dark matter, neutrino astrophysics, cosmic rays - Statistical mechanics: galaxy clustering, large-scale structure, cosmological perturbation theory - Numerical methods: N-body simulations, smoothed particle hydrodynamics (SPH), adaptive mesh refinement (AMR) Mathematical Physics (general tools): - Differential equations: Sturm-Liouville theory, Green's functions, solitons - Functional analysis: Hilbert spaces, Banach spaces, distributions, Sobolev spaces - Operator theory: unbounded operators, spectral theory, C*-algebras - Probability theory: stochastic processes, stochastic differential equations, large deviations - Topology: algebraic topology, differential topology, Morse theory - Differential geometry: symplectic geometry, Poisson geometry, Kähler manifolds - Lie groups and algebras: representation theory, structure theory, Haar measure - Variational principles: calculus of variations, optimal control theory, conservation laws - Integrable systems: inverse scattering transform, Lax pairs, Painlevé transcendents - Nonlinear dynamics: bifurcation theory, pattern formation, synchronization - Numerical analysis: finite element methods, spectral methods, wavelets This expanded map provides a more comprehensive overview of the mathematical tools and concepts used in various fields of modern physics. It includes additional subtopics and highlights the interconnections between different areas of mathematics and physics. However, it's important to note that this map is still not exhaustive, and there are many more specialized mathematical techniques used in specific research areas within each field of physics. Here is a detailed map of the mathematics used in various fields of artificial intelligence: # Mathematics of Artificial Intelligence ## Machine Learning - Linear Algebra - Vectors and Matrices - Eigenvalues and Eigenvectors - Matrix Decomposition (SVD, QR, etc.) - Probability and Statistics - Probability Distributions - Maximum Likelihood Estimation - Gradient Descent - Stochastic Gradient Descent - Convex Optimization - Kullback-Leibler Divergence - Cross-Entropy - Graph Theory (for Graph Neural Networks) ## Deep Learning - All topics from Machine Learning, plus: - Derivatives and Gradients - Chain Rule - Partial Derivatives - Numerical Computation - Numerical Stability - Floating Point Arithmetic - Differential Equations (for Recurrent Neural Networks) ## Computer Vision - Linear Algebra (for Transformations and Projections) - Probability and Statistics (for Image Noise Modeling) - Fourier Analysis (for Frequency Domain Processing) - Differential Geometry (for Manifold Learning) - Topology (for Persistent Homology) ## Natural Language Processing - Probability and Statistics (for Language Modeling) - Information Theory (for Text Compression and Encoding) - Graph Theory (for Knowledge Graphs and Semantic Networks) - Computational Linguistics - Formal Languages and Automata Theory - Parsing and Context-Free Grammars ## Reinforcement Learning - Probability and Statistics (for Markov Decision Processes) - Dynamic Programming - Optimal Control Theory ## Robotics - Linear Algebra (for Kinematics and Dynamics) - Differential Equations (for System Modeling) - Optimization (for Path Planning and Control) - Probability and Statistics (for Localization and Mapping) - Geometry and Trigonometry (for Spatial Reasoning) ## Explainable AI and Interpretability - Causality and Counterfactuals - Information Theory (for Feature Importance and Relevance) - Graph Theory (for Causal Graphs and Bayesian Networks) ## AI Ethics and Fairness - Probability and Statistics (for Bias Detection and Mitigation) - Game Theory (for Fairness in Multi-Agent Systems) - Social Choice Theory (for Aggregating Preferences and Decisions) ## AI Safety and Security - Probability and Statistics (for Risk Assessment and Uncertainty Quantification) - Game Theory (for Adversarial Reasoning and Robustness) - Cryptography and Information Security This map covers the main mathematical areas used in various AI fields, but it is not exhaustive. Many AI applications draw from multiple mathematical disciplines, and the boundaries between fields can be fluid. Additionally, as AI research progresses, new mathematical tools and techniques may be developed or adapted from other areas of mathematics. Certainly! Here's an expanded and more detailed map of the mathematics used in artificial intelligence: # Mathematics of Artificial Intelligence ## Machine Learning - Linear Algebra - Vectors and Matrices - Vector Spaces and Subspaces - Linear Independence and Basis - Eigenvalues and Eigenvectors - Matrix Decomposition (SVD, QR, LU, Cholesky) - Tensor Algebra - Probability and Statistics - Probability Distributions (Discrete and Continuous) - Joint, Marginal, and Conditional Probabilities - Bayes' Theorem and Bayesian Inference - Hypothesis Testing and p-values - Maximum Likelihood Estimation - Expectation, Variance, and Covariance - Monte Carlo Methods - Markov Chains and Hidden Markov Models - Gaussian Processes - Gradient Descent and its Variants (Batch, Stochastic, Mini-Batch) - Momentum and Accelerated Methods (Nesterov, AdaGrad, RMSprop, Adam) - Convex Optimization - Constrained Optimization (Lagrange Multipliers, KKT Conditions) - Combinatorial Optimization - Evolutionary Algorithms and Genetic Algorithms - Entropy (Shannon, Rényi, Tsallis) - Kullback-Leibler Divergence and Jensen-Shannon Divergence - Cross-Entropy and Perplexity - Channel Coding and Source Coding - Graph Representation (Adjacency Matrix, Adjacency List) - Graph Traversal (BFS, DFS) - Shortest Paths (Dijkstra, Bellman-Ford, Floyd-Warshall) - Minimum Spanning Trees (Kruskal, Prim) - Network Flow (Ford-Fulkerson, Push-Relabel) - Spectral Graph Theory - Big O Notation and Asymptotic Analysis - P vs. NP and NP-Completeness - Approximation Algorithms - Randomized Algorithms ## Deep Learning - All topics from Machine Learning, plus: - Limits and Continuity - Derivatives and Gradients - Partial Derivatives and Gradient Vector - Directional Derivatives - Chain Rule and Backpropagation - Higher-Order Derivatives (Hessian, Jacobian) - Vector Calculus (Divergence, Curl, Laplacian) - Integral Calculus - Numerical Computation - Numerical Stability and Conditioning - Floating Point Arithmetic and Precision - Numerical Differentiation and Integration - Numerical Optimization - Numerical Linear Algebra - Ordinary Differential Equations (ODEs) - Partial Differential Equations (PDEs) - Stability Analysis and Lyapunov Theory - Dynamical Systems and Bifurcation Theory - Topology - Metric Spaces and Normed Spaces - Continuity and Homeomorphisms - Topological Spaces and Separation Axioms - Compactness and Connectedness - Manifolds and Differential Topology ## Computer Vision - All relevant topics from Machine Learning and Deep Learning, plus: - Fourier Analysis - Fourier Series and Transforms - Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT) - Short-Time Fourier Transform (STFT) - Wavelets and Multiresolution Analysis - Curves and Surfaces - Geodesics and Exponential Map - Curvature and Second Fundamental Form - Lie Groups and Lie Algebras - Topology - Homology and Cohomology - Projective Geometry - Homogeneous Coordinates - Projective Transformations - Camera Models and Calibration - Computational Geometry - Convex Hull - Voronoi Diagrams and Delaunay Triangulations - Arrangements and Duality - Geometric Data Structures (kd-trees, Range Trees, BSP Trees) ## Natural Language Processing - All relevant topics from Machine Learning and Deep Learning, plus: - Computational Linguistics - Formal Languages and Automata Theory - Regular Languages and Finite Automata - Context-Free Languages and Pushdown Automata - Context-Sensitive Languages and Linear Bounded Automata - Recursively Enumerable Languages and Turing Machines - Parsing and Context-Free Grammars - Constituency Parsing and Phrase Structure Grammars - Dependency Parsing and Dependency Grammars - Probabilistic Parsing and Probabilistic Context-Free Grammars - Lexical Semantics and Word Sense Disambiguation - Discourse Analysis and Rhetorical Structure Theory - Information Retrieval - Vector Space Model and TF-IDF - Latent Semantic Analysis (LSA) and Indexing (LSI) - Probabilistic Models (BM25, Language Models) - Evaluation Metrics (Precision, Recall, F1-score, MAP, NDCG) - Text Mining and Information Extraction - Named Entity Recognition (NER) - Relation Extraction - Event Extraction - Coreference Resolution - Sentiment Analysis and Opinion Mining ## Reinforcement Learning - All relevant topics from Machine Learning and Deep Learning, plus: - Markov Decision Processes (MDPs) - States, Actions, Transitions, Rewards - Policies and Value Functions - Bellman Equations and Dynamic Programming - Policy Iteration and Value Iteration - Temporal Difference Learning - Q-Learning and SARSA - Eligibility Traces and TD(λ) - Actor-Critic Methods - Monte Carlo Methods - Monte Carlo Estimation - Monte Carlo Control - Off-Policy and On-Policy Methods - Normal Form and Extensive Form Games - Nash Equilibrium and Correlated Equilibrium - Stackelberg Games and Mechanism Design - Evolutionary Game Theory - Optimal Control Theory - Pontryagin's Maximum Principle - Hamilton-Jacobi-Bellman Equation - Linear-Quadratic Regulator (LQR) - Model Predictive Control (MPC) - Multi-Agent Systems - Coordination and Cooperation - Communication and Signaling - Distributed Optimization and Consensus - Swarm Intelligence and Emergent Behavior ## Robotics - All relevant topics from Machine Learning, Deep Learning, and Reinforcement Learning, plus: - Kinematics and Dynamics - Forward and Inverse Kinematics - Jacobians and Manipulator Equations - Lagrangian and Hamiltonian Mechanics - Rigid Body Dynamics - Contact Dynamics and Friction Models - PID Control - Feedback Linearization - Adaptive Control - Robust Control (H-infinity, Sliding Mode) - Optimal Control (LQR, MPC) - Estimation and Filtering - Kalman Filter and Extended Kalman Filter - Particle Filter and Unscented Kalman Filter - Simultaneous Localization and Mapping (SLAM) - Structure from Motion (SfM) - Motion Planning - Configuration Space and Obstacle Avoidance - Sampling-Based Planning (RRT, PRM) - Graph-Based Planning (A*, D*, Dijkstra) - Optimization-Based Planning (CHOMP, TrajOpt) - Topological Planning and Homotopy Classes - Grasping and Manipulation - Force Closure and Form Closure - Grasp Stability and Grasp Metrics - Dexterous Manipulation and In-Hand Manipulation - Nonprehensile Manipulation and Pushing ## Explainable AI and Interpretability - All relevant topics from Machine Learning and Deep Learning, plus: - Causal Inference - Causal Graphs and Bayesian Networks - Interventions and Counterfactuals - Causal Effect Estimation (Propensity Scores, Instrumental Variables) - Causal Discovery and Structure Learning - Interpretable Models - Decision Trees and Rule-Based Models - Linear Models and Generalized Additive Models - Sparsity and Regularization (Lasso, Ridge, Elastic Net) - Monotonic Constraints and Shape Constraints - Feature Importance and Attribution - Permutation Importance and Drop-Column Importance - Shapley Values and SHAP (SHapley Additive exPlanations) - Integrated Gradients and Layer-Wise Relevance Propagation (LRP) - Saliency Maps and Class Activation Mapping (CAM) - Counterfactual Explanations - Nearest Counterfactual Explanations - Contrastive Explanations and Pertinent Negatives - Counterfactual Feature Importance - Uncertainty Quantification - Bayesian Inference and Posterior Distributions - Confidence Intervals and Credible Intervals - Calibration and Reliability Diagrams - Ensemble Methods and Bayesian Model Averaging ## AI Ethics and Fairness - All relevant topics from Machine Learning, Deep Learning, and Explainable AI, plus: - Fairness Metrics - Demographic Parity and Equalized Odds - Equal Opportunity and Predictive Parity - Individual Fairness and Group Fairness - Counterfactual Fairness and Path-Specific Effects - Bias Detection and Mitigation - Statistical Parity Difference and Disparate Impact - Equalized Odds Difference and Disparate Mistreatment - Bias Amplification and Bias Propagation - Debiasing Techniques (Reweighting, Adversarial Debiasing, Calibration) - Privacy and Security - Differential Privacy and Randomized Response - Secure Multiparty Computation and Homomorphic Encryption - Adversarial Examples and Robustness - Model Inversion and Membership Inference Attacks - Accountability and Transparency - Model Cards and Datasheets - Algorithmic Impact Assessments - Audit Trails and Provenance Tracking - Participatory Design and Stakeholder Engagement - Social Choice Theory - Voting Rules and Aggregation Methods - Arrow's Impossibility Theorem and Condorcet Paradox - Ranking and Scoring Rules - Fairness Criteria (Anonymity, Neutrality, Monotonicity) ## AI Safety and Security - All relevant topics from Machine Learning, Deep Learning, and AI Ethics, plus: - Robustness and Adversarial Attacks - Adversarial Examples and Perturbations - Poisoning Attacks and Backdoor Attacks - Robustness Metrics (Adversarial Accuracy, Empirical Robustness) - Adversarial Training and Robust Optimization - Formal Verification and Program Synthesis - Satisfiability Modulo Theories (SMT) and Constraint Solving - Model Checking and Temporal Logic - Inductive Program Synthesis and Automated Reasoning - Verified AI Systems and Certified Algorithms - AI Alignment and Value Learning - Inverse Reinforcement Learning and Preference Learning - Cooperative Inverse Reinforcement Learning and Assistance Games - Value Iteration Networks and Reward Modeling - Corrigibility and Interruptibility - AI Governance and Policy - AI Regulation and Standardization - Liability and Accountability Frameworks - Intellectual Property and Patent Law - International Cooperation and Global Coordination This expanded map provides a more comprehensive overview of the mathematical foundations and techniques used across various subfields of artificial intelligence. It covers additional topics in each area and delves deeper into the specific mathematical tools and theories employed. However, it's important to note that even this expanded map is not exhaustive, as the field of AI is vast and rapidly evolving. New mathematical approaches and interdisciplinary connections are continually being developed and applied to advance the state of the art in AI research and applications. Certainly! Here's an even more comprehensive map of the mathematics used in artificial intelligence, with additional theoretical concepts: # Mathematics of Artificial Intelligence ## Machine Learning - Linear Algebra - ... - Functional Analysis and Hilbert Spaces - Spectral Theory and Singular Value Decomposition (SVD) - Matrix Calculus and Matrix Differentials - Probability and Statistics - ... - Concentration Inequalities (Hoeffding, Chernoff, Bernstein) - Extreme Value Theory and Heavy-Tailed Distributions - Copulas and Multivariate Dependence Modeling - Stochastic Processes and Martingales - Empirical Process Theory and Uniform Convergence - ... - Submodular Optimization and Greedy Algorithms - Stochastic Optimization and Sample Average Approximation - Distributed Optimization and Consensus Algorithms - Multiobjective Optimization and Pareto Optimality - Monotone Operator Theory and Variational Inequalities - ... - Rényi Entropy and f-Divergences - Information Bottleneck and Information Geometric Methods - Rate-Distortion Theory and Lossy Compression - Network Information Theory and Coding for Distributed Systems - ... - Random Graphs and Percolation Theory - Expander Graphs and Spectral Graph Theory - Graph Embedding and Representation Learning - Higher-Order Interactions and Hypergraphs - ... - Parameterized Complexity and Fixed-Parameter Tractability - Hardness of Approximation and Inapproximability Results - Communication Complexity and Information Complexity - Circuit Complexity and Boolean Function Analysis ## Deep Learning - ... - Variational Calculus and Euler-Lagrange Equations - Calculus of Variations and Optimal Control Theory - Stochastic Calculus and Itô Calculus - Numerical Computation - ... - Randomized Numerical Linear Algebra - Stochastic Gradient Methods and Variance Reduction Techniques - Automatic Differentiation and Differentiable Programming - ... - Stochastic Differential Equations (SDEs) and Fokker-Planck Equation - Delay Differential Equations and Neutral Functional Differential Equations - Infinite-Dimensional Dynamical Systems and Operator Semigroups - Topology - ... - Algebraic Topology and Homotopy Theory - Differential Topology and Morse Theory - Persistent Homology and Topological Data Analysis ## Computer Vision - ... - Spherical Harmonics and Zonal Kernels - Gabor Filters and Wavelet Transforms - Scattering Transform and Invariant Representations - Integral Geometry - Radon Transform and Hough Transform - Geometric Probability and Integral Geometry Descriptors - Tensor Voting and Perceptual Organization - Algebraic Curves and Surfaces - Polynomial Optimization and Sum-of-Squares Programming - Semidefinite Programming and Convex Algebraic Geometry ## Natural Language Processing - ... - Category Theory and Categorical Grammars - Monoidal Categories and Pregroup Grammars - Lambek Calculus and Type-Logical Grammars - Categorical Compositional Distributional Semantics - Algebraic Topology and Topological Text Analysis - Persistent Homology for Text Representation - Topological Signatures and Invariants for NLP - Morse Theory for Document Summarization - Quantum Information Theory and Quantum NLP - Quantum Language Models and Quantum Embeddings - Quantum Entanglement and Quantum Contextuality in NLP - Quantum-Inspired Algorithms for NLP Tasks ## Reinforcement Learning - ... - Stochastic Games and Markov Games - Nash Q-Learning and Correlated Q-Learning - Extensive-Form Games and Counterfactual Regret Minimization - Mean Field Games and Mean Field Control - Robust Control and H-infinity Control - Minimax Optimal Control and Worst-Case Analysis - Disturbance Rejection and Attenuation - Robust Markov Decision Processes and Robust Dynamic Programming - Stochastic Optimal Control - Stochastic Maximum Principle and Backward Stochastic Differential Equations (BSDEs) - Risk-Sensitive Control and Utility Theory - Stochastic Model Predictive Control and Scenario Optimization ## Robotics - ... - Riemannian Geometry and Geometric Mechanics - Lie Groups and Lie Algebras for Robot Kinematics - Riemannian Metrics and Geodesics for Robot Motion - Symplectic Geometry and Hamiltonian Mechanics - Topological Robotics and Motion Planning - Configuration Spaces and Obstacle Avoidance - Homology and Cohomology for Topological Path Planning - Morse Theory and Morse-Smale Complexes for Navigation - Contact Mechanics and Tribology - Hertzian Contact and Elasticity Theory - Friction Models and Stick-Slip Phenomena - Lubricant Dynamics and Hydrodynamic Lubrication ## Explainable AI and Interpretability - ...