## Tags - Part of: [[Mathematics]], [[Natural science]] - Related: - Includes: [[Physics]] - Additional: ## Main resources - [Applied mathematics - Wikipedia](https://en.wikipedia.org/wiki/Applied_mathematics#:~:text=Applied%20mathematics%20is%20the%20application,mathematical%20science%20and%20specialized%20knowledge.) <iframe src="https://en.wikipedia.org/wiki/Applied_mathematics" allow="fullscreen" allowfullscreen="" style="height:100%;width:100%; aspect-ratio: 16 / 5; "></iframe> ## Landscapes written by AI (may include factually incorrect information) **Gigantic Map of Applied Mathematics** Applied Mathematics is a vast and dynamic field that uses mathematical methods and models to solve real-world problems in science, engineering, business, and other areas. Below is a comprehensive map of the various branches and subfields within applied mathematics. --- ### 1. **Differential Equations** - **Ordinary Differential Equations (ODEs)** - Linear ODEs - Nonlinear ODEs - Systems of ODEs - Stability Theory - Perturbation Methods - Boundary Value Problems - **Partial Differential Equations (PDEs)** - Elliptic PDEs - Parabolic PDEs - Hyperbolic PDEs - Analytical Methods - Separation of Variables - Transform Methods - Green's Functions - Numerical Methods - Finite Difference Methods - Finite Element Methods - Spectral Methods - **Dynamical Systems** - Chaos Theory - Bifurcation Theory - Nonlinear Dynamics - Discrete Dynamical Systems ### 2. **Numerical Analysis** - **Numerical Linear Algebra** - Solving Linear Systems - Eigenvalue Problems - Singular Value Decomposition - Iterative Methods - **Numerical Methods for Differential Equations** - ODE Solvers (Euler, Runge-Kutta Methods) - PDE Solvers (FDM, FEM, FVM) - **Optimization** - Linear Programming - Nonlinear Optimization - Convex Optimization - Integer Programming - Stochastic Optimization - **Approximation Theory** - Interpolation - Least Squares Approximation - Orthogonal Polynomials - **Error Analysis and Stability** ### 3. **Computational Mathematics** - **Algorithm Design and Analysis** - Computational Complexity - Parallel Computing - High-Performance Computing - **Scientific Computing** - Simulation and Modeling - Computational Fluid Dynamics (CFD) - Computational Electromagnetics - **Computational Geometry** - Mesh Generation - Computational Topology - Geometric Algorithms - **Symbolic Computation** - Computer Algebra Systems - Automatic Differentiation ### 4. **Mathematical Physics** - **Classical Mechanics** - Newtonian Mechanics - Lagrangian and Hamiltonian Mechanics - **Quantum Mechanics** - Schrödinger Equation - Quantum Field Theory - Quantum Statistical Mechanics - **Statistical Mechanics** - Thermodynamics - Phase Transitions - Critical Phenomena - **Relativity** - Special Relativity - General Relativity - **Mathematical Methods** - Special Functions - Integral Transforms (Fourier, Laplace) - Green's Functions ### 5. **Probability and Statistics** - **Probability Theory** - Random Variables and Distributions - Law of Large Numbers - Central Limit Theorem - Stochastic Processes - Markov Chains - Poisson Processes - Martingales - **Statistical Inference** - Estimation Theory - Hypothesis Testing - Bayesian Methods - **Regression and Time Series Analysis** - Linear and Nonlinear Regression - ARIMA Models - Spectral Analysis - **Multivariate Statistics** - Principal Component Analysis - Factor Analysis - Cluster Analysis - **Statistical Computing** - Monte Carlo Methods - Bootstrap and Resampling Techniques ### 6. **Discrete Mathematics** - **Graph Theory** - Network Flows - Graph Algorithms - Connectivity and Traversability - **Combinatorics** - Enumerative Combinatorics - Combinatorial Designs - Ramsey Theory - **Cryptography** - Symmetric and Asymmetric Encryption - Cryptographic Protocols - Hash Functions - **Discrete Optimization** - Scheduling Algorithms - Knapsack Problems - Travelling Salesman Problem - **Game Theory** - Cooperative and Non-Cooperative Games - Nash Equilibrium - Mechanism Design ### 7. **Mathematical Biology and Bioinformatics** - **Population Dynamics** - Predator-Prey Models - Age-Structured Models - **Epidemiology** - SIR Models - Disease Spread Simulation - **Systems Biology** - Gene Regulatory Networks - Metabolic Pathway Analysis - **Bioinformatics** - Sequence Alignment - Phylogenetic Trees - Genomic Data Analysis - **Computational Neuroscience** - Neural Modeling - Brain Connectivity ### 8. **Financial Mathematics** - **Mathematical Finance** - Option Pricing (Black-Scholes Model) - Interest Rate Models - Portfolio Optimization - **Risk Management** - Value at Risk (VaR) - Credit Risk Models - **Actuarial Science** - Life Contingencies - Pension Fund Mathematics - Insurance Risk Models ### 9. **Operations Research** - **Linear and Nonlinear Programming** - **Integer and Combinatorial Optimization** - **Network Models** - Shortest Path Problems - Max Flow-Min Cut Theorems - **Queueing Theory** - Service Systems - Inventory Models - **Simulation Modeling** - **Decision Analysis** - Multi-Criteria Decision Making - Utility Theory ### 10. **Control Theory** - **Linear Control Systems** - State-Space Representation - Transfer Functions - **Nonlinear Control Systems** - Feedback Linearization - Lyapunov Stability - **Optimal Control** - Pontryagin's Maximum Principle - Dynamic Programming - **Robust and Adaptive Control** - H-infinity Methods - Model Predictive Control ### 11. **Signal Processing** - **Digital Signal Processing** - Filtering Techniques - Fast Fourier Transform (FFT) - **Time-Frequency Analysis** - Wavelet Transforms - Short-Time Fourier Transform - **Image Processing** - Image Enhancement - Computer Vision - **Speech Processing** - Speech Recognition - Signal Compression ### 12. **Information Theory** - **Entropy and Mutual Information** - **Data Compression** - Lossless and Lossy Compression - **Error-Correcting Codes** - Linear Codes - Cyclic Codes - **Channel Capacity** - Shannon's Theorems - Noisy Channel Coding ### 13. **Fluid Dynamics** - **Navier-Stokes Equations** - Incompressible Flow - Turbulence Modeling - **Aerodynamics** - Lift and Drag Calculations - Boundary Layer Theory - **Computational Fluid Dynamics (CFD)** - Mesh Generation - Solver Algorithms - **Hydrodynamics** - Wave Motion - Ship Hull Design ### 14. **Solid Mechanics** - **Elasticity** - Stress-Strain Relationships - Elastic Deformations - **Plasticity** - Yield Criteria - Hardening Models - **Fracture Mechanics** - Crack Propagation - Fatigue Analysis - **Finite Element Analysis** - Structural Analysis - Modal Analysis ### 15. **Materials Science** - **Crystallography** - Lattice Structures - Defect Analysis - **Phase Transformations** - Solidification - Diffusion Processes - **Materials Modeling** - Molecular Dynamics - Continuum Mechanics ### 16. **Applied Algebra and Geometry** - **Coding Theory** - Error Detection and Correction - Cryptographic Codes - **Cryptography** - Elliptic Curve Cryptography - RSA Algorithm - **Computational Algebraic Geometry** - Gröbner Bases - Algebraic Varieties ### 17. **Functional Analysis** - **Hilbert and Banach Spaces** - Inner Product Spaces - Normed Vector Spaces - **Operator Theory** - Spectral Theory - Compact Operators - **Applications** - Quantum Mechanics - Signal Processing ### 18. **Mathematical Modeling** - **Physical Systems Modeling** - Mechanical Systems - Electrical Circuits - **Biological Modeling** - Cellular Processes - Ecosystem Dynamics - **Environmental Modeling** - Climate Models - Pollution Dispersion - **Social Systems Modeling** - Economic Models - Agent-Based Models ### 19. **Optimization** - **Convex Analysis** - Convex Sets and Functions - Duality Theory - **Nonlinear Programming** - Karush-Kuhn-Tucker Conditions - Barrier and Penalty Methods - **Global Optimization** - Branch and Bound Methods - Evolutionary Algorithms - **Heuristic Methods** - Genetic Algorithms - Simulated Annealing - Particle Swarm Optimization ### 20. **Machine Learning and Data Science** - **Statistical Learning Theory** - **Supervised and Unsupervised Learning** - Regression and Classification - Clustering Algorithms - **Neural Networks and Deep Learning** - Convolutional Neural Networks - Recurrent Neural Networks - **Reinforcement Learning** - Q-Learning - Policy Gradients - **Natural Language Processing** - Text Mining - Sentiment Analysis - **Data Mining** - Association Rules - Anomaly Detection ### 21. **Nonlinear Analysis** - **Fixed Point Theorems** - Banach Fixed Point Theorem - Brouwer Fixed Point Theorem - **Variational Methods** - Calculus of Variations - Critical Point Theory - **Topological Methods** - Degree Theory - Morse Theory ### 22. **Quantum Computing** - **Quantum Algorithms** - Shor's Algorithm - Grover's Algorithm - **Quantum Information Theory** - Qubits and Quantum Gates - Entanglement - **Quantum Cryptography** - Quantum Key Distribution - Post-Quantum Cryptography ### 23. **Mathematical Ecology** - **Ecosystem Modeling** - Food Web Dynamics - Resource Management - **Population Genetics** - Gene Frequency Models - Evolutionary Game Theory ### 24. **Network Science** - **Complex Networks** - Small-World Networks - Scale-Free Networks - **Network Dynamics** - Epidemic Spreading - Information Diffusion - **Social Network Analysis** - Community Detection - Centrality Measures ### 25. **Applied Complex Analysis** - **Conformal Mapping** - Schwarz-Christoffel Transformation - Applications in Aerodynamics - **Complex Dynamics** - Julia and Mandelbrot Sets - Holomorphic Dynamics - **Applications** - Fluid Mechanics - Electrostatics ### 26. **Mathematical Geosciences** - **Seismic Modeling** - Wave Propagation - Inverse Problems - **Climate Modeling** - Atmospheric Dynamics - Ocean Modeling - **Geophysical Fluid Dynamics** - Earth's Rotation Effects - Geostrophic Flows ### 27. **Signal and Image Processing** - **Medical Imaging** - MRI and CT Reconstruction - Tomography - **Remote Sensing** - Satellite Image Analysis - Geographic Information Systems (GIS) ### 28. **Artificial Intelligence and Robotics** - **Robotics Kinematics and Dynamics** - **Path Planning Algorithms** - **Machine Vision** - Object Recognition - Scene Reconstruction ### 29. **Economic Mathematics** - **Econometrics** - Time Series Econometrics - Panel Data Analysis - **Game Theory in Economics** - Market Equilibria - Auction Theory - **Financial Engineering** - Derivatives Pricing - Risk Hedging Strategies ### 30. **Environmental Mathematics** - **Pollution Modeling** - Air and Water Quality Models - **Renewable Energy Modeling** - Solar and Wind Energy Forecasting - **Sustainability Metrics** - Carbon Footprint Analysis - Resource Optimization ### 31. **Transportation and Logistics** - **Traffic Flow Theory** - Macroscopic and Microscopic Models - **Routing Problems** - Vehicle Routing Problem (VRP) - Traveling Salesman Problem (TSP) - **Supply Chain Optimization** - Inventory Management - Distribution Networks ### 32. **Mathematical Education** - **Mathematical Pedagogy** - Curriculum Development - Educational Technology - **Assessment and Evaluation** - Standardized Testing - Performance Metrics ### 33. **Applied Number Theory** - **Cryptanalysis** - Prime Number Theorems - Integer Factorization - **Algorithmic Number Theory** - Modular Arithmetic - Diophantine Equations ### 34. **Data Analytics** - **Big Data Techniques** - Distributed Computing (Hadoop, Spark) - **Statistical Analysis of Large Data Sets** - **Data Visualization** - Infographics - Interactive Dashboards ### 35. **Mathematical Sociology** - **Social Modeling** - Opinion Dynamics - Cultural Diffusion - **Agent-Based Modeling** - Simulation of Social Systems ### 36. **Mathematical Chemistry** - **Molecular Modeling** - Quantum Chemistry Calculations - Molecular Dynamics Simulations - **Reaction Kinetics** - Rate Equations - Catalysis Modeling ### 37. **Medical Mathematics** - **Biomedical Signal Processing** - ECG and EEG Analysis - **Physiological Modeling** - Cardiovascular System Models - Pharmacokinetics ### 38. **Decision Sciences** - **Risk Analysis** - Probabilistic Risk Assessment - **Behavioral Decision Theory** - Cognitive Biases - Decision Under Uncertainty ### 39. **Topological Data Analysis** - **Persistent Homology** - **Shape Recognition** - **Data Simplification Techniques** ### 40. **Fractals and Multifractals** - **Fractal Geometry** - Hausdorff Dimension - Self-Similarity - **Applications** - Image Compression - Modeling Natural Phenomena ### 41. **Mathematical Logic and Foundations** - **Set Theory** - **Model Theory** - **Proof Theory** - Automated Theorem Proving - Formal Verification ### 42. **Computational Biology** - **Proteomics** - Protein Structure Prediction - **Systems Biology** - Cellular Automata Models ### 43. **Applied Topology** - **Topological Insulators** - **Robotics** - Configuration Spaces ### 44. **Energy Mathematics** - **Power Systems Analysis** - Load Flow Studies - Stability Analysis - **Renewable Energy Modeling** ### 45. **Meteorology and Oceanography** - **Weather Prediction Models** - **Ocean Circulation Models** - **Climate Change Projections** ### 46. **Mathematical Psychology** - **Cognitive Modeling** - **Psychometric Analysis** ### 47. **Computational Economics** - **Agent-Based Computational Economics** - **Computational General Equilibrium Models** ### 48. **Mathematical Linguistics** - **Statistical Natural Language Processing** - **Computational Semantics** ### 49. **Mathematical Criminology** - **Crime Modeling** - Hotspot Analysis - Predictive Policing ### 50. **Astrodynamics** - **Orbital Mechanics** - Trajectory Optimization - **Space Mission Design** --- This map captures the breadth and depth of applied mathematics, highlighting its interdisciplinary nature and its crucial role in solving complex problems across various domains. Each field is interconnected, often relying on methods and theories from others, demonstrating the integrated fabric of applied mathematics.