Mathematics

The language and mindset that separates mathematical thinkers from everyone else — formal logic, proof techniques, algebraic fluency, trigonometric reasoning, and the set-theoretic foundations that underpin every branch of modern mathematics. These skills haven't changed since Euclid because the rules of logical reasoning are timeless.

• Algebra: The Language of Mathematics(30 concepts)

  • Polynomial Arithmetic & Long Division
  • Algebraic Expressions & Simplification
  • What Is a Function?
  • Exponential Functions & Growth
  • The Imaginary Unit & Complex Arithmetic
  • Arithmetic Sequences & Series
  • Factoring Polynomials
  • Linear Equations & Systems
  • Function Notation & Evaluation
  • The Number e & Natural Exponentials
  • The Complex Plane & Geometric Interpretation
  • Geometric Sequences & Series
  • Composition of Functions
  • Graphs of Polynomial Functions
  • Quadratic Equations
  • Logarithmic Functions
  • Polar Form of Complex Numbers
  • Sigma Notation & Summation Techniques
  • Inverse Functions
  • Inequalities & Absolute Value
  • Rational Expressions & Equations
  • Logarithm Properties & Equations
  • Euler's Formula
  • Recursive Sequences & the Fibonacci Numbers
  • Transformations of Functions
  • From Words to Equations: Algebraic Modeling
  • Graphs of Rational Functions: Asymptotes & Holes
  • Exponential & Logarithmic Modeling
  • Roots of Unity & De Moivre's Theorem
  • Algebraic Modeling: From Pattern to Prediction
• Set Theory & Logic(25 concepts)

  • Sets, Elements & Membership
  • Binary Relations
  • Comparing Set Sizes with Bijections
  • Propositional Logic: Syntax & Semantics
  • The Peano Axioms & Natural Numbers
  • Set Operations: Union, Intersection & Complement
  • Properties of Relations: Reflexivity, Symmetry, Transitivity
  • Countable Sets & Hilbert's Hotel
  • Tautology, Satisfiability & Validity
  • Constructing the Integers
  • Subsets & Power Sets
  • Equivalence Relations & Equivalence Classes
  • Cantor's Diagonal Argument
  • Natural Deduction: Rules of Inference
  • Constructing the Rationals
  • Cartesian Products & Ordered Pairs
  • Partial Orders & Hasse Diagrams
  • Cantor's Theorem & the Hierarchy of Infinities
  • First-Order Predicate Logic
  • The Real Numbers via Dedekind Cuts
  • Set Identities & Set-Theoretic Proofs
  • Functions as Relations: Injection, Surjection & Bijection
  • The Continuum Hypothesis
  • Soundness & Completeness
  • Why Foundations Matter: From Paradox to Axioms
• Trigonometry & Analytic Geometry(30 concepts)

  • Angles, Arcs & Radian Measure
  • The Pythagorean Identity & Fundamental Identities
  • Vectors: Magnitude & Direction
  • Distance, Midpoint & Slope
  • Parabolas: Focus & Directrix
  • Polar Coordinates: Distance & Angle
  • The Unit Circle
  • Sum & Difference Formulas
  • Vector Addition & Scalar Multiplication
  • Equations of Lines & Circles
  • Ellipses: Orbits & Eccentricity
  • Graphing Polar Equations
  • The Six Trigonometric Functions
  • Double-Angle & Half-Angle Formulas
  • The Dot Product
  • Geometric Transformations as Coordinate Mappings
  • Hyperbolas: Asymptotes & Applications
  • Parametric Equations & Curves
  • Graphing Sinusoidal Functions
  • Verifying Trigonometric Identities
  • Vector Projections & Decomposition
  • Composition of Transformations
  • Classifying Conics from the General Equation
  • Eliminating the Parameter
  • Inverse Trigonometric Functions
  • Solving Trigonometric Equations
  • Applications: Force, Navigation & Work
  • Coordinate Proofs in Geometry
  • Conic Sections in Science & Engineering
  • Applications of Polar & Parametric Curves
• Mathematical Thinking & Proof(30 concepts)

  • Mathematical Statements & Propositions
  • Anatomy of a Proof
  • Proof by Contradiction
  • Weak (Ordinary) Induction
  • Polya's Four Steps
  • Mathematical Notation Fluency
  • Predicates & Open Sentences
  • Direct Proof
  • Existence & Uniqueness Proofs
  • Induction in Action: Sums, Inequalities & Divisibility
  • Conjecture & Test
  • Proof Writing Style & Conventions
  • Logical Connectives & Truth Tables
  • Proof by Contrapositive
  • Proof by Cases
  • Strong Induction
  • Working Backwards
  • Translating Between Formal & Informal Mathematics
  • Universal & Existential Quantifiers
  • Biconditional Statements & If-and-Only-If Proofs
  • Without Loss of Generality (WLOG)
  • Structural Induction
  • Reasoning by Analogy
  • Critiquing & Debugging Proofs
  • Logical Equivalence & Complex Negation
  • Common Proof Errors & Logical Fallacies
  • Combining Proof Techniques
  • The Well-Ordering Principle
  • Getting Unstuck: Persistence & Strategy Switching
  • Writing Your First Complete Proofs

The mathematics of change, motion, and the infinite — limits, derivatives, integrals, differential equations, and the rigorous foundations of analysis. Every time your phone predicts your ETA, every time a neural network adjusts its weights, every time an engineer models a bridge under stress, calculus is the language being spoken. This pillar takes you from computing derivatives to proving why calculus works.

• Differential Equations(30 concepts)

  • Separable Equations
  • Homogeneous Equations and the Characteristic Equation
  • Introduction to Systems and Matrix Formulation
  • Definition and Basic Properties of the Laplace Transform
  • Equilibrium Points and Their Classification
  • Population Dynamics: Logistic and Lotka-Volterra Models
  • First-Order Linear Equations and Integrating Factors
  • Superposition and the Structure of Solutions
  • The Eigenvalue Method for Linear Systems
  • Solving IVPs with the Laplace Transform
  • Linearization Near Equilibria
  • Electrical Circuits: RLC Equations
  • Exact Equations
  • Complex Roots and Oscillatory Solutions
  • Phase Portraits for 2D Linear Systems
  • Step Functions and the Shifting Theorems
  • Lyapunov Stability and Direct Method
  • Vibrations, Forcing, and Resonance
  • Existence and Uniqueness of Solutions
  • Method of Undetermined Coefficients
  • Repeated and Defective Eigenvalues
  • The Dirac Delta Function and Impulse Response
  • Bifurcation Theory
  • Introduction to the Heat Equation
  • Slope Fields and Qualitative Analysis
  • Variation of Parameters
  • The Matrix Exponential
  • The Convolution Theorem
  • Limit Cycles and Introduction to Chaos
  • Introduction to the Wave Equation
• Real Analysis(30 concepts)

  • The Ordered Field Axioms
  • Convergence of Sequences: The Epsilon-N Definition
  • Open and Closed Sets
  • Epsilon-Delta Continuity and Sequential Continuity
  • The Derivative: Definition and Basic Properties
  • Darboux Sums and the Definition of the Integral
  • The Completeness Axiom and Least Upper Bounds
  • Monotone Convergence and Bounded Sequences
  • Compactness
  • Uniform Continuity
  • Rolle's Theorem and the Mean Value Theorem (Proofs)
  • Integrability Criteria
  • The Archimedean Property
  • Cauchy Sequences and Completeness
  • The Heine-Borel Theorem
  • The Extreme Value Theorem (Proof)
  • Taylor's Theorem with Remainder
  • Properties of the Riemann Integral
  • The Nested Interval Property
  • Limsup and Liminf
  • Connectedness
  • The Intermediate Value Theorem (Proof)
  • L'Hopital's Rule (Rigorous Proof)
  • The Fundamental Theorem of Calculus (Proof)
  • Constructing the Real Numbers
  • Rigorous Treatment of Infinite Series
  • Perfect Sets and the Cantor Set
  • Classifying Discontinuities
  • Pathological Functions and the Limits of Differentiability
  • Limitations of the Riemann Integral and Preview of Lebesgue
• Calculus I: Differentiation(30 concepts)

  • Tangent Line Approximation
  • Rolle's Theorem
  • Derivatives of Trigonometric Functions
  • Intuitive Notion of Limits
  • The Derivative as a Limit
  • Related Rates
  • Differentials and Small Changes
  • The Mean Value Theorem
  • Derivatives of Inverse Trigonometric Functions
  • The Epsilon-Delta Definition of a Limit
  • Basic Differentiation Rules
  • Optimization Problems
  • Newton's Method
  • Monotonicity and the First Derivative Test
  • Derivatives of Exponential Functions
  • Limit Laws and Algebraic Techniques
  • The Product and Quotient Rules
  • Curve Sketching with Derivatives
  • Error Analysis and Propagation
  • Concavity and Inflection Points
  • Derivatives of Logarithmic Functions
  • The Squeeze Theorem
  • The Chain Rule
  • L'Hopital's Rule
  • The Extreme Value Theorem and Critical Point Analysis
  • Higher-Order Approximation and Quadratic Models
  • Consequences and Extensions of the MVT
  • Logarithmic Differentiation
  • Continuity and the Intermediate Value Theorem
  • Implicit Differentiation
• Complex Analysis(30 concepts)

  • The Complex Plane and Its Geometry
  • Contour Integrals
  • Taylor Series in the Complex Plane
  • The Residue Theorem
  • Mobius Transformations
  • Complex Analysis in Signal Processing
  • Complex Differentiability and Analyticity
  • Cauchy's Theorem
  • Laurent Series
  • Techniques for Computing Residues
  • Mapping Between Standard Domains
  • Two-Dimensional Fluid Dynamics
  • The Cauchy-Riemann Equations
  • The Cauchy Integral Formula
  • Removable Singularities
  • Evaluating Real Trigonometric Integrals
  • The Schwarz-Christoffel Transformation
  • Electrostatics and Potential Theory
  • Elementary Complex Functions
  • Liouville's Theorem and the Fundamental Theorem of Algebra
  • Poles and Their Orders
  • Evaluating Improper Real Integrals via Residues
  • The Riemann Mapping Theorem
  • Connections to Quantum Mechanics
  • Conformal Mapping: Angle Preservation
  • The Maximum Modulus Principle
  • Essential Singularities and the Casorati-Weierstrass Theorem
  • Summation of Series Using Residues
  • Applications of Conformal Mapping to Physics
  • Connections to Number Theory and the Riemann Zeta Function
• Multivariable Calculus(30 concepts)

  • The Dot Product and Projections
  • Partial Derivatives
  • Double Integrals over Rectangular and General Regions
  • Line Integrals of Scalar Functions
  • Conservative Vector Fields and Potential Functions
  • Flux Calculations in Physics
  • The Cross Product
  • The Gradient Vector
  • Changing the Order of Integration
  • Line Integrals of Vector Fields and Work
  • Green's Theorem
  • Circulation and Rotational Flow
  • Equations of Lines and Planes in Space
  • Directional Derivatives
  • Double Integrals in Polar Coordinates
  • Surface Integrals and Flux
  • Stokes' Theorem
  • Center of Mass and Moments of Inertia
  • Quadric Surfaces
  • The Multivariable Chain Rule
  • Triple Integrals in Cartesian, Cylindrical, and Spherical Coordinates
  • The Curl of a Vector Field
  • The Divergence Theorem
  • Fluid Flow and Stream Functions
  • Parametric Curves and Surfaces
  • Tangent Planes and Linear Approximation
  • Change of Variables and the Jacobian
  • The Divergence of a Vector Field
  • The Generalized Stokes' Theorem: A Unified Perspective
  • Electrostatics and Heat Flow
• Calculus II: Integration & Series(30 concepts)

  • Riemann Sums and Area Approximation
  • Integration by Substitution
  • Area Between Curves
  • Improper Integrals with Infinite Limits
  • Sequences and Their Limits
  • Power Series and Radius of Convergence
  • The Definite Integral as a Limit
  • Integration by Parts
  • Volumes of Revolution: Disk and Washer Methods
  • Improper Integrals with Discontinuous Integrands
  • Geometric and Telescoping Series
  • Taylor and Maclaurin Series
  • The Fundamental Theorem of Calculus, Part I
  • Trigonometric Substitution
  • Volumes of Revolution: Shell Method
  • Comparison Tests for Improper Integrals
  • Comparison and Integral Tests
  • Manipulating Known Series
  • The Fundamental Theorem of Calculus, Part II
  • Partial Fraction Decomposition
  • Arc Length and Surface Area
  • The Trapezoidal Rule
  • The Ratio and Root Tests
  • Taylor's Theorem and the Remainder
  • Properties of the Definite Integral
  • Integration Strategy and When Techniques Fail
  • Work, Force, and Physical Applications
  • Simpson's Rule and Error Analysis
  • Alternating Series and Absolute vs. Conditional Convergence
  • Applications of Taylor Series

The mathematics of structure, symmetry, and transformation — vectors, matrices, eigenvalues, groups, rings, fields, and computational methods. Machine learning is linear algebra wearing a trenchcoat; cryptography is finite fields in disguise; physics is group theory made physical. This pillar reveals the hidden architecture that connects seemingly unrelated mathematical objects.

• Abstract Algebra: Groups & Symmetry(30 concepts)

  • The Group Axioms
  • Group Homomorphisms
  • Group Actions
  • Cyclic Groups
  • Cosets & Lagrange's Theorem
  • The Rubik's Cube Group
  • Symmetry Groups of Geometric Objects
  • Kernel & Image of a Homomorphism
  • Orbits & Stabilizers
  • Direct Products of Groups
  • Normal Subgroups
  • Crystallography & Space Groups
  • Permutation Groups
  • Isomorphisms & Structural Equivalence
  • The Orbit-Stabilizer Theorem
  • The Chinese Remainder Theorem for Groups
  • Quotient Groups
  • Group Theory in Coding Theory
  • Subgroups
  • The First Isomorphism Theorem
  • Burnside's Lemma
  • Fundamental Theorem of Finitely Generated Abelian Groups
  • Simple Groups
  • Symmetry in Physics & Noether's Theorem
  • Order of Elements & Generators
  • Automorphisms
  • Symmetry Groups of 3D Objects
  • Applications of Abelian Group Theory
  • Composition Series & Jordan-Holder Theorem
  • The Landscape of Group Theory
• Abstract Algebra: Rings, Fields & Polynomials(25 concepts)

  • Rings: Definition & Examples
  • Polynomial Rings: Structure & Arithmetic
  • Field Extensions
  • Constructing Finite Fields
  • Galois Groups
  • Ideals
  • Irreducibility of Polynomials
  • Algebraic vs. Transcendental Elements
  • Classification of Finite Fields
  • Fixed Fields & the Galois Correspondence
  • Quotient Rings
  • The Eisenstein Criterion & Irreducibility Tests
  • Degree of an Extension & the Tower Law
  • The Frobenius Endomorphism
  • The Fundamental Theorem of Galois Theory
  • Ring Homomorphisms & the First Isomorphism Theorem
  • Factoring Polynomials Over Different Fields
  • Splitting Fields
  • Finite Fields in Cryptography
  • Solvability by Radicals
  • Integral Domains & Fields of Fractions
  • Unique Factorization Domains & PIDs
  • Algebraic Closure
  • Reed-Solomon Codes & Error Correction
  • The Insolvability of the Quintic
• Linear Algebra I: Vectors, Matrices & Systems(25 concepts)

  • Linear Equations & Systems
  • Vectors & Vector Operations
  • Matrix Operations
  • The Determinant: Definition & Properties
  • The Dot Product
  • Augmented Matrices & Row Operations
  • Span & Linear Combinations
  • Linear Transformations
  • Cofactor Expansion
  • Orthogonality
  • Gaussian Elimination
  • Linear Independence
  • Kernel & Image
  • Determinants & Row Operations
  • The Gram-Schmidt Process
  • Row Echelon & Reduced Row Echelon Form
  • Basis of a Vector Space
  • The Rank-Nullity Theorem
  • Geometric Interpretation of Determinants
  • Orthogonal Projections
  • Existence & Uniqueness of Solutions
  • Dimension
  • Matrix Inverses
  • Cramer's Rule
  • Least Squares Approximation
• Numerical Methods & Computation(30 concepts)

  • IEEE 754 Floating-Point Representation
  • The Bisection Method
  • Numerical LU & QR Factorization
  • Newton-Cotes Quadrature
  • Euler's Method
  • Lagrange Interpolation
  • Rounding Error & Error Propagation
  • Newton-Raphson Method
  • Computing the SVD Numerically
  • Gaussian Quadrature
  • Runge-Kutta Methods
  • Newton's Divided Differences
  • Catastrophic Cancellation
  • The Secant Method
  • Iterative Methods: Jacobi & Gauss-Seidel
  • Richardson Extrapolation
  • Stability & Stiff Systems
  • Spline Interpolation
  • Conditioning & Condition Numbers
  • Convergence Rates & Order of Convergence
  • The Conjugate Gradient Method
  • Adaptive Integration Methods
  • Adaptive Step-Size Control
  • Least Squares Fitting
  • Designing Numerically Stable Algorithms
  • Hybrid Methods & Practical Root-Finding
  • Preconditioning & Practical Considerations
  • Numerical Differentiation
  • Multistep Methods & Method Selection
  • The Fast Fourier Transform
• Linear Algebra II: Eigentheory & Applications(30 concepts)

  • Eigenvalues & Eigenvectors: Definition
  • Symmetric Matrices & Their Properties
  • Constructing the SVD
  • LU Decomposition
  • Principal Component Analysis (PCA)
  • General Vector Spaces
  • The Characteristic Polynomial
  • The Spectral Theorem
  • Geometric Interpretation of SVD
  • QR Factorization
  • Google's PageRank Algorithm
  • Isomorphism of Vector Spaces
  • Eigenspaces & Multiplicity
  • Quadratic Forms
  • Low-Rank Approximation & the Eckart-Young Theorem
  • Cholesky Factorization
  • Computer Graphics Transformations
  • Change of Basis
  • Diagonalization
  • Positive Definite Matrices
  • The Pseudoinverse
  • Schur Decomposition
  • Markov Chains
  • Direct Sums
  • Complex Eigenvalues & Rotations
  • Applications of Spectral Theory
  • SVD in the Wild
  • Choosing the Right Decomposition
  • Least Squares Regression Revisited
  • Dual Spaces & Linear Functionals

The mathematics of the countable — combinatorics, graphs, prime numbers, computability, and the information-theoretic limits of what can be known, computed, and communicated. Every algorithm has a counting argument at its heart; every network is a graph; every secure transaction depends on number theory. This pillar connects pure mathematical elegance to the digital infrastructure civilization depends on.

• Combinatorics & Counting(30 concepts)

  • The Product Rule
  • Permutations of Distinct Objects
  • The Pigeonhole Principle
  • The Inclusion-Exclusion Sieve
  • Ordinary Generating Functions
  • Linear Recurrences with Constant Coefficients
  • The Sum Rule & Disjoint Cases
  • Combinations & the Binomial Coefficient
  • Generalized & Infinite Pigeonhole
  • Euler's Totient Function
  • Exponential Generating Functions
  • The Characteristic Equation Method
  • Bijective Counting
  • Permutations with Repetition
  • Ramsey Theory & Ramsey Numbers
  • Counting Surjections
  • Coefficient Extraction Techniques
  • Non-Homogeneous Recurrences
  • Overcounting & Symmetry Corrections
  • Multisets & Stars-and-Bars
  • Extremal Combinatorial Problems
  • Derangements
  • Solving Recurrences via Generating Functions
  • Catalan Numbers
  • Building Counting Models
  • The Balls-in-Bins Framework
  • The Probabilistic Method (Introduction)
  • Mobius Inversion on Posets
  • The Partition Function
  • Stirling Numbers
• Graph Theory(30 concepts)

  • Graphs, Vertices, and Edges
  • Trees & Equivalent Characterizations
  • Euler Tours & Circuits
  • Chromatic Number & Proper Coloring
  • Network Flows & Capacity Constraints
  • The Adjacency Matrix & Its Spectrum
  • Degree Sequences
  • Spanning Trees & Kirchhoff's Theorem
  • Euler Paths & Extensions
  • The Chromatic Polynomial
  • The Max-Flow Min-Cut Theorem
  • The Graph Laplacian
  • The Handshaking Lemma
  • Cayley's Formula & Prufer Sequences
  • Hamiltonian Cycles
  • The Four-Color Theorem
  • Hall's Marriage Theorem
  • Spectral Properties & Graph Structure
  • Graph Isomorphism
  • Cut Vertices & Bridges
  • Ore's & Dirac's Theorems
  • Kuratowski's Theorem & Planarity Testing
  • Maximum Bipartite Matching
  • Expander Graphs
  • Special Graph Families
  • Menger's Theorem
  • Applications of Euler & Hamilton Theory
  • Euler's Formula for Planar Graphs
  • Konig's Theorem
  • The Cheeger Inequality
• Number Theory(30 concepts)

  • The Division Algorithm
  • Primes & Irreducibility
  • Congruences & Modular Arithmetic
  • Quadratic Residues Modulo p
  • Pythagorean Triples
  • Simple Continued Fractions
  • The Euclidean Algorithm
  • The Fundamental Theorem of Arithmetic
  • Modular Inverses & Linear Congruences
  • The Legendre & Jacobi Symbols
  • Fermat's Method of Infinite Descent
  • Convergents & Best Approximations
  • Bezout's Identity
  • The Sieve of Eratosthenes
  • The Chinese Remainder Theorem
  • The Law of Quadratic Reciprocity
  • Pell's Equation
  • Periodic Continued Fractions & Quadratic Irrationals
  • Linear Diophantine Equations
  • The Distribution of Primes
  • Fermat's Little Theorem
  • Gauss's Lemma
  • Fermat's Last Theorem
  • Irrationality Proofs via Continued Fractions
  • Properties of GCD and LCM
  • Arithmetic Functions & Multiplicativity
  • Euler's Theorem & the Totient
  • Applications of Quadratic Residue Theory
  • Local-Global Principles & Obstructions
  • Diophantine Approximation & Hurwitz's Theorem
• Cryptography & Information Theory(25 concepts)

  • Caesar & Substitution Ciphers
  • The RSA Cryptosystem
  • Entropy & Information Content
  • Linear Codes & the Hamming Bound
  • Cryptographic Hash Functions
  • The Vigenere Cipher & Polyalphabetic Systems
  • Diffie-Hellman Key Exchange
  • Mutual Information & Conditional Entropy
  • Hamming Codes
  • Zero-Knowledge Proofs
  • The Enigma Machine & Its Cryptanalysis
  • Elliptic Curve Cryptography
  • Shannon's Source Coding Theorem
  • Reed-Solomon Codes
  • The Quantum Threat: Shor's Algorithm
  • The One-Time Pad & Perfect Secrecy
  • Digital Signatures
  • Channel Capacity & the Noisy Channel Coding Theorem
  • Decoding Algorithms
  • Lattice-Based Cryptography
  • From Classical to Modern: The Paradigm Shift
  • Computational Hardness & Security Proofs
  • KL Divergence & Information Geometry
  • Capacity-Approaching Codes: Turbo & LDPC
  • The Post-Quantum Cryptographic Landscape
• Mathematical Logic & Computation(25 concepts)

  • First-Order Logic: Syntax & Semantics
  • The ZFC Axiom System
  • Turing Machines
  • Godel Numbering & Arithmetization of Syntax
  • Decidable Theories
  • Formal Proofs & Natural Deduction
  • Ordinal Numbers
  • Recursive Functions & Lambda Calculus
  • Self-Reference & the Diagonal Lemma
  • Undecidable Theories & Peano Arithmetic
  • The Completeness Theorem
  • Cardinal Numbers & Cardinal Arithmetic
  • The Church-Turing Thesis
  • Godel's First Incompleteness Theorem
  • P vs NP: The Bounded Analog
  • The Compactness Theorem
  • The Axiom of Choice & Equivalents
  • The Halting Problem
  • Godel's Second Incompleteness Theorem
  • Logic-Complexity Connections
  • The Lowenheim-Skolem Theorem
  • Independence Results & the Set-Theoretic Multiverse
  • Reductions & Rice's Theorem
  • Implications & Legacy of Incompleteness
  • Independence & Proof Barriers

Where mathematics meets reality — probability, stochastic processes, statistics, optimization, game theory, and mathematical modeling. Your intuition about risk is systematically wrong; this pillar recalibrates it. From predicting stock prices to designing clinical trials to training AI systems to modeling pandemics, these are the tools that turn mathematical theory into real-world impact.

• Game Theory & Decision Mathematics(30 concepts)

  • Normal Form Games & Payoff Matrices
  • Game Trees & Sequential Play
  • Coalition Games & Characteristic Functions
  • The Revelation Principle
  • Utility Theory & Axioms
  • Evolutionarily Stable Strategies
  • Dominant Strategies & Iterated Elimination
  • Backward Induction
  • The Shapley Value
  • VCG Mechanisms
  • Risk Aversion & Certainty Equivalents
  • Replicator Dynamics
  • Best Response & Nash Equilibrium
  • Subgame Perfect Equilibrium
  • The Core
  • Auction Theory
  • Prospect Theory
  • Population Games & Dynamics
  • Mixed Strategy Equilibria
  • Information Sets & Imperfect Information
  • The Nucleolus
  • Market Design
  • Decisions Under Ambiguity
  • Biological Applications
  • Nash's Existence Theorem
  • Signaling & Screening Games
  • Voting Power & Weighted Voting Games
  • Impossibility Theorems in Mechanism Design
  • Multi-Criteria Decision Making
  • Social Norms & Cultural Evolution
• Mathematical Statistics(30 concepts)

  • Point Estimation
  • Maximum Likelihood Estimation
  • The Neyman-Pearson Framework
  • Confidence Intervals
  • Rank-Based Tests
  • The Curse of Dimensionality
  • Bias & Consistency
  • Fisher Information
  • Likelihood Ratio Tests
  • Linear Regression Theory
  • Bootstrap Methods
  • Ridge Regression
  • Efficiency & the Cramer-Rao Bound
  • Asymptotic Properties of MLE
  • P-Values & Statistical Significance
  • Inference in Regression
  • Permutation Tests
  • LASSO Regression
  • Sufficient Statistics
  • Bayesian Updating & Conjugate Priors
  • Power & Sample Size Determination
  • Analysis of Variance (ANOVA)
  • Kernel Density Estimation
  • Cross-Validation & Model Selection
  • The Rao-Blackwell Theorem
  • Frequentist vs. Bayesian Perspectives
  • Multiple Testing Corrections
  • Regression Diagnostics & Model Assessment
  • Nonparametric Regression
  • Connections to Machine Learning
• Optimization Theory(30 concepts)

  • Convex Sets
  • Gradient Descent
  • Lagrange Multipliers
  • Linear Program Formulation
  • Quadratic Programming
  • Integer Programming
  • Convex Functions
  • Newton's Method
  • KKT Conditions
  • The Simplex Method
  • Semidefinite Programming
  • Branch-and-Bound
  • Jensen's Inequality
  • Convergence Rate Analysis
  • Lagrangian Duality
  • LP Duality
  • Conic Optimization
  • Cutting Planes & Polyhedral Theory
  • Convex Combinations & Hulls
  • Line Search Methods
  • Duality Applications
  • Complementary Slackness & Sensitivity
  • Interior Point Methods
  • Approximation Algorithms
  • Strong Convexity & Smoothness
  • Quasi-Newton Methods
  • Saddle Point Theory
  • LP Applications & Network Flows
  • Disciplined Convex Programming
  • Metaheuristics
• Probability Theory(30 concepts)

  • Sample Spaces & Events
  • Probability Mass Functions
  • Probability Density Functions & CDFs
  • Joint Probability Distributions
  • Weak Law of Large Numbers
  • Moment Generating Functions
  • Sigma-Algebras
  • Expectation & Variance
  • Uniform Distribution
  • Marginal Distributions
  • Strong Law of Large Numbers
  • MGF Properties & Computation
  • Kolmogorov Axioms
  • Bernoulli & Binomial Distributions
  • Exponential Distribution
  • Conditional Distributions
  • Central Limit Theorem
  • Applications to Sums of Random Variables
  • Conditional Probability
  • Poisson Distribution
  • Normal Distribution
  • Covariance & Correlation
  • Modes of Convergence
  • Characteristic Functions
  • Bayes' Theorem & Independence
  • Geometric & Negative Binomial Distributions
  • Transformations of Random Variables
  • Independence of Random Variables
  • CLT Applications & Refinements
  • Generating Functions as Proof Tools
• Stochastic Processes(30 concepts)

  • Transition Matrices
  • The Poisson Process
  • One-Dimensional Random Walks
  • Martingale Definition & Examples
  • The Wiener Process
  • Queuing Network Applications
  • State Classification
  • Rate Matrices & Kolmogorov Equations
  • Gambler's Ruin
  • Sub- and Supermartingales
  • Properties of Brownian Motion
  • Population Genetics Models
  • Stationary Distributions
  • Birth-Death Processes
  • Higher-Dimensional Random Walks
  • Optional Stopping Theorem
  • Ito's Lemma
  • Stock Price Models
  • Ergodicity
  • Queuing Theory Fundamentals
  • Recurrence & Transience Analysis
  • Martingale Convergence Theorems
  • Stochastic Differential Equations
  • Monte Carlo Methods
  • Absorbing States & First Passage Times
  • Multi-Server & Network Queues
  • Donsker's Invariance Principle
  • Martingales in Finance
  • The Black-Scholes Framework
  • MCMC & Advanced Simulation
• Mathematical Modeling(30 concepts)

  • Problem Formulation & Assumptions
  • SIR & Compartmental Models
  • Difference Equations
  • PDE Classification
  • Random Number Generation
  • Climate & Energy Balance Models
  • Dimensional Analysis & Scaling
  • Predator-Prey Models
  • Leslie Matrix Models
  • The Heat Equation
  • Monte Carlo Estimation
  • Epidemiological Modeling & Intervention
  • Model Validation & Verification
  • Equilibria & Stability Analysis
  • The Logistic Map & Chaos
  • The Wave Equation
  • Variance Reduction Techniques
  • Financial Risk Modeling
  • Sensitivity Analysis
  • Phase Portraits & Qualitative Analysis
  • Lyapunov Exponents & Sensitive Dependence
  • Fourier Series
  • MCMC Methods
  • Network Dynamics & Spreading Processes
  • Model Types & Tradeoffs
  • Bifurcation Theory
  • Fractals & Strange Attractors
  • Laplace's Equation & Boundary Value Problems
  • Simulation Study Design & Validation
  • Optimization in Practice