Mathematics

Game Theory Curriculum Game Theory Curriculum This curriculum provides a structured overview of the study of game theory. Foundations Introduction to Game Theory: Basic concepts: players, strategies, payoffs Types of games: cooperative vs. non-cooperative, static vs. dynamic Examples of games: prisoner's dilemma, rock-paper-scissors Applications of game theory in various fields Static Games of Complete Information: Normal form representation of games Dominant strategies Iterated elimination of strictly dominated strategies ...

Optimization Curriculum Optimization Curriculum This curriculum provides a structured overview of the study of optimization. Foundations Introduction to Optimization: Basic concepts: objective function, constraints, feasible region Types of optimization problems: linear programming, nonlinear programming, convex optimization Applications of optimization in various fields Formulation of optimization problems Linear Programming: The standard form of linear programming problems Basic feasible solutions The simplex method Du...

Numerical Analysis Curriculum Numerical Analysis Curriculum This curriculum provides a structured overview of the study of numerical analysis. Foundations Introduction to Numerical Analysis: Sources of error: approximation, round-off, truncation Error analysis: absolute, relative, and percentage error Floating-point arithmetic Stability and convergence Root-Finding Methods: Bisection method Newton's method Secant method Convergence analysis of iterative methods Error bounds ...

Probability and Statistics Curriculum Probability and Statistics Curriculum This curriculum provides a structured overview of the study of probability and statistics. Probability Basic Concepts: Sample spaces and events Axioms of probability Conditional probability Independence Bayes' Theorem Random Variables: Discrete and continuous random variables Probability mass functions (PMF) and probability density functions (PDF) Cumulative distribution functions (CDF) Expectation and variance ...

Number Theory Curriculum Number Theory Curriculum This curriculum provides a structured overview of the study of number theory. Foundations Divisibility: The division algorithm Greatest common divisor (GCD) Euclidean algorithm Bézout's identity Least common multiple (LCM) Prime Numbers: Definition and properties of prime numbers The Fundamental Theorem of Arithmetic Distribution of prime numbers The Prime Number Theorem Tests for primality ...

Topology Curriculum Topology Curriculum This curriculum provides a structured overview of the study of topology. General Topology Basic Concepts: Definition of a topological space Examples of topological spaces Open sets, closed sets, and neighborhoods Basis and subbasis for a topology Continuous functions Important Properties of Spaces: Separation axioms (T0, T1, T2, T3, T4) Compactness Connectedness Countability Importa...

Non-standard Analysis Curriculum Non-standard Analysis Curriculum This curriculum provides a structured overview of the study of non-standard analysis. Foundations Introduction to Non-standard Analysis: Motivation and history Limitations of standard analysis The concept of hyperreal numbers Infinitesimals and infinitely large numbers The Hyperreal Number System: Construction of the hyperreals Properties of the hyperreals Order structure and arithmetic operations Transfer principle ...

Harmonic Analysis Curriculum Harmonic Analysis Curriculum This curriculum provides a structured overview of the study of harmonic analysis. Foundations Fourier Series: Trigonometric polynomials Convergence of Fourier series Dirichlet and Fejér kernels Applications to differential equations The Fourier Transform: The Fourier transform on L^1 Properties of the Fourier transform Convolution The Plancherel Theorem Function Spaces: ...

Functional Analysis Curriculum Functional Analysis Curriculum This curriculum provides a structured overview of the study of functional analysis. Foundations Vector Spaces: Definition and examples of vector spaces Subspaces, linear independence, and basis Dimension of a vector space Quotient spaces Normed Spaces: Definition and properties of a norm Examples of normed spaces (e.g., l^p spaces, C[a,b]) Convergence and completeness in normed spaces Banach spaces ...

Complex Analysis Curriculum Complex Analysis Curriculum This curriculum provides a structured overview of the study of complex analysis. Foundations Complex Numbers: The complex plane Polar form of complex numbers Powers and roots of complex numbers Stereographic projection Analytic Functions: Functions of a complex variable Limits and continuity Derivatives and the Cauchy-Riemann equations Harmonic functions Elementary Functions: ...

Real Analysis Curriculum Real Analysis Curriculum This curriculum provides a structured overview of the study of real analysis. Foundations The Real Number System: Axiomatic approach to real numbers Ordered field properties Completeness axiom (least upper bound property) Archimedean property Density of rational and irrational numbers Sequences and Series: Convergence of sequences Limit theorems for sequences Monotone sequences and Cauchy sequences Convergence of series Te...

Differential Equations Curriculum Differential Equations Curriculum This curriculum provides a structured overview of the study of differential equations. First-Order Ordinary Differential Equations Introduction to Differential Equations: Definitions and terminology (order, linearity, etc.) Initial value problems Direction fields Existence and uniqueness of solutions Applications of differential equations Methods for Solving First-Order ODEs: Separable equations Linear equations (integrating factors) Exact equations ...

Calculus Curriculum Calculus Curriculum This curriculum provides a structured overview of the study of calculus. Differential Calculus Limits and Continuity: Definition of a limit Techniques for evaluating limits One-sided limits Continuity of a function The Intermediate Value Theorem Derivatives: Definition of the derivative The power rule, product rule, quotient rule, and chain rule Derivatives of trigonometric, exponential, and logarithmic functions Implicit differentiation ...

Sequences, Series, and Mathematical Induction Sequences, Series, and Mathematical Induction This curriculum provides a structured overview of the study of sequences, series, and the principle of mathematical induction. Sequences and Series Sequences: Definition of a sequence Types of sequences (arithmetic, geometric, etc.) Explicit and recursive formulas for sequences Finding terms of a sequence Applications of sequences Series: Definition of a series Sigma notation (summation notation) Finite and infinite series ...

Combinatorics Curriculum Combinatorics Curriculum This curriculum provides a structured overview of the study of combinatorics. Basic Counting Principles The Fundamental Counting Principle: Counting the number of ways to perform a sequence of tasks Applications to simple counting problems Using tree diagrams to visualize counting problems Permutations: Definition of a permutation Permutations of n objects taken r at a time (nPr) Permutations with repetition Circular permutations Applications of permutations ...