Linear Algebra [MATH221, MATH403]

• Abstraction of Vector Spaces, Quotient Spaces, Dual Spaces, Abstraction of Linear Maps, Factorization of Linear Maps, Exact Sequences, Metrics, Norms, Inner Products
• Matrices over a Basis, Strassen Algorithm, Trace & Determinant, Matrix Calculus, Eigendecomposition, Genuine/Generalized Eigenvectors, Adjoint Operators, Singular Value Decomposition
• Lie Groups, Positive Definite Matrices, Stochastic Matrices, Duality Theorem, Numerical Methods, Tensors, Exterior/Symmetric Algebras

Multivariate Calculus [MATH222]

• Total Derivatives/Differentiation, Partial/Directional Derivatives, Gradients, Iterated Derivatives, Hessian
• Taylor Approximations, Extrema, Lagrange Multipliers, Inverse/Implicit Function Theorem
• Integration, Change of Basis, Improper Integrals, Line/Surface Integrals, Green's Theorem, Stoke's Theorem, Divergence Theorem

Probability Theory [MATH 340, MATH 640]

• Probability Spaces, Measure, Discrete/Continuous Random Variables, Independence, Bayes' Formula, Expectation, Variance, Covariance, Correlation, Sums of Distributions, Joint/Marginal/Conditional Distributions, Multivariate Gaussian Distributions, Poisson Arrival Process, Markov Chains, Monte Carlo Algorithms, Branching Processes

Ordinary Differential Equations

• Phase Spaces/Flows, Existence/Uniqueness Theory, Basic Methods of Solution, Homogeneous/Inhomogeneous Linear DEQs, Reduction of Order, Variation of Constants, Series Solutions, Singular Points, Linear Systems of DEQs, Laplace Transforms, Euler Method, Mline Method, Runge-Kutta Methods

Stochastic Processes [MATH 391]

• Discrete-Time Markov Processes: Semigroups, Transition Probabilities, Irreducibility, Stopping Times, Periodicity, Stationary Measure, Reversed Markov Process, Reversibility (Detailed Balance), Metropolis-Hastings Algorithm, Ergodicity
• Poisson Processes: Memoryless Distributions, Limits of Binomials
• Continuous-Time Markov Processes: Semigroups, Generators, Irreducibility, Strong Markov Property, Stationary Measure, Reversed Markov Process, Reversibility (Detailed Balance), Ergodicity

Concentration of Measure [MATH 391]

• High-Dimensional Geometry
• Variance Bounds & Poincare Inequalities: Markov Semigroups, Dirichlet Forms
• Subgaussian Concentration, (Modified) log-Sobolev Inequalities, Martingale Method, Entropy

Real Analysis [MATH 531, 532]

• Construction of the Real Numbers, Limits of Sequences/Functions, Convergence/Divergence of Series, Continuity, Differentiability, Asymptotic Behavior, Infinitesimality, Mean Value Theorem, Intermediate Value Theorem, Complex Analysis, Primitives, Riemannian Integral

Abstract Algebra

• Group Classes, Generating Sets, Group Actions, Cosets, Orbits, Abelian Groups, Rings/Algebras, Commutative Rings, Ideals & Quotient Rings, Fields, Affine/Projective Spaces, Convexity, Quadrics, Tensor Algebras, Representation Theory, Lie Groups/Algebras

Differential Geometry

Partial Differential Equations

Point Set Topology

• Topologies, Induced Topologies, Hausdorff Spaces, Box/Product Topologies, Metric Topologies, Quotient Topologies, Continuity, Homeomorphisms, Connectedness, Compactness, Countability, Separability, Urysohn Metrization Theorem

Number Theory.pdf

• Divisibility Theory, Primes, Congruences, Euler's Theorem, Totient Function, Primitive Roots, Cryptography, Fermant's Last Theorem, Integers as Sums of Squares, Fibonacci Numbers, Continued Fractions

Algebraic Topology.pdf

• Homotopy, Fundamental Group, Homeomorphism Groups

Smooth Manifolds.pdf

• Topological Manifolds, (Non-Diffeomorphic) Smooth Structures, Smooth Maps, Lie Groups, Partitions of Unity, Tangent Vectors/Spaces, Pushforwards, Tangent Bundles, Lie Algebras, Category Theory, Cotangent Bundle, Pullbacks, Integration, Tensor Fields, Differential Forms, Orientations

Measure Theory.pdf

Functional Analysis.pdf