Teaching

This course discusses three of the most important types of stochastic processes: Markov chains (in both discrete and continuous time), martingales (the mathematical model of "fair games"), and Brownian motion (random continuous motion). Applications include random walk, queueing theory, and branching processes, and may also include other areas such as optimal stopping or stochastic integration.

This is an accelerated version of MATH 20300. MATH 20300 covers the construction of the real numbers, the topology of $\mathbb{R}^n$ including the Bolzano-Weierstrass and Heine-Borel theorems, and a detailed treatment of abstract metric spaces, including convergence and completeness, compact sets, continuous mappings, and more.

This course is an introduction to the theory of ordinary differential equations in Euclidean space. Topics covered include: first-order equations of one variable, solving higher order systems via reduction of order, linear ODEs in arbitrary dimension, real Jordan form and the matrix exponential, variation of parameters, existence and uniqueness of solutions for Lipschitz vector fields, local analysis near equilibria, stability of solutions, introduction to dynamical systems and the global analysis of flows.

Calculus of several variables. Vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques. Double integrals and line integrals in the plane; exact differentials and conservative fields; Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications.

The second half of multi-variable calculus. Covers double integrals and line integrals in the plane; exact differentials and conservative fields; Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; and applications.

Six-week review of one-variable calculus, emphasizing material not on the high-school AB syllabus: integration techniques and applications, improper integrals, infinite series, applications to other topics, such as probability and statistics, as time permits.

Followed by the first half of multi-variable calculus. Covers vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques, and double integrals.

Differentiation and integration of functions of one variable, with applications. Informal treatment of limits and continuity. Differentiation: definition, rules, application to graphing, rates, approximations, and extremum problems. Indefinite integration; separable first-order differential equations. Definite integral; fundamental theorem of calculus. Applications of integration to geometry and science. Elementary functions. Techniques of integration. Polar coordinates. L'Hôpital's rule. Improper integrals. Infinite series: geometric, p-harmonic, simple comparison tests, power series for some elementary functions.

Six-week review of one-variable calculus, emphasizing material not on the high-school AB syllabus: integration techniques and applications, improper integrals, infinite series, applications to other topics, such as probability and statistics, as time permits.

Followed by the first half of multi-variable calculus. Covers vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques, and double integrals.