Global Arc

Combinatorics is the study of enumeration and structure of discrete objects. These structures are widespread throughout mathematics, including geometry, topology and algebra, as well as computer science, physics and optimization. This course will give an introduction to modern techniques in the field, and how they relate to objects such as polytopes, permutations and hyperplane arrangements.

Games in extensive form, pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristic-function form, imputations, solution concepts; related topics and applications. Prerequisite: MAT202 or 204 or 217 or equivalent. MAT215 or equivalent is recommended.

Sequence of independent trials, applications to number theory and analysis, Monte Carlo method. Markov chains, ergodic theorem for Markov chains. Entropy and McMillan theorem. Random walks, recurrence and non-recurrence; connection with the linear difference equations. Strong laws of large numbers, random series and products. Weak convergence of probability measures, weak Helly theorems, Fourier transforms of distributions. Limit theorems of probability theory. Prerequisite: MAT203 or 218 or equivalent.

Linear programs, duality, Dantzig's simplex method; theory of dual linear systems; matrix games, von Neumann's minimax theorem, simplex solution; algorithms for assignment, transport, flow; brief introduction to nonlinear programming.

An introduction to classical results in analytic number theory, presenting fundamental theorems with detailed proofs and highlighting the tight connections between them. Topics covered might include: the prime number theorem, Dirichlet L-functions, zero-free regions, sieve methods, representation by quadratic forms, and Gauss sums. Prerequisites: MAT335 and MAT345.

Topics introducing various aspects of number theory, including analytic and algebraic number theory, L-functions, and modular forms. See Course Offerings listing for topic details. Prerequisites: MAT 215, 345, 346 or equivalent.

The theory of Lebesgue integration in n-dimensional space. Differentiation theory. Hilbert space theory and applications to Fourier Transforms, and partial differential equations. Introduction to fractals. This course is the third semester of a four-semester sequence, but may be taken independently of the other semesters. Prerequisites: MAT215 or 218 or equivalent.

Introduction to the study of ordinary differential equations; explicit solutions, general properties of solutions, and applications. Topics include explicit solutions of some non-linear equations in two variables by separation of variables and integrating factors, explicit solution of simultaneous linear equations with constant coefficients, explicit solution of some linear equations with variable forcing term by Laplace transform methods, geometric methods (description of the phase portrait), and the fundamental existence and uniqueness theorem.

Introduction to incompressible fluid dynamics. The course will give an introduction to the mathematical theory of the Euler equations, the fundamental partial differential equation arising in the study of incompressible fluids. We will discuss several topics in analysis that emerge in the study of these equations: Lebesgue and Sobolev spaces, distribution theory, elliptic PDEs, singular integrals, and Fourier analysis. Content varies from year to year. See Course Offerings listing for topic details.

This course will cover the standard material in a first course on commutative algebra. Topics include: ideals in and modules over commutative rings, localization, primary decomposition, integral dependence, Noetherian rings and chain conditions, discrete valuation rings and Dedekind domains, completion; and dimension theory.