This course covers classical methods in applied mathematics and math modeling, including dimensional analysis, asymptotics, regular and singular perturbation theory for ordinary differential equations, random walks and the diffusion limit, and classical solution techniques for PDE. The techniques will be applied to models arising throughout the natural sciences.

Continuation of Math 623. Introduction to functional analysis; elementary theory of Hilbert and Banach spaces; functional analytic properties of Lp-spaces, appli-cations to Fourier series and integrals; interplay between topology, and measure, Stone-Weierstrass theorem, Riesz representation theorem. Further topics depending on instructor.

Complex number field, elementary functions, holomorphic functions, integration, power and Laurent series, harmonic functions, conformal mappings, applications.

A continuation of Math 611. Topics in group theory (e.g., Sylow theorems, solvable and simple groups, Jordan-Holder and Schreier theorems, finitely generated Abelian groups). Topics in ring theory (matrix rings, prime and maximal ideals, Noetherian rings, Hilbert basis theorems). Modules, including cyclic, torsion, and free modules, direct sums, tensor products. Algebraic closure of fields, normal, algebraic, and transcendental field extensions, basic Galois theory. Prerequisite: Math 611 or equivalent.

This course is an introduction to stochastic processes. The course will cover Monte Carlo methods, Markov chains in discrete and continuous time, martingales, and Brownian motion. Theory and applications will each play a major role in the course. Applications will range widely and may include problems from population genetics, statistical physics, chemical reaction networks, and queueing systems, for example.

Not available at this time

This course is an introduction to differential geometry, where we apply theory and computational techniques from linear algebra, multivariable calculus and differential equations to study the geometry of curves, surfaces and (as time permits) higher dimensional objects.

This proof-based course covers the fundamentals of linear optimization and polytopes and the relationship between them. The course will give a rigorous treatment of the algorithms used in linear optimization. The topics covered in linear optimization are graphical methods to find optimal solutions in two and three dimensions, the simplex algorithm, duality and Farkas? lemma, variation of cost functions, an introduction to integer programming and Chvatal-Gomory cuts.

Introduction to the application of computational methods to models arising in science and engineering, concentrating mainly on the solution of partial differential equations. Topics include finite differences, finite elements, boundary value problems, fast Fourier transforms. Prerequisite: MATH 551 or consent of instructor.

Introduction to computational techniques used in science and industry. Topics selected from root-finding, interpolation, data fitting, linear systems, numerical integration, numerical solution of differential equations, and error analysis. Prerequisites: MATH 233 and 235, or consent of instructor, and knowledge of a scientific programming language.