Global Arc

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.

Topics in algebra selected from areas such as representation theory of finite groups and the theory of Lie algebras. Prerequisite: MAT 345 or MAT 346.

Introduction to Algebraic Geometry; no previous knowledge of the topic is assumed. Familiarity with commutative algebra is helpful but will cover the necessary background

Elements of smooth manifold theory, tensors, and differential forms, Riemannian metrics, connection and curvature; selected applications to Hodge theory, curvature in topology and general relativity.

Knot theory involves the study of smoothly embedded circles in three-dimensional space. There ar lots of different techniques to study knots: combinatorial invariants, algebraic topology, hyperbolic geometry, Khovanov homology and mathematical gauge theory. This course will cover some of the modern techniques and recent developments in the field.

The course will target the following topics: The definition of knots in the 3-sphere, first invariants; algebraic knots and links in the 3-sphere; classification of algebraic knots, Puiseux pairs, iterated torus knots; fibred links, monodromy, the case of algebraic links; higher dimensional algebraic knots, Milnor theory of complex isolated hypersurface singularities.

Advanced course in Graph Theory. Further study of graph coloring, graph minors, perfect graphs, graph matching theory. Topics covered include: stable matching theorem, list coloring, chi-boundedness, excluded minors and average degree, Hadwiger's conjecture, the weak perfect graph theorem, operations on perfect graphs, and other topics as time permits.