Analytical Track
10:00 MWF // 151 Sloan
Instructor Nikolai Makarov
Course Secretary: Stacey Croomes

Loosely speaking, a special function is a function that, on the one hand, is more complicated than the elementary functions (such as exp, log, cos, sin, etc.) but, on the other hand, appears often enough to deserve a name. Because of their remarkable properties, special functions have been used for centuries. The subject has been continuously developed, with contributions by a host of mathematicians, including Euler and Gauss. In the past thirty years, the discovery of new special functions and applications to new areas of mathematics has initiated a resurgence of interest in the field. Today special functions are used in various fields, for instance, in combinatorics, number theory, differential equations, and mathematical physics. The aim of the course is to give an introduction to some of the classical special functions, with their most important properties in mind. The historical perspective will be kept in view and at the same time the aim is to give an up-to-date picture of the development and some of the problems that still may be unsolved, such as the Riemann hypothesis.

This is a topics course on the interactions of foliations and geometric structures on 3-manifolds. It is suitable for graduate students with a general background in geometry/topology, and we expect to reach current developments in the theory. We expect to be able to discuss a number of open problems which would make a suitable Ph.D. thesis topic. Undergraduate students require instructor's permission to register.

The course will study properties of subsets of finite sets. Topics will include Sperner systems, shadows and the shifting operation, intersecting families, the four functions theorem, isoperimetric problems, and one-factorization of the complete uniform hypergraphs.