Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.

Tentative schedule for Spring 2015

Fall 2014

date	speaker	title	host(s)
September 12	Moon Duchin (Tufts University)	TBA	Dymarz and WIMAW
September 19	Greg Smith (Queen's University)	Nonnegative sections and sums of squares	Erman
September 26	Jack Xin (UC Irvine)	TBA	Jin
October 3	Pham Huu Tiep (University of Arizona)	Adequate subgroups	Gurevich
October 10	Alejandro Adem (UBC)	TBA	Yang
October 17	Tentatively reserved until 9/5		Stovall
October 24	Almut Burchard (University of Toronto)	TBA	Stovall
October 31	Bao Chau Ngo (University of Chicago)	TBA	Gurevich
November 7	Reserved for possible job interview
November 14	Reserved for possible job interview
November 21	Reserved for possible job interview
November 28	University holiday
December 5	Reserved for possible job interview
December 12	Reserved for possible job interview

Abstracts

September 19: Gregory G. Smith (Queen's University)

Nonnegative sections and sums of squares

A polynomial with real coefficients is nonnegative if it takes on only nonnegative values. For example, any sum of squares is obviously nonnegative. For a homogeneous polynomial with respect to the standard grading, Hilbert famously characterized when the converse holds, that is when every nonnegative homogeneous polynomial is a sum of squares. After reviewing some history of this problem, we will examine this converse in more general settings such as global sections of a line bundles. This line of inquiry has unexpected connections to classical algebraic geometry and leads to new examples in which every nonnegative homogeneous polynomial is a sum of squares. This talk is based on joint work with Grigoriy Blekherman and Mauricio Velasco.

October 3: Pham Huu Tiep (Arizona)

Adequate subgroups

The notion of adequate subgroups was introduced by Thorne. It is a weakening of the notion of big subgroups used in generalizations of the Taylor-Wiles method for proving the automorphy of certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown recently by Guralnick, Herzig, Taylor, and Thorne that if the degree is small compared to the characteristic then all absolutely irreducible representations are adequate. We will discuss extensions of this result obtained recently in joint work with R. M. Guralnick and F. Herzig. In particular, we show that almost all absolutely irreducible representations in characteristic p of degree less than p are adequate. We will also address a question of Serre about indecomposable modules in characteristic p of dimension less than 2p-2.