All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.
|September 12||Moon Duchin (Tufts University)||Geometry and counting in the Heisenberg group||Dymarz and WIMAW|
|September 19||Gregory G. 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|
September 12: Moon Duchin (Tufts University)
Geometry and counting in the Heisenberg group
The growth function of a finitely-generated group enumerates how many words can be spelled with each possible number of letters-- this should be thought of as a sort of volume growth in any geometric model of the group. A major theorem of Gromov tells us exactly which groups have growth in the polynomial range: those that are (virtually) nilpotent. But we can still wonder how regular the growth of a nilpotent group is: is it actually a polynomial? Or could it exhibit some transcendentality together with pretty slow growth?
I'll talk about some themes and techniques in the study of group growth and outline a geometry of numbers for nilpotent groups, including a recent result with M. Shapiro settling a long-standing question: the Heisenberg group -- the simplest non-abelian nilpotent group -- has rational growth in any generating set.
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)
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.