Colloquia/Fall18

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)	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)	G-equations and Front Motion in Fluid Flows	Jin
October 3	Pham Huu Tiep (University of Arizona)	Adequate subgroups	Gurevich
October 10	Alejandro Adem (UBC)	TBA	Yang
October 17	Roseanna Zia (Cornell University)	TBA	Spagnolie
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 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.

September 26: Jack Xin (UC Irvine)

G-equations and Front Motion in Fluid Flows

G-equations are level set Hamilton-Jacobi equations (HJE) for modeling flame fronts in turbulent combustion where a fundamental problem is to characterize the turbulent flame speeds s_T. The existence of s_T is connected with the homogenization of HJE, however classical theory does not apply due to the non-coercive and non-convex nature of the level set Hamiltonian. We shall illustrate the asymptotic properties of s_T from both Eulerian and Lagrangian perspectives in the case of two dimensional periodic incompressible flows, in particular cellular flows.

Analytical and numerical results demonstrate that G-equations capture well the enhancement, slow down and quenching phenomena observed in fluid experiments. We also comment on s_T in chaotic flows. This is joint work with Yifeng Yu and Yu-Yu Liu.

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.

Past Colloquia

Spring 2014

Fall 2013

Spring 2013

Fall 2012