**Seminars and Colloquiums**

for the week of October 16, 2017

for the week of October 16, 2017

*SPEAKERS*

Jen Berg, Rice University, Monday

Michael Wise, UTK, Wednesday
**
**Jerzy Dydak, UTK, Wednesday

Ryan Unger, Thursday

Graduate Student Industrial/Post-Doc Panel, Friday

Tuoc Phan, UTK, Friday

*TEA TIME*

*3:00 pm – 3:30 pm
Monday, Tuesday,
Wednesday
Ayres 401
Hosted By: Ibrahim Aslan & Pawel Grzegrzolka*

**Monday, October 16th**

**ALGEBRA SEMINAR**

TITLE: Obstructions to integral points on affine surfaces

SPEAKER: Jen Berg, Rice University

TIME: 3:35pm-4:25pm

ROOM: Ayres 113

In 1970, Manin showed that the Brauer group (an object that encodes reciprocity laws) can obstruct the existence of rational points on varieties, even when there exist points everywhere locally. This obstruction provided a single framework that encompassed all known counterexamples to the Hasse principle for rational points at that time.

In 2009, Colliot-Thelene and Xu showed that the Brauer group is relevant to obstructions to integral points on non-proper (e.g. affine) varieties as well. In this talk, we will consider this obstruction to integral points for a class of surfaces over a number field defined by an equation of the form x^2?ay^2 = P(t), where P(t) is a separable polynomial and a is not a square. We show that, unlike their smooth proper compactifications, these affine surfaces can have Brauer groups that are neither generated by quaternion algebras nor their higher dimensional generalizations.

Moreover, we exhibit a family of surfaces for which the Brauer-Manin obstruction does not explain the failure of the existence of integral points.

**Wednesday, October 18th **

**COMPUTATIONAL and APPLIED MATHEMATICS (CAM) SEMINAR
**TITLE: A Multilevel Preconditioner for the Interior Penalty Discontinuous Galerkin Method

SPEAKER: Mike Wise, UTK

TIME: 3:35-4:25pm

ROOM: Ayres 113

In this talk, we present a multilevel preconditioner of Brix, Pinto, and Dahmen (2008) for symmetric interior penalty discontinuous Galerkin approximations of second-order elliptic boundary value problems that gives rise to uniformly bounded condition numbers without additional regularity assumptions on the solution. The underlying triangulations are shape regular, but may have hanging nodes provided a certain grading condition is met. We highlight the key role of their decomposition of the discontinuous trial space into a conforming subspace and a non-conforming subspace that is controlled by the jumps across the edges and track dependence on the penalty parameters in the estimates. We conclude with a discussion of numerical experiments which serve to validate the theory.

**TOPOLOGY/ GEOMETRY SEMINAR
**TITLE: Profinite structures on residually finite groups

SPEAKER: Jerzy Dydak, UTK

TIME: 3:35-4:25pm

ROOM: Ayres 405

Given a countable group $G$ one can put a large scale (coarse) structure on it by selecting an increasing sequence $\{F_n\}_{n\ge 1}$ of finite subsets of $G$ whose union is $G$ and declaring a cover $\mathcal{U}$ of $G$ to be uniformly bounded if and only if there is $n\ge 1$ such that $\mathcal{U}$ refines $\{g\cdot F_n\}_{g\in G}$. It is well-known and easy to show that the coarse structure obtained that way is unique and is equal to the bounded metric structure given by any word metric on $G$ if $G$ is finitely generated.

In this talk we consider a dual situation in case of residually finite countable groups $G$. Given a decreasing sequence $\{G_n\}_{n\ge 1}$ of subgroups of $G$ of finite index whose intersection consists of the neutral element $e_G$ only, we define a small scale (uniform) structure on $G$ (which we call a profinite structure on $G$) by declaring a cover $\mathcal{U}$ of $G$ to be uniform if and only if there is $n\ge 1$ such that $\{g\cdot G_n\}_{g\in G}$ refines $\mathcal{U}$. One can give a characterization of profinite structures on $G$ in terms of topological group structures on $G$.

A natural question is if profinite structures are unique on countable groups. It turns out the answer is no. However, in case of finitely generated residually finite groups $G$ there is only one profinite structure.

This is joint work with Joanna Furno (University of Houston) and James Keesling (University of Florida).

**Thursday, October 19th**

**GEOMETRIC ANALYSIS SEMINAR
**TITLE: The Yamabe Problem I

SPEAKER: Ryan Unger

TIME: 5-6pm

ROOM: Ayres 404

In 1960 H. Yamabe conjectured that, given a compact Riemannian manifold, there exists a conformal deformation of the metric to one of constant scalar curvature. The combined works of Yamabe, N. Trudinger, T. Aubin, and R. Schoen gave an affirmative solution in 1984. Here we review Yamabe's original paper and give the proof of Yamabe's conjecture in the case when the Yamabe energy is nonpositive.

**Friday, October 20th**

**GRADUATE STUDENT INDUSTRIAL/POST-DOC PANEL**

TIME: 3:30

ROOM: Ayres 308H

These former UT math graduate students will share information about their careers as well as advice for current graduate students. There will be a reception with time for informal conversations following the panel discussion. Come and hear advice from folks who have successfully completed their UT graduate degrees, find out about possible career paths, and network. Panelist include: Greg Clark, Software Engineer at Google, Kai Kang, Post-doc at National Institutes of Health (NIH), Tyler Massaro, Post-doc at Duke Clinical Research Institution, James Sunkes, Research Assistant at Dynetics, Matt Turner, Principal Mathematical Statistician at Equifax, Fei Xing, Data Scientist at Mathematica Policy Research Group.

**COLLOQUIUM**

Title: Regularity theory for nonlinear partial differential equations and applications.

Speaker: Tuoc Phan

Time: 3:35pm - 4:30pm

Room: Ayres 405

In this talk, we will discuss several recent results on regularity theory in Sobolev spaces for several classes of elliptic and parabolic nonlinear PDEs. Important ideas and techniques will be highlighted. Connections and applications of the results to other areas will be discussed or briefly mentioned. In particular, a solution to an open question on the existence and uniqueness of smooth global-time solutions of a system of nonlinear parabolic equations called cross-diffusion equations will be given.

*If you are interested in giving or arranging a talk for one of our seminars or colloquiums, please review our calendar. *

*If you have questions, or a date you would like to confirm, please contact colloquium AT math DOT utk DOT edu *

**Past notices:**