The Brouwer Fix-Point Theorem
Yago Antolin Pichel, Vanderbilt University
Location: Stevenson 1206
Play Hex, Prove Brouwer
Weakly Commensurable Groups
Andrei Rapinchuk, University of Virginia
Location: Stevenson 5211
The notion of weak commensurability (of Zariski-dense subgroups of semi-simple algebraic groups) was introduced in the ongoing joint work with Gopal Prasad on length-commensurable and isospectral locally symmetric spaces. We were able to determine when two arithmetic subgroups are weakly commensurable. This led to various geometric results, some of which are related to the famous question "Can one hear the shape of a drum?" Tea at 3:30 pm in SC 1425.
"Stories from Another Pocket" (after Karel Capek and others).
Dmitry Burago, Penn State
Location: Stevenson 1310
This year, I have been delivering a number of talks under almost the same title. However, the talks are quite different. I have prepared about twenty topics, with two-three slides for each. For each talk, I select about eight topics; the choice depends on the audience, how long the talk is etc. The topics are united only by the fact they were of interest to me in the past several years. For each topic, I give only key definitions, one or two theorems and several open problems (which may form the most important part of the talk). The talk is supposed to be accessible to (reasonable) graduate students. We will not go into (almost:) any technicalities.
Graduate Student Tea
Location: Stevenson 1425
An Introduction to Markov Chain Monte Carlo Methods
Jorge Roman, Vanderbilt University
Location: Stevenson 1307
The need to approximate an intractable integral with respect to a probability distribution P is a problem that frequently arises across many different disciplines. A popular alternative to numerical integration and analytical approximation methods is the Monte Carlo (MC) method which uses computer simulations to estimate the integral. In the MC method, one generates independent and identically distributed (iid) samples from P and then uses sample averages to estimate the integral. However, in many situations, P is a complex high-dimensional probability distribution and obtaining iid samples from it is either impossible or impractical. When this happens, one may still be able to use the increasingly popular Markov chain Monte Carlo (MCMC) method in which the iid draws are replaced by a Markov chain that has P as its stationary distribution. In this talk, I will give a brief introduction to the MC and MCMC methods. The focus will be on the MCMC method and its applications to Bayesian statistics.
Famous Proofs of the Pythagorean Theorem
Tim Ferguson, Vanderbilt University
Location: Stevenson 1206
You have probably heard of the Pythagorean theorem, but can you explain why it is true? I will explain several different proofs of this famous theorem, including proofs by Euclid and President Garfield. If time permits, I will also discuss another famous result attributed to the Pythagoreans: the proof of the irrationality of the square root of 2.
Bach-Maxwell Equations and Extremal Kahler Metrics
Caner Koca, Vanderbilt University
Location: Stevenson 1312
The Bach-Maxwell Equations on a 4-dimensional compact oriented manifold can be thought of as a conformally invariant version of the classical Einstein-Maxwell Equations in general relativity. Riemannian metrics which solve the BM equations have interesting geometric properties. In this talk, I will introduce these equations and give several variational characterizations. I will also show that extremal Kahler metrics are among the solutions and discuss their role in this variational setting.
The 3D Index of a Cusped Hyperbolic 3-Manifold
Stavros Garoufalidis, Georgia Tech
Location: Stevenson 1432
The 3D index of Dimofte-Gaiotto-Gukov is a partially defined function on the set of ideal triangulations of 3-manifolds with torus boundary, which is partially invariant under 2-3 moves. It turns out that an ideal triangulation has 3D index if and only if it is 1-efficient. Moreover, the 3D index descends to a topological invariant of cusped hyperbolic manifolds. Parts are joint work with Hodgson-Rubinstein-Segerman.
Analyticity of Solutions to the Yamabe Flow on Non-Compact Manifolds
Yuanzhen Shao, Vanderbilt University
Location: Stevenson 1307
The Yamabe flow can be considered as an alternative approach to the famous Yamabe problem. Nowadays there is increasing interest in studying the Yamabe flow on non-compact manifolds. We show by means of continuous maximal regularity theory and the implicit function theorem that in every conformal class containing at least one real analytic metric, solutions to the Yamabe flow immediately become analytic jointly in time and space. In comparison with the existing results, we do not ask for a uniform bound on the curvatures of the initial metric. We will also briefly discuss a generalization of our results on singular manifolds.