# Graduate Colloquium Fall 2015

McHenry Building - Room 4130

Refreshments served at 3:30 in room 4161

For further information, please contact Gabriel Martins or call 831-459-2969

**Wednesday, October 7, 2015**

** Rings with Anti-Involution and Their Orthogonal Units
Rob Carman, University of California, Santa Cruz
** I'll begin this talk by defining and discussing some basic theory of rings with an anti-involution and their orthogonal unit groups. Then I'll describe a few such rings used in the study of the representation theory of finite groups (the Burnside ring, character ring, and trivial source ring) and explain what is known about their orthogonal unit groups. The goal of this talk is to present some topics from representation theory to those unfamiliar with the field as well as describe a bit of my recent work.

**Wednesday, October 14, 2015
** In this talk I'll discuss the principle of least action, one of the most fundamental concepts in physics. Then I'll use it to derive Newton's second law and later the equation for geodesics in a Riemannian manifold. If time permits, I'll discuss the Palais-Smale condition, a very important tool for proving existence of action minimizers.

*An Introduction to the Calculus of Variations*Gabriel Martins,

**University of California, Santa Cruz**

**Wednesday, October 21, 2015
** Combinatorial games are two player perfect information games with no chance elements. Familiar examples are Tic-Tac-Toe, Chess and Go. In this talk I will introduce the mathematical theory of `short’ games which, more or less, are finite combinatorial games in which the two players alternate turns and the first player unable to make a legal move loses. We’ll begin with examples to motivate a quick development of the axiomatic mathematical theory. Short games have a rich combinatorial and algebraic structure; they form a partially ordered abelian group. Finally, I’ll introduce the game Upset-Downset which is played on any finite poset and give a partial solution to a subclass of Upset-Downset games. If time permits we’ll discuss how to extend Upset-Downset to be played on arbitrary graphs.

*How to play Short Games*Charles Petersen,

**University of California, Santa Cruz**

**Wednesday, October 28, 2015**

**Equidistributed Sequences**

**Victor Bermudez, University of California, Santa Cruz**

In 1916 Hermann Weyl developed the theory of equidistributed sequences providing another proof of the Kronecker’s Approximation Theorem: the sequence formed by the fractional parts of the multiples of an irrational number, is dense in the unit interval. Equidistributed sequences appear in analytic number theory, numerical integration and functional analysis. They relate measure theory and topology in an interesting way: they show how different they are. I will focus on this aspect and, time allowing, I will mention some other results that emerge when putting equidistribution in the realm of topological groups.

**Wednesday, November 4, 2015
**

*Random Matrix Theory and Determinantal Random Point Field*

**Zheng Zhou, University of California, Santa Cruz**

The study of random matrices, in particular the properties of their eigenvalues, has emerged from the applications, first in data analysis and later as statistical models for heavy-nuclei atoms in 1950's. People are interested in the joint distributions and fluctuations of eigenvalues of specific matrix ensembles, and especially the ones with determinantal correlation kernels. I'll give some basic definitions and well-known results, as well as the fundamental methods that are used in this field.

**Wednesday, November 11, 2015 ****No Colloquium* VETERANS DAY HOLIDAY*

**Wednesday, November 18, 2015
** TBA

**Wednesday, November 25, 2015 ** ****No Colloquium* THANKSGIVING HOLIDAY*

**Wednesday, December 2, 2015**

**Introduction to Bisets and Biset Functors**

**Deniz Yilmaz**

Representation theory of finite groups associates two classical rings, the representation ring and the Burnside ring, to a given group. These rings share common properties coming from induction, inflation, conjugation, deflation, and restriction maps; there are some other objects having these properties too. Biset functors, introduced by Bouc, give a unified treatment of these objects. The aim of the talk is to introduce bisets and biset functors.