Graduate Student Advisory Committee (GSAC) Colloquium Schedule:

Welcome back.

We'll celebrate the beginning of a new school year by eating pizza and discussing plans for upcoming events and colloquia. Nelson Beebe will give a brief introduction to the computing facilities in the math department.

Virtual solvability is not a geometric property.

Considering the Cayley graph of a group up to quasi-isometry gives a natural way to think about the geometry of a group independent of a choice of generators. Gromov showed that being virtually nilpotent is a geometric property (meaning if a group is quasi-isometric to a virtually nilpotent group then it must be virtually nilpotent itself). This initiated a investigation as to what other group properties are geometric. This seminar will showcase an example of Dioubina's and demonstrate that virtual solvability is not a geometric property.

The Mathematics of Doodling

Start doodling on a piece of scratch paper, and before you know it you'll have discovered winding numbers, Euler characteristics and geometric probability. Come find out how, so that you can properly warn your (future) children.

Quasigroup automorphisms and the Norton-Stein complex

A quasigroup, generalizing the concept of a group, is a set with a binary operation such that multiplication by any fixed element on the left or right is a bijection. An open problem is the determination of the possible cycle types of automorphisms of finite quasigroups. We use an algebraic topological approach to eliminate a large class of impossible cycle types: namely, if d>1 is the largest power of two dividing the order of a finite quasigroup Q, we show that each automorphism of Q must contain a cycle of length not divisible by d in its disjoint cycle decomposition. The proof is obtained by considering the action induced by the automorphism on a certain orientable surface originally described by D. A. Norton and S. K. Stein. Joint work with Jonathan D. H. Smith of Iowa State University.

Groups Acting on Graphs

Groups can be described axiomatically as a set with an associative binary operation with an identity, and where every element has an inverse. But this abstract definition misses out on a wonderfully intuitive way to represent groups geometrically as symmetries of graphs. In this talk I will discuss three theorems, two by Cayley and one by Frucht. When given a group Cayley's two theorems demonstrate how to generate a symmetry group of a graph. An extension of Frucht's theorem shows that every finitely generated group is realizable as the full symmetry group of a locally finite graph. These results show that viewing groups via their actions on graphs is a powerful tool in group theory. I promise lots of pretty pictures.

Talk canceled.

There will be no colloquium this week.

Fall Break!

There will be no colloquium this week.

Lie Groups by Example

Lie Groups are a relatively natural mathematical object -- many mathematicians use them regularly, often without realizing it. The most basic examples are matrix groups such as the rotations in three dimensions and the two by two invertible real matrices. Many of the familiar tools of linear algebra are shadows of much more sophisticated results from the study of Lie groups. The goal of this talk is to introduce some of the basic tools of Lie theory in a way understandable to anyone familiar with linear algebra. It will culminate with one of the loveliest results in mathematics, the classification of Lie algebras.

Classification of Complete Discrete Valuation Rings

On the rational numbers, any nontrivial absolute value is equivalent to either the usual (Euclidean) absolute value or the p-adic absolute value. Completing the rational numbers with respect to the p-adic metric gives the p-adic numbers. In this talk, I will describe the properties of the p-adics and use this construction to motivate a discussion of some basic algebraic structures--namely, discrete valuation rings and Witt vectors--used in local class field theory.

Computation of invariants for Harish-Chandra modules of SU(p,q) by combining algebraic and geometric methods.

We will discuss the infinite dimensional representation theory of the real Lie group SU(p,q). An infinite dimensional representation gives rise to a Harish-Chandra module X. The module X is in some sense a version of the representation with improved algebraic properties. To X are attached various geometric objects: an associated variety, an associated cycle and various Springer fibers. Geometric and algebraic methods are combined to compute specific invariants for X. This talk will emphasize exposition and will target a general mathematical audience.

Page's CUSUM for changes in linear models

Has there been a change in the growth of plants or animals after an ecological catastrophe? Are certain models for the calculation of asset prices or derivatives still valid after a shock in the market? Those and more questions motivate the work in the field of change-point analysis. In this talk a linear model is considered which has applications in a whole variety of research fields. We will present a method for the detection of abrupt parameter changes in this linear model that is based on an idea of E.S. Page that was already published in 1954 but hasn't been used in this context so far. After the construction of this method it will be compared to an already existing approach in a small simulation study.

Extreme Value Asymptotics under Strong Mixing Conditions

One main issue of change-point analysis in econometrics is to test wether the volatility is stable or if it changes in the observation period. This testing problem requires certain extreme value asymptotics for weighted test statistics of dependent random variables. In this talk I will present a Darling-Erdos type limit theorem for standardized tied-down partial sums of mixing random variables. Various examples of random sequences which are strongly mixing or even absolutely regular are presented, too.

A Derivation of the Navier-Stokes Equations

The Navier-Stokes Equations are some of the most commonly used equations in fluid dynamics. They can be used to study the atmosphere, ocean currents, air flow around a wing, blood flow, and other applications. Extensions of the Navier-Stokes Equations can be used to study things such as two-phase fluid flow. This talk will follow a derivation of the Navier-Stokes Equations starting from the basic ideas of conservation of mass and conservation of momentum. Along the way we will introduce the ideas of material derivatives and stress tensors, and will discuss the simplifications based on assumptions about the fluid that lead to the Navier-Stokes Equations in their commonly seen form.

Talk canceled.

There will be no colloquium this week.

El Farol Bar and Game Theory

In this talk, I will present the El Farol Bar problem and the necessary game theory terminology to understand certain aspects. Examples of combinatorial games and strategic forms of games will be provided to contrast the difference to the El Farol Bar problem. A brief description of the Minority Game will wrap up the discussion.

Points on Algebraic Varieties

We give a concrete introduction to the functor of points of an algebraic variety. No background in algebraic geometry will be assumed.