# Spring 2005 Abstracts

• January 26
Speaker - Arlo Caine
Title - Geometry and Differential Equations

Free Food! Need I say more? Okay, maybe its not quite free. You'll probably feel guilty if you take your bagel, cookie, and cup of coffee and then not stick around to hear my talk. So the small price you will have to pay is to listen to fun talk about the geometry of simple surfaces and how non-linear differential equations can naturally arise in their study. I'll try to give you intuitive way of thinking about Gaussian curvature and we'll see what happens. First and second year students in particular, this one's for you.

• February 2
Speaker - No Speaker
Title - Day of Celebration

• February 9
Speaker - Derek Habermas
Title - Rotation Systems, or How to Build a Graph a Home

Some graphs can be drawn in the plane (or, equivalently, on a sphere) without having to cross the edges. These are called planar graphs. Not all graphs are planar; for example, K_4, the complete graph on four vertices (complete means that every vertex is connected to every other vertex by exactly one edge each), is NOT a planar graph. But it CAN be drawn (with no crossing edges) on a torus! This begs the question: for a given graph, on what surfaces can it be drawn without crossing edges? We shall explore this marriage of graph theory and topology with transparencies, colored chalk, props, and bagels.

• February 16
Title - Introduction to Derived Categories

In this presentation we will look at the various concepts related to deived categories and familiarity of some of them from topology.

• February 23
Title - Derived Equivalences of Group Algebras

To study the representations of a group G over a field F, one studies the module category of the group algebra FG. The case of our interest is the one in which the group G is finite and characteristic of the field F divides the order of the group. Morita equivalence is a strong relation between two categories. It was observed, that there were groups G and H for which the module categories of FG and FH shared interesting properties even though they were not Morita equivalent. This led to the study of the notion of derived equivalence of these categories. In the late 80's, Rickard gave a constructive criterion to check derived equivalence of the module categories of FG and FH. We will discuss this criterion and illustrate how to use it by sketching a proof of derived equivalence of FA_4 and FA_5, where F is a field of characteristic 2.

• March 2
Speaker - Cam McLeman
Title - An Introduction to Kolyvagin Systems

The purpose of this talk is to define Euler systems (of which Kolyvagin systems are a special type) and motivate their study. Specifically, we will see how studying them gives strong results on the structure of the ideal class group of some special (i.e. cyclotomic) number fields. As an example of an application, these class groups give many conditions on a prime for which Fermat's last theorem fails, so we can in part view this study as a number-theoretic attempt to prove Fermat's Last Theorem without resorting to elliptic curves or modular forms.

• March 9
Title - Feigenbaum's Route to Chaos

This talk introduces the idea of a bifurcation and explains how it relates to the analysis of hydrodynamic turbulence. A map undergoes a bifurcation when its dynamics change as a result of varying one of its parameters. The period-doubling bifurcation causes a map with a fixed point to develop a period two point. After successive period-doubling bifurcations, certain maps become chaotic. One famous example is Feigenbaum's cascade. The talk concludes with a heuristic discussion of how these ideas occur in turbulence. The pure at heart should not be scared, calculus is the only prerequisite for the talk.

• March 23
Speaker - Luis Garcia Naranjo
Title - Integration of the Free Rigid Body Equations

The free rigid body is one of the most classical non-linear integrable mechanical systems. Its full discussion brings together many different areas of mathematics. In this talk I will derive Euler's equations using Euler-Poincare reduction, and I will discuss how the motion can be described as motion along geodesics of SO(3) endowed with a left invariant riemannian metric. Finally I will discuss how the solutions can be expressed in terms of elliptic funcions. This should be a good introduction to the ideas in geometric mechanics and integrability.

• March 30
Speaker - Rob Pawlowski
Title - Computing Minimal Projective Resolutions

The computation of projective resolutions of modules over rings has been a problem in ring theory and homological algebra for many years. For example, Ext-groups are defined using projective resolutions. For a finite dimensional k-algebra such as the group algebra of a finite group, we have more information which allows us to write the algebra as the quotient of the path algebra of a quiver (to be defined in the talk). The first terms of the projective resolution can be determined using quiver information. This partial resolution can be extended to a (not necessarily minimal) resolution using the Anick-Green resolution which relies upon Gröbner bases theory, higher overlaps, and differential maps which are recursively defined. To make this resolution minimal, a concept called "one-point extension will be introduced. This entire procedure is algorithmic and has been implemented by E. Green et. al in C and also has been implemented in GAP. A general overview and examples of this procedure will be given.

• April 6
Speaker - Sheldon Joyner
Title - The p-adic Riemann Zeta Function

The standard Riemann zeta function has an integral representation which can be realized as a Mellin transform. A p-adic analog can be defined in a similar fashion, viewed in an appropriate sense as a p-adic Mellin transform. In order to explain this, we describe a theory of a p-adic integration, beginning with an introduction to the p-adic numbers themselves.

• April 13
Title - The Discrete Laplacian

We consider the sum of the lower eigenvalues of the discrete Laplacian (adjacency matrix) on a finite subset of the discrete torus. I will discuss a lower bound that involves the boundary of the domain.

• April 20
Speaker - Anthony Fernandez
Title -

Teacher: Which is bigger, 2/3 or 3/4?

Middle Schol Student: Well, I guess that they are both the same size because they both have one piece missing.

As seen in the above exchange and in research, Middle school students have difficulties with fractions. In this talk I will examine different interpretations of fractions and provide examples from research. Towards the end you will work on activities that will aim at tying all the ideas discussed.

• April 27
Speaker - Jim Martin
Title - Using RefWorks to Effectively Manage your References

RefWorks is a Web-based bibliography and database manager that allows for the creation of personal databases by importing references from text files or online databases. You can use these references in writing papers and RefWorks will automatically format the paper and the bibliography in seconds. RefWorks has been licensed by the UA libraries and is available to all UA faculty students and staff from both on or off campus.

Jim Martin, Science-Engineering Librarian will provide a hands-on demonstration of RefWorks in how to get started building a personal reference database by importing BibTeX references from MathSciNet.

FYI: Registering for an individual account is recommended before the Colloquium on April 27th. To Register, go to www.library.arizona.edu, click on RefWorks at the bottom of the page At the RefWorks Login Center, and click on "sign-up for an individual account."

• May 4
Speaker - Derek Habermas
Title - Reflected Gray Codes And Their Application To At Least Three Things

Years ago in this very department, in the Applied Mathematics Graduate Program's Brown Bag seminar, a tradition was born: The tradition of making the last talk in the semester a "fun" talk. Past years' "fun" Brown Bags have seen blues musicians and armor clad warriors. This year the mantle passes to us, and we will uphold this fine tradition by playing with puzzles. We may also learn some things about Gray codes: why they were invented, what advantages they have, connections to graph theory, and how they can be used to solve puzzles like the Tower of Hanoi.