2009 Summer Graduate Student Seminar

Date: July 28, 2009
Time: 2:15-3:45 pm
Location: Room 323

Title: Nilpotent orbit in semisimple Lie algebras
Speaker #1:  Brandon Samples
Abstract: Nilpotent orbits under the action of the adjoint group on a semisimple lie algebra are studied for their interesting geometric properties as well as for their relationship with the structure of the Lie Algebra. In this talk, I will begin by introducing Lie Groups and Lie Algebras and give an example of a nilpotent orbit in the case of sl(n, C). Finally, I will end with some fundamental results that I have been learning this summer.

Title: The Party Problem
Speaker #2: Theresa Brons
Abstract: The famous "Party Problem" will be used to introduce Ramsey theory in general.  The talk will segue into a discussion of Ramsey theory on the integers by introducing van der Waerden's theorem concerning the existence of arbitrarily long monochromatic arithmetic progressions in any finite coloring of a set of consecutive integers. Later, a generalization of arithmetic progressions will be briefly considered.

Date: July 21, 2009
Time: 2:15-3:45 pm

Title: Bulls, Bears, and Snails: an Introduction to Mathematical Financial Markets Modeling
Speaker #1:  David Prager
Abstract: With the economic turmoil of the past year, financial issues
have received increased attention in scholarly and popular circles.  Over
the past 40 years, mathematical models have become one of the most
important tools for managing and navigating financial markets.  This talk
will present three of the most common models used in mathematical finance
and briefly describe some specific applications.  This will be a general
talk and should be accessible to everyone.

Title: Using an Integrate and Fire Model to simulate populations of neurons
Speaker #2: Rebecca Gaff
Abstract: This talk is about the research project I have been working on this summer with Andrew Sornborger. I will describe the model we have been using to simulate two populations of excitatory neurons which tracks the voltage, conductance, and calcium levels in each neuron. I will then talk about what we've found from this model and how the data compares to what they've seen in their experiments with zebrafish brains. Finally, I'll mention what we'd like to see from the model in the future.

Title: Hamiltonicity in Graph Products
Speaker #3: Joe Tenini
Abstract: Determining if a given graph possesses a spanning cycle is a classically difficult problem. We summarize the current results on this general problem and then restrict our attention to the class of graphs which are non-trivial Cartesian products. The talk will conclude with some interesting open problems in this area.

Date: July 14, 2009
Time: 2:15-3:45 pm

Title: Finding Intervals
Speaker #1: Stacy Musgrave
Abstract: I will give a brief introduction to Lebesgue measure and apply a corollary of the Lebesgue Differentiation Theorem from Real Analysis to prove two results related to intervals in difference and sumsets.

Title: The Happy Ending Problem
Speaker #2: Maury LeBlanc
Abstract: How many points in the plane in general position are required to guarantee that there exists a subset of 4 points that form the vertices of a convex quadrilateral? To guarantee the existence of a subset of 5 points that form the vertices of a convex pentagon?  I will discuss this problem first proposed by Esther Klein in 1935 and display proofs of known cases. This talk is very elementary and should be accessible to everyone.

Title: On the Aggregation Model of Marine Particles by Quadrature Method of Moments
Speaker #3: Louis Yang Liu 
Abstract: This talk is about a project with Adrian Burd on the aggregation model of particles in marine environment. We mainly consider the growth-aggregation process of marine particles in a mathematical model, which can be described by an integro-differential equation of size distribution from coagulation theory, and use the quadrature method of moments to solve the equation numerically by Matlab programming.

Date: July 7, 2009
Time: 2:15-3:45 pm

Title: Title: Roth's (Big) Theorem
Speaker #1: David Krumm
Abstract: In 1955 Klaus Roth proved one of the fundamental finiteness theorems in the field of diophantine geometry, and was awarded a Fields medal for it. In this talk I'll present the interesting history of the theorem,  and some applications of both Roth's theorem and a later generalization to the problem of studying integral points on varieties.

Title: Circle Packing: Rigidity and Approximation
Speaker #2: Jennifer Belton
Abstract: I will give a brief introduction to circle packing, and I will identify some of its connections with other branches of mathematics. I will focus on the relationship between circle packing and complex analysis. I will conclude with an overview of my current research.

Title: Integral sets on a circle
Speaker #3: Nham Ngo
Abstract: Solymosi proved that any planar integral set of n elements has a diameter of at least cn, for some constant c. In this talk, we will look at the Solymosi's theorem from a different point of view. This induces some interesting properties of integral sets on a circle.:

Date: June 30, 2009
Time: 2:15-3:45 pm

Title: The Jones Polynomial and Khovanov Homology
Speaker #1: Whitney Montgomery
Abstract: The Jones Polynomial is a knot invariant that was discovered in 1983. It is described by a skein relation, but has other definitions.  We will use the Kauffman bracket description to understand how Mikhail Khovanov's bigraded cohomology theory has the jones polynomial as its Euler characteristic.

Title: Edge-colorings of Kn Which Forbid Rainbow Cycles
Speaker #2: Laura Nunley
Abstract: An introduction to a paper soon-to-be-published in Utilitas Mathematica by Adam Gouge, Dean Hoffman and Peter Johnson, Laura Nunley, and Luke Paben, entitled Edge-colorings of Kn Which Forbid Rainbow Cycles. In this talk, I hope to share one of the major results of the paper: It is known that the greatest number of colors that can appear on the edges of Kn in a coloring that forbids rainbow K3’s, and thus all rainbow cycles, is n − 1. We characterize all such colorings (with n − 1 colors): for n \geq 3 the essentially different such colorings are in natural one-to-one correspondence with the full binary trees with n leafs.
Notes: http://www.math.uga.edu/~lnunley/talknotes.pdf

Title: Continued Fractions
Speaker #3: Josh Wood
Abstract:  I will define finite and infinite simple continued fractions.  I will prove some basic theorems about convergence issues in the infinite case.  We will see that continued fractions give a simple proof that the ratios of successive Fibonacci numbers converge to the golden ratio.  Finally, I will list some features that continued fractions have which make them a "natural" candidate for representing the real numbers.

Date: June 23, 2009
Time: 2:15-3:45 pm

Title: Kazhdan-Lusztig polynomials
Speaker #1: Wenjing Li
Abstract: We study the Kazhdan-Lusztig polynomials following the
fundamental paper of Kazhdan and Lusztig: Representations of Coxeter
groups and Hecke algebras. Kazhdan-Lusztig polynomials were introduced in
this paper and they have been very important in Lie theory.

Title: An introduction to algebraic curves and their moduli.
Speaker #2: Maxim Arap
Abstract: This presentation will introduce the notion of an algebraic
curve and the notion of the moduli space of smooth curves. Examples
will be the main focus of the presentation. The presentation will
assume no previous background in algebraic geometry.

Title: Knot Theory and Algebra
Speaker #3: Matt Mastin
Abstract: The fundamental question of knot theory is to classify the isotopy classes of maps from the circle into the 3-sphere. While at first glance this question may not seem algebraic, many of the tools used in the field come from algebraic constructions. We will begin with definitions and discuss Reidemeister's Theorem, the main ingredient in studying knots through algebra. The remainder of the talk will focus on constructions of knot invariants.

Date: June 16, 2009
Time: 2:15-3:45 pm

Title: Periodic Semi-Bounded Apollonian Circle Packings
Speaker #1: John Doyle
Abstract:  In this talk, I will introduce some properties of Apollonian circle packings that contain lines, or circles of curvature zero.  The "generic" case of such a packing is known as a semi-bounded packing, in which there is precisely one line.  I will then define what it means for a packing to be periodic and give a necessary condition for a semi-bounded packing to be periodic.

Title: Introduction to integral Appolonian Circle Packings
Speaker #2: Michael Berglund
Abstract: In this talk, I will introduce the basis of Apollonian Circle Packings and Some of the nice properties enjoyed by integral packings.  I will then describe characterizations of packings and some of our methods of determining which some example packings are.

Title: L^p Spaces and Lambda^p Sets
Speaker #3: Alex Rice
Abstract: The first half of the talk will be exclusively recollection for any alumnus of a graduate real analysis course, and extremely accessible to others, as we will define and discuss L^p spaces and their notation, norm, and inner product. This will provide the necessary background to discuss restriction phenomena, an important concept in harmonic analysis. The question is as follows:

We know that if A is a set of finite measure, and p>2, then f in L^p(A) implies f in L^2(A). Is there a usefully nonstringent condition we can put on f such that the converse holds?

Date: June 11, 2009
Time: 2:15-3:45 pm

Title: Structure in 1-dimensional difference sets
Speaker #1: Kate Thompson
Abstract: This talk will discuss the motivations of the Geometric
Combinatorics VIGRE group, and will provide examples of applications
of the work done to one-dimensional cases--focusing mainly on
arithmetic progressions and reproducing a result of Croot.

Title: Structure in k-dimensional difference sets
Speaker #2: Nathan Walters
Abstract: This talk will take the results of the previous talk to
higher dimensions.  We also generalize the concept of arithmetic
progressions to differences coming from polynomials, and conclude with
a general theorem (proved two weeks ago by the presenters!) for
specified structure in difference sets.

Date: June 9, 2009
Time: 2:00-3:30 pm

Title: On the parametrization of rings of low rank
Speaker: Jim Stankewicz
Abstract: We explore the ways in which we can parametrize rings which are also free modules of low rank over the integers and how this fits into a more general theory. This talk is based primarily on a set of lectures by Manjul Bhargava at the Arizona Winter School in March 2009.