Department of Mathematics
Graduate Colloquium Fall 2005

Graduate Colloquium
Fall 2005
Tuesdays, 4:35 - 5:35 PM, JWB 335
Math 6960-003
(credit hours available!)

The goal of this Colloquium is to encourage interaction among graduate students, specifically between graduate students who are actively researching a problem and those who have not yet started their research. Speakers will discuss their research or a related introductory topic on a level which should be accessible to nonspecialists. The discussions will be geared toward graduate students in the beginning of their program, but all are invited to attend. This invitation explicitly includes undergraduate students.

Talks will be held on Tuesdays at 4:35 pm in JWB 335, unless otherwise noted.

September 6:
Speaker: Jon Forde
Title: Introduction to Delay Differential Equations
When the rates of change in a differential equation depends not only on the current state of the system, but also on some past state, it is called a delay differential equation. I will introduce this type of equation, and see how the addition of a delay affects the behavior of solutions and the analysis of the equations. In general, simple delay differential equations can have quite complicated behavior, and their analysis can touch many other branches of mathematics, such as functional analysis, topology and abstract algebra.

September 13:
Speaker: Peter Trapa
Title: Representation Theory and Geometry
Decomposing a function of a real variable into sines and cosines is a familiar mathematical tool. There are similar ways of decomposing functions of several variables. But in higher dimensional situations various other "special" functions naturally arise besides sines and cosines. (For example, Bessel functions appear in applications to cylindrically symmetric problems. Spherical harmonics appear when the symmetry is spherical.) One might ask: is there a unified approach to these various special functions? The purpose of this talk is to give a positive answer using representation theory. We will also touch on a modern approach to the subject which, perhaps surprisingly, is from a purely algebraic geometric perspective.

September 20:
Speaker: Kim Montgomery
Title: Multifrequency Forcing of a Nonlinear Oscillator Model of the Inner Ear
Hair cells are auditory cells that translate sound-induced mechanical stimuli into electrical auditory nerve signals. The hair bundles of the hair cells have been shown to respond actively to stimuli near their preferred frequency. This active motion may play a role in the production of otoacoustic emissions, sounds emitted by the cochlea. I will discuss a coupled oscillator model useful in understanding otoacoustic emission data and predicting the source of otoacoustic emissions.

September 27:
Speaker: Joro Todorov
Title: Minimal Model Program
Minimal model program is one of the major discoveries in algebraic geometry and it gives an effective approach toward the classification of higher dimensional algebraic varieties. We will discuss the main ideas in minimal model program, "run it" in dimension two, and state some exiting new results in higher dimension.

October 4:
Speaker: An Le
Title: Mathematical Methods in Solving Diffusion Equations
In this talk, we will discuss diffusion equations. In a natural way, we will see how to obtain solutions to such equations. Using the integrating factor method and the eigenfuction expansion method we will establish an explicit solution formula. In spite of its technical title, the talk is comprehensible to everyone with calculus background.

October 11:
Speaker: Marian Bocea
Title: Partial Differential Equations with $L^{1}$ Data
Several issues regarding the existence of solutions for elliptic PDEs with right hand side in $L^{1}$ will be discussed. Time permitting, a number of striking recent results regarding systems of PDEs with $L^{1}$ data will also be presented.

October 18:
Speaker: Yoshihiro Iwao
Title: Additive Abelian Groups
An elliptic curve is a nonsingular curve of genus 1. The equation for such a curve can be transformed to a long Weierstrass form. One of the most important facts about elliptic curves is that the points on the curve form an (additive) abelian group. In this talk, we give the additive law of this group.

October 25:
Speaker: Alex Aue
Title: Markov Chain Monte Carlo Methods
We will discuss how popular simulation methods such as the Metropolis algorithm and the Gibbs sampler work. Theoretical results, which are based on Markov chain theory, will be explained with examples from statistical physics. Special emphasis is put to the so--called Ising model frequently used to describe the magnetization of iron.

November 1:
Speaker: John Zobitz
Title: Consistent Linear Regression
Linear regression is frequently used to determine relationships between variables and/or as an estimator of a dependent variable. The values of the regression coefficients (the slope and intercept) are heavily dependent on the setup of the regression. In this talk I will develop three commonly used models for linear regression, point out where these three models fail to be a consistent estimator, and mathematically quantify why they are not consistent.

November 8:
Speaker: Sam Isaacson
Title: Introduction to Stochastic Chemical Kinetics
Stochastic chemical kinetics provides a physical theory to account for molecular noise in the chemical reaction process. The amount of a given chemical species is no longer modeled as a continuously varying chemical concentration, but instead as an integer valued, continuous time Markov Process. We will discuss the chemical master equation, which describes the probability of having a given amount of each chemical species at a given moment in time. The Gillespie method for creating realizations of the chemical master equation will also be introduced. Time permitting, we will investigate the connection between stochastic chemical kinetics and deterministic chemical kinetics.

Note: Only basic differential equations and linear algebra is assumed. No previous knowledge of stochastic processes is needed.

November 15:
Speaker: Talk Cancelled This Week

November 22:
Speaker: Nathan Albin
Title: A Mathematical Model for Shape-memory Alloys
Shape-memory alloys (SMAs) are metals with two very interesting properties. First of all, below a certain critical temperature, the SMA exhibits pseudo-elastic properties, deforming easily almost like rubber. Secondly, when heated, the SMA returns to its original manufactured shape. Such materials have numerous applications, for example in aerospace, medicine and robotics. In this talk, we will discuss a mathematical model which describes how SMAs work.

November 29:
Speaker: Oana Veliche
Title: Acyclic and Total Acyclic Complexes
A complex is a chain of modules linked by homomorphisms such that the composition of any consecutive ones is 0. To every finitely generated module over a noetherian ring one can attach an acyclic complex of finite free modules bounded to the right, called minimal free resolution. We will discuss how the existence of "extendable to the right" free resolutions into acyclic or total acyclic complexes have an impact on the ring. We will consider several examples and present some results.

December 6:
Speaker: Zachary Kilpatrick
Title: Traveling Fronts in an Inhomogeneous Neural Network
A major simplification concerning the structure of the cortex is to assume that it is homogeneous and isotropic, with synaptic connections determined by the distance between neurons. This is reasonable at the macroscopic level, but periodic inhomogeneities modify this picture on a microscopic level. We are concerned with how this changes conditions for wave propagation. Activity waves can occur in cortex after sensory stimulation and during seizures and migraines. We analyze the Wilson-Cowan equation in 1-D using spatial averaging, perturbation methods, and the Fredholm alternative to determine the form of solutions. This places conditions on average wave speed and length scale of wave front solutions that will propagate or abort in the presence of periodic inhomogeneities.

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