Graduate Colloquium Fall 2005

Fall 2005

Tuesdays, 4:35 - 5:35 PM, JWB 335

Math 6960-003

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

**Abstract:**

*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

**Abstract:**

*
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

**Abstract:**

*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

**Abstract:**

*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

**Abstract:**

*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

**Abstract:**

*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

**Abstract:**

*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

**Abstract:**

*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

**Abstract:**

*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

**Abstract:**

*
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.
*

155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, T:+1 801 581 6851, F:+1 801 581 4148