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.
November 15:
Speaker:
Talk Cancelled This Week
Title:
Abstract:
November 22:
Speaker:
Nathan Albin
Title: A Mathematical Model for Shape-memory Alloys
Abstract:
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
Abstract:
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
Abstract:
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.