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Graduate Colloquium 2001 - 2002

August 27, 2001
Speaker: Eric Cytrynbaum (Graduate Student)
Title: Periodic Stimulus and the Single Cell
Abstract: The response of an isolated cardiac cell to a periodic stimulus is traditionally studied in terms of the duration of the action potential (APD) immediately following each stimulus. Experiments demonstrate that the rhythms generated by such stimulus protocols can vary widely, depending on frequency and amplitude of the stimulus. A theoretical approach based on the APD is capable of explaining some of the experimental observations but a complete understanding of the parameter dependence is beyond the scope of the APD approach. We discuss some fundamental problems with the APD approach and propose a new one-dimensional map that relies on the presence of a one-dimensional slow manifold in the dynamics. This slow manifold map extends the understanding offered by the APD approach to include an explanation of Wenckebach rhythms. In addition, the bifurcation structure of the map provides a unified description of the parameter dependence that agrees fairly well with experimental observation.

Some familiarity with two-dimensional ODEs and one-dimensional maps will be helpful.

Click here to view overheads and a movie for this talk.

September 4, 2001
Speakers: Aaron Bertram (Professor of Mathematics), Graeme Milton (Professor of Mathematics), Cindi Phillips (Mathematics Department Accountant), and Anurag Singh (Graduate Student)
Title: Preparing a Successful Grant Proposal
Abstract: This talk is for graduate students who may be interested in getting some perspective on the grant process, for associate professors who may be applying for a grant for the first time, and for other interested faculty. The focus will be on applying for a regular NSF individual investigator mathematics research grant, rather than on a large multi-investigator grant. Topics that will be covered include: The talk will end with a discussion, with input from other faculty, to provide other perspectives. Click here for a summary of the talk.

September 11, 2001
Speaker: Peter Trapa (Professor of Mathematics)
Title: Representations of Sn and GLn
Abstract: This talk is about the interaction between the representation theory of the group of permutations on n letters and the group of n x n invertible real matrices. For finite dimensional representations of GL(n, R), it's completely elementary and dates back to Schur. For infinite dimensional representations, the interaction is of a completely different (and more sophisticated) flavor. It involves the geometry of the desingularization of the variety of n-dimensional nilpotent matrices.

September 18, 2001
Speaker: Fumitoshi Sato (Graduate Student)
Title: Symmetric Functions
Abstract: This talk is about symmetric functions which are the generalization of symmetric polynomials. The theory of symmetric functions is one of the most classical parts of algebra, going back to the 16th century. I will give a modern viewpoint of this classical material.

September 25, 2001
Speaker: Robert Guy (VIGRE Graduate Fellow)
Title: The Stokes Paradox
Abstract: Many classical problems in fluid dynamics motivated the development of a branch of mathematics called perturbation theory. One such classic problem is to find the force acting on a solid body moving in a fluid, given the velocity of the body.

From a reduction of the equations governing viscous fluid flow, in 1851, Stokes solved for the velocity field around a steady sphere with a uniform flow at infinity. From this, the force on the sphere can be computed, giving the famous Stokes relation. However, a solution does not exist in two dimensions. This is known as the Stokes Paradox. In 1889, Whitehead tried to improve on the Stokes relation and found a correction did not exist in 3D, similar to the problem encountered in 2D. This became known as the Whitehead paradox. The problem was temporarily solved by Oseen in 1910 and Lamb in 1911, but the reasoning used was questionable. The problem was not resolved until the late 1950s. The solution requires formal perturbation theory, specifically matched asymptotic expansions and matching via an intermediate scale.

I will go over the conceptual ideas of the problem and the reasoning behind the various attempts at solutions. I will use this to introduce ideas from perturbation theory, and demonstrate them on a simplified model problem. The level of the talk will be appropriate for first and second year graduate students.

October 2, 2001
Speaker: Paul Roberts (Professor of Mathematics)
Title: Intersection Theory and Commutative Algebra
Abstract: One of the basic problems in Intersection Theory is how to define the intersection multiplicity of two algebraic sets which meet at a point. The definition should give a measure of the order of tangency at the point and generalize the classical definitions for intersections of curves in the plane. Attempts to define intersection multiplicities in a general algebraic setting have led to several interesting conjectures and questions in Commutative Algebra. In this talk, I will discuss intersections of curves in the plane and what properties the multiplicity should satisfy, show how they can be generalized higher dimensions, and describe some questions which have developed from these ideas.

October 9, 2001
Speaker: Brynja Kohler (Graduate Student)
Title: An Exploration of Immunological Memory through Mathematical Models
Abstract: After an initial priming infection, the immune system of an individual changes in such a way that the immune response to a secondary infection by the same pathogen will be faster and more efficient. This adaptive feature of memory in the immune system is the basis for vaccination. In this talk, I will discuss some aspects of immunological memory that have been explored through mathematical modeling.

October 16, 2001
Speaker: Matthew Rudd (VIGRE Graduate Fellow)
Title: An Introduction to Evolution Equations
Abstract: Evolution equations occur throughout mathematics, both as useful tools and as fundamental objects of study. We will see the elementary examples from calculus provide helpful intuition for evolution problems in more complicated situations, such as dynamical systems on Euclidian space, periodic orbits on compact manifolds, and partial differential equations. We will use some concrete examples to illustrate the ideas common to these different settings.

October 23, 2001
Speaker: Brad Peercy (Graduate Fellow)
Title: Some Basics of Bifurcation Theory and an Application of a Hopf Bifurcation in Cardiac Arrhythmias
Abstract: Last week, Matt Rudd talked about evolution equations. I will begin with his basic evolution equation (u' = ku, u(0) = u0) and discuss what happens under parameter variations including which parameter variations are bifurcations. I will then discuss what a bifurcation is in general and consider slightly more complicated systems. Specifically, I will develop the Hopf bifurcation. I will end with an application of the Hopf bifurcation to cardiac arrhythmias. For those of you who find the study of modeling cardiac arrhythmias distasteful, I will not spend any time on the modeling aspect of the system I will present.

October 30, 2001
Speaker: Boas Erez (Professor of Mathematics)
Title: How to Win a Nobel Prize in Mathematics
Abstract: The Nobel prize celebrates its 100th anniversary this year. No mathematician has ever won this prize for work in mathematics as such. Using this observation as a starting point, I shall discuss several questions related to the practice of mathematics that might be of interest to anyone interested in understanding what it is to do research in mathematics. In the course of the discussion, I shall give examples of what I consider to be great achievements in the fields in which I work.

November 6, 2001
Speaker: Andrej Cherkaev (Professor of Mathematics)
Title: Control of Damage and Dissipation in Structures
Abstract: When a construction fails, most of its material is still in perfect shape: The failure is related to instabilities in the materials. The resistance of a design can be significantly increased if we could control the process of damage by using special structures. We discuss the underlying principles of increasing the stability and some corresponding protective structures that realize these principles. Namely, we show how to fairly distribute a "partial damage" over the structure and how to increase the "wave resistance" that is the excitation of high-speed waves which radiate and dissipate the energy of the impact.

November 13, 2001
Speaker: Larsen Louder (VIGRE Graduate Fellow)
Title: An Introduction to Hyperbolic Geometry With Some Neat Applications
Abstract: What does an automorphism of a surface "do"? any surface of genus greater than one can be given a hyperbolic structure. After going over the basic notions of hyperbolic geometry, we'll learn how to construct hyperbolic structures on surfaces and see what the space of all such structures looks like. With that completed, we'll try to answer the question above.

November 16, 2001 (Special Colloquium)
Speaker: Carl Cowen, Professor of Mathematics, Purdue University
Title: Rearranging the Alternating Harmonic Series
Note: Professor Cowen is particularly interested in meeting with students informally before and after the talk.
Abstract: The commutative property of addition is so familiar to all of us as school children that it comes as a shock to those studying college level mathematics that NOT all 'natural extensions' of the law are true! One of the first instances that we see the failure of an extended commutative law of addition is in infinite series. Often in the introduction to infinite series in calculus, on sees
Riemann's Theorem:
A conditionally convergent series can be rearranged to sum to any number.

Unfortunately, the usual proof of this theorem does not indicate what the sum of a given rearrangement is. In this talk, we will examine the best known conditionally convergent series, the alternating harmonic series, and show how to find the sum of any rearrangement in which the positive terms and the negative terms are each in their usual order.

November 20, 2001
Speaker: Aaron Bertram (Professor of Mathematics)
Title: Algebraic Curves: What are they and what are they good for?
Abstract: What is an algebraic curve? Well, that depends upon your "ground field". If you like the complex numbers, an algebraic curve is a Riemann surface, which is, topologically, an oriented compact manifold of dimension two (a many-holed torus). When the ground field is finite, an algebraic curve resembles the ring of integers in a number field-a number-theoretic object. And when the ground field is the rational numbers, then the elliptic curves (one-holed tori) become fascinating for their unpredictable numbers of rational points. I'd like to touch on all of these, explain a little bit of history, and explain some of their practical uses.

November 27, 2001
Speaker: Bobby Hanson (Graduate Student)
Title: This Graph is Tangled!
Abstract: About 20 years ago, John H. Conway and Cameron Gordon were able to prove that every embedding of the complete graph on six vertices into three-space has a pair of linked triangles and that every embedding of the complete graph on seven vertices has a knotted Hamiltonian cycle. We say that graphs of this type are intrinsically linked or intrinsically knotted. We will prove the first result above, and briefly outline the proof of the second. We will also try to generalize the results by asking the questions, "Which graphs are intrinsically linked (knotted)?"

December 4, 2001
Speaker: Aaron Fogelson (Professor of Mathematics)
Title: Mathematical Explorations of Blood Clotting
Abstract: Thrombosis is the formation of clots within blood vessels and is the immediate cause of most heart attacks and many other severe cardiovascular problems. The main components of this process are platelet aggregation and coagulation. Platelet aggregation involves processes of cell-cell and cell-substrate adhesion and cell signaling and response, all within the moving blood. Coagulation involves a tightly-regulated network of enzyme reactions with the important feature that many of the key reactions occur on surfaces (e.g. platelet surfaces), not in the bulk fluid, while transport of the reactants occurs in the fluid. Coagulation results in the formation of a polymer mesh around the aggregating platelets. I will introduce listeners to the biology of blood clotting , will outline several aspects of our efforts to model these complex dynamic biological systems, and will discuss some of the modeling and computational challenges in this work. Gory pictures and movies will be shown.

March 5, 2002
Speaker: Graeme Milton (Professor of Mathematics)
Title: A Variety of Problems in Composite Materials
Abstract: In this lecture, I will discuss a variety of problems that currently interest me. One is creep in composites; a second is extrapolating measurements of the real and imaginary parts of the dielectric constant beyond the measured frequency interval; a third is exploring the possible stress-strain pairs in linear composites (this work was initiated with Sergey Serkov and Sasha Movchan); and a fourth is exploring the properties of a whole new class of composites called partial differential microstructures. These problems involve a variety of mathematical tools and many open questions remain.

March 12, 2002
Speaker: Jesse Ratzkin (VIGRE Assistant Professor)
Title: A (Brief) Survey of Constant Mean Curvature Surfaces in Euclidean Space
Abstract: Constant mean curvature surfaces model a film dividing two regions of space with a pressure difference across the interface. (No pressure difference corresponds to zero mean curvature.) People have studied these surfaces for over 200 years (going back to Gauss and Lagrange), yet they remain somewhat mysterious. I will begin this talk by explaining several notions of mean curvature and some of the basic tools one uses to study constant mean curvature surfaces (particularly Alexandrov reflection). During the reminder of the talk, I will survey some of the existence and classification results of the last 10-15 years. Hopefully, this lecture will be accessible to anyone familiar with vector calculus and some basic geometry.

March 19, 2002
Speaker: Kenneth Chu (Graduate Student)
Title: The Good and Evil of the Axiom of Choice
Abstract: The Axiom of Choice (AC) is undeniably the most (well, the only) controversial axiom of Set Theory, the foundation of modern mathematics. In this talk, there are three points I would like to make: Personally, I find it mildly interested to be "AC-alert", yet I am starting to feel that I am being perceived as "AC-lunatic" instead. I am planning on using this talk to vindicate myself (it might very well do the exact opposite). Come see for yourself; it is going to be fun.

March 26, 2002
Speaker: Mladen Bestvina (Professor of Mathematics)
Title: How to Average Trees
Abstract: I will describe the recent work of Billera (combinatorialist), Holmes (biologist), and Vogtmann (topologist) on the following problem from biology: Genetic relationships between species can be described in terms of phylogenetic* trees - these are (usually) binary trees with leaves representing species. Various ambiguities lead to several candidate trees and the problem is to "average" them to a single tree.

The set of possible trees has the structure of a metric space which happens to be nonpositively curved. In spaces of nonpositive curvature, there is a good notion of a "centroid" of a set of points, and in the space of trees, it can be found by a fast algorithm that provides a solution (according to the biologists) to the above problem.

This space of trees is also closely related to certain polyhedra called associahedra studied 40 years ago by Stasheff. The vertices correspond to different ways of parenthesizing the expression 1 + 2 +...+ n and their cardinality is a Catalan number.

*All Greek and Latin words will be defined upon request.

April 2, 2002
Speaker: Javier Fernandez (VIGRE Assistant Professor)
Title: A Window into Mirror Symmetry
Abstract: Enumerative geometry is a classical subject in algebraic geometry that, roughly speaking, deals with counting the number of geometric objects satisfying some conditions. For example, we may want to find the number of lines (or quadrics or cubics or...) through a given number of points.

As can be expected, some problems have been solved in ancient times, while some others are still unsolved. Surprisingly, around 1990, a new insight to many problems that had been intractable came from physics. A conjectural theory known as "mirror symmetry" allowed the computation (even though not at a mathematically satisfying level of rigor) of many enumerative objects.

In this talk, I will try to sketch some of the ideas that underlie this fascinating interplay between theoretical physics and geometry. I will assume no technical knowledge and most formulas have been cut out in order to make the material understandable to all graduate students.

April 9, 2002
Speaker: Florian Enescu (Professor of Mathematics)
Title: An Incursion Into Commutative Algebra on the Footsteps of a Simple Example
Abstract: Consider f, g, h three polynomials in two variables, with complex coefficients. It is true that there exist a, b, c polynomials in two variables with complex coefficients such that (fgh)2 = a*f3 + b*g3 + c*h3. The only proof I know uses some nontrivial commutative algebra techniques. I will use this elegant example to talk about some standard objects in commutative algebra such as local rings, localization, systems of parameters, dimension, integral closure, etc. The talk will illustrate how commutative algebra can sometimes be used in establishing elementary statements as the one above.

April 16, 2002
Speaker: Grigory Mikhalkin (Professor of Mathematics)
Title: What is Tropical Algebraic Geometry?
Abstract: Consider the set of real numbers with two operations, taking the maximum for addition and taking the sum for multiplication. This is an example of a Tropical Semiring. (The name "Tropical" was given by French computer scientists in the honor of the Brazilian mathematician Imre Simon who pioneered the subject.) Note that 1 + 1 = 1 tropically.

The talk is devoted to the geometric objects described by tropical equations. It turns out that the tropical objects are handier than their classical algebro-geometric counterparts. For instance, tropical algebraic curves are graphs. A curious fact is that, even though this geometry is based only on a semiring, it resembles in many ways complex algebraic geometry (which is based on an algebraically closed field).

April 23, 2002
Speaker: Mark Avery (Graduate Student)
Title: The American Option Valuation Problem
Abstract: Suppose you own one share of Microsoft stock and speculate that the price will go down. One way of safeguarding your investment is to purchase an (American) put option, which gives you the right to sell your stock for some prescribed price at any time before a certain date. How much would you be willing to pay for the option, and when is it optimal to exercise it?

First, we will derive the famous Black-Scholes formula for pricing European put options. Second, we will show how the American option problem can be formulated as a free boundary problem. Lastly, we will reformulate the problem as a linear complimentarity problem and solve it numerically using the projected SOR algorithm. [an error occurred while processing this directive]