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 inrtoductory 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 4:30pm in JTB 130, unless otherwise noted.
 
 
 

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.

Sept. 4
Speaker:  Aaron Bertram, Graeme Milton, Cindy Phillips, and Anurag Singh
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-investgator grant. Topics that will be covered include: 
 

• why should I apply for a grant?, 
• what are my chances for success?
• how do I get started?, 
• what are the important things I should know about? 
• what strategy should I use?
• can I see some examples of successful grants? 
• what should I put in the budget?
• how do I justify the budget?
• how will my grant be reviewed?
• what are some of the things the reviewers will be looking for in my proposal?
• what happens after I hear from the program officer?

The talk will end with a discussion, with input from other faculty,
to provide other perspectives.

Click here for a summary of this talk.
 

Sept. 11
Speaker:  Peter Trapa
Title:  Representations of S_n and GL_n
Abstract:  This talk is about the interaction between the representation

theory of the group of permutations on  n  letters and the group of
$n \times n$  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.

Sept. 18
Speaker:  Fumitoshi Sato
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 16th century.  I will give a modern viewpoint of this classical material.

Sept. 25
Speaker:  Bob Guy
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 1950's.  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.

 

Oct. 2
Speaker:  Paul Roberts
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 dimension, and describe some questions which have developed from these ideas.

 

Oct. 9
Speaker:  Brynja Kohler
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 sceondary 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.

 

Oct. 16
Speaker:  Matthew Rudd
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 that elementary examples from calculus provide helpful intuition for evolution problems in more complicated situations, such as dynamical systems on Euclidean 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.

 

Oct. 23
Speaker:  Brad Peercy
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.

 
 

Oct. 30
Speaker:  Boas Erez
Title:  How to win a Nober 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 I work in.

 

Nov. 6
Speaker:  Andrej Cherkaev
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.

 

Nov. 13
Speaker:  Larsen Louder
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.

Nov. 16 (SPECIAL COLLOQUIUM)

Place and Time: 3:00-4:00  JTB 110

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, one 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.
 

Nov. 20
Speaker:  Aaron Bertram
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.

 

Nov. 27
Speaker:  Bobby Hanson 
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 these results by asking the questions, "Which graphs are intrinsically linked (knotted)?"
        I also want to advertise the other knot theory talk I will be giving that day.  I will be presenting an introduction to knot theory at the undergraduate colloquium in AEB 350 at 12:55.  This first talk requires no technical mathematics.

 

Dec. 4
Speaker:  Aaron Fogelson
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.