Department of Mathematics
Graduate Colloquium: Spring 2007

Graduate Colloquium
Spring 2007
Tuesdays, 4:35 - 5:35 PM, JWB 335
Math 6960-001
(credit hours available!)

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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.

January 9
No Talk

January 16
Speaker: Talk Cancelled This Week

January 23
Speaker: Mladen Bestvina
Title: Scissors Congruence and Hilbert's Third Problem
It was known to Euclid that any two polygons with equal areas can be obtained from one another by cutting up into smaller polygons and rearranging the pieces. The analogous problem for polyhedra made the famous Hilbert's list of outstanding problems in mathematics (1900). It was solved in the negative by Max Dehn shortly thereafter. I will explain Dehn's solution. The solution involves tensor products, and I will give a crash course on this.

January 30
Speaker: Elijah Newren
Title: Investigating the Numerical Stability of the Immersed Boundary method
The Immersed Boundary method is a method for solving fluid-structure interaction equations, in particular the coupled equations of motion of a viscous, incompressible fluid, and one or more massless, elastic surfaces or objects immeresed in the fluid. It has been applied in a wide variety of problems, from modelling blood flow in the heart to mechanical properties of cells to insect flight. However, it has suffered from a severe (and unknown) timestep restriction needed to maintain stability. Much effort has been expended to alleviate this restriction, including the development of various implicit methods, but the problem has remained and is not well understood. In this talk, I will discuss this problem and the common beliefs and conjectures in the IB community about the causes of instability, show that those beliefs and conjectures turned out to be misleading, and present an unconditionally stable discretization.

February 6
Speaker: Jeremy Pecharich
Title: Ricci Flow and Surfaces
In the 1980's Richard Hamilton developed the Ricci flow equations for the metric on a manifold. Using these equations he proved that any compact 3-manifold with postive Ricci curvature is topologically a 3-sphere or a quotient of a 3-sphere by a finite group. This is close to the Poincare conjecture but lacks any information about the fundamental group. In 2003, G. Perelman used the Ricci flow to prove the Poincare Conjecture. The techniques and analysis of his argument are very difficult but the fundamental ideas are much easier to understand on surfaces.
We will show using the Ricci flow a very classical result in geometry, that every surface has a canonical geometry. If time permits we will also discuss some of the difficulties that arise in three dimensions and the advances that Perelman made in Ricci flow. No background in geometry or topology will be necessary since no proofs will be presented. The only prerequisite should be a good knowledge of undergraduate analysis.

February 13
Speaker: Tommasso Centeleghe
Title: Primes of the form X^2+nY^2 and Ideal Class Group
In 1640 Fermat claimed to have a proof of the fact that an odd prime p is equal to the sum of two squares if and only if p is congruent to 1 modulo 4. Few years later he gave similar statements for primes of the form x^2+2y^2 and x^2+3y^2, where the moduli involved in the congruences are, respectively, 8 and 3. In the next century Euler became interested in Fermat's assertions and it took forty years to him to find solid proofs of those. Euler was the first who sistematically studied the problem of writing a prime number p in the form x^2+ny^2 and he had the first insight of the Quadratic Reciprocity Law, proved later by Gauss. Other matematicians, such as Lagrange, Legendre, Dirichlet and Gauss, studied the related problem of classifying binary quadratic forms with integer coefficients under the action of GL_2(Z) and they realized what is the obstruction to the existence of statements similar to the ones Fermat gave for n>3 (this is what it is called Ideal Class Group). Nowadays the complete answer to the problem of representing a prime number p into the form x^2+ny^2 is provided by Class Field Theory which also underlines a beautiful link to the theory of Complex Multiplication of elliptic curves. In this talk we won't explain the full solution of the problem but rather we will give a modern proof of Fermat assertions and give a description of the class groups involved by means of integral conjugacy classes of certain two by two matrices with integer coefficients.

February 20
Speaker: Peter Trapa
Title: Canonical forms of nilpotent matrices
A useful theorem from linear algebra is the existence of the Jordan Canonical Form of a nilpotent matrix. It says that any n-by-n matrix X such that some power of X is the zero matrix can be conjugated into a matrix with zeros everywhere, except for some collection of ones just above the diagonal. Consider the following generalization. Fix integers n_1,..., n_k, set n_{k+1} = n_1, and fix X_1,..., X_k with each X_i an (n_i)-by-(n_{i+1}) matrix. Suppose further that some power of the product X_1 X_2...X_k is the zero matrix. (Notice that if k = 1 and n = n1, we recover the original setting above.) Then one can ask: is there a generalization of Jordan Canonical Form for every such sequence of matrices? (There is a natural notion of "conjugacy" in the general setting that I'll leave for you to define.) It turns out that such a canonical form exists, and that this form appears in unexcepted places. I'll finish by explaining a few of them.

February 27
Speaker: Talk Cancelled This Week

March 6
Speaker: Jon Forde
Title: Applications of Delays
Things are not always as simple as we applied mathematicians might like. For instance, the dynamics of a situation are not always instantaneous, and we must therefore take delays into account, even though it makes our lives as analysts more difficult. We will discuss several situations from physics and biology in which a delay might be necessary in a mathematical model. Then we will see how to implement such a delay and explore the consequences with some concrete models, simulations and analysis.

March 13
Speaker: Bob Guy
Title: Traveling waves and shocks in a viscoelastic generalization of Burgers equation
In many physical systems there are conserved quantities such as mass, energy, and momentum. In mathematical models of these systems the equations that govern these quantities are called conservation laws, which are often quasilinear, hyperbolic PDEs. The solutions to these systems can develop discontinuities (shocks) in finite time from smooth initial data. To make mathematical sense of shocks, the definition of solution must be expanded to include weak solutions. Another way to handle shocks is to regularize the problem by adding a small viscosity. This prevents shocks from forming, and limits to the discontinuous solution as the viscosity goes to zero. These ideas are illustrated using Burgers equation, u_{t} + u u_{x}=0, as a model problem. We then consider what happens if we add a small viscoelastic stress rather than a viscous stress to Burgers equation, and some unexpected behaviors emerge. With the addition of viscoelasticity, the system is no longer conservative, and so to analyze the behavior we use techniques from singular perturbation theory. This is joint work with V. Camacho and J. Jacobsen of Harvey Mudd College.

March 20
Spring Break

March 27
Speaker: Giao Huynh
Title: Disease Dynamic - Competition and Evolution
In this talk I will present disease competition models. I will first briefly introduce different types of disease dynamics. Then SIR model is mainly used to analyze different aspects of competitions between diseases, such as with/without super infection. Competition between diseases can be looked at as an evolutionary game. The questions here are: who will win the game? who will coexist? what are the tradeoffs? I will talk about different tools that are used to analyze the models which will help to address these questions.

April 3
Speaker: Talk Cancelled This Week

April 10
Speaker: Yael Algom-Kfir
Title: Tilings of the Hyperbolic Plan and their Applications to the Word Problem
A presentation of a group G is a description of the elements in G as words in a fixed alphabet a, b, c... (called generators) with the equivalence relation that w=w' if they differ by a finite sequence of relations: r_1=1, r_2=1, ...

In 1912, Max Dehn posed the following question known as the word problem: Given a finite group presentation is there an algorithm to decide if a given word in the generators is the identity in G. In the same paper, Dehn showed that there's an elegant solution to this problem for a particular class of groups called surface groups.

To demonstrate Dehn's algorithm, we will analyze a particular tiling of the plane and show that the group of the two holed torus has a solvable word problem. If time permits, we'll discuss directions in geometric group theory which evolved from these ideas.

Hopefully, the talk will be accessible to anyone, no background required.

April 17 - Distinguished Lecture
Speaker: William Casselman, University of British Columbia
Title: An Illustrated History of Written Numbers
This talk will present some snapshots from the history of numerals, mostly photographs of various systems, dating from Sumer (3400 B.C.) to the early Renaissance.

April 24
End of Year Meeting

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