Graduate Colloquium: Spring 2007

Spring 2007

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

Math 6960-001

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

**Title:**

**Abstract:**

*
*

**January 23**

**Speaker: **
Mladen Bestvina

**Title:** Scissors Congruence and Hilbert's Third Problem

**Abstract:**

**January 30**

**Speaker: **
Elijah Newren

**Title:** Investigating the Numerical Stability of the Immersed Boundary
method

**Abstract:**

**February 6**

**Speaker: **
Jeremy Pecharich

**Title:** Ricci Flow and Surfaces

**Abstract:**

**February 13**

**Speaker: **
Tommasso Centeleghe

**Title:** Primes of the form X^2+nY^2 and Ideal Class Group

**Abstract:**

**February 20**

**Speaker: **
Peter Trapa

**Title:** Canonical forms of nilpotent matrices

**Abstract:**

**February 27**

**Speaker: **
Talk Cancelled This Week

**Title:**

**Abstract:**

*
*

**March 6**

**Speaker: **
Jon Forde

**Title:** Applications of Delays

**Abstract:**

**March 13**

**Speaker: **
Bob Guy

**Title:** Traveling waves and shocks in a viscoelastic generalization of Burgers equation

**Abstract:**

**March 20**

**Spring Break **

**March 27**

**Speaker: **
Giao Huynh

**Title:** Disease Dynamic - Competition and Evolution

**Abstract:**

**April 3**

**Speaker: **
Talk Cancelled This Week

**Title:**

**Abstract:**

*
*

**April 10**

**Speaker: **
Yael Algom-Kfir

**Title:** Tilings of the Hyperbolic Plan and their Applications to the Word Problem

**Abstract:**

**April 17 - Distinguished Lecture**

**Speaker: **
William Casselman, University of British Columbia

**Title:** An Illustrated History of Written Numbers

**Abstract:**

**April 24**

**End of Year Meeting**

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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