Department of Mathematics
Graduate Colloquium Spring 2006

Graduate Colloquium
Spring 2006
Tuesdays, 4:35 - 5:35 PM, JWB 335
Math 6960-001, Class Number 4665

(register for the appropriate number of units)

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.

January 24:
Speaker: Qiang Song
Title: What's a stack?
This talk is intended to be an introduction for everyone to the notion of an algebraic stack, which was first developed by A.Grothendieck and M.Artin in 1960s in order to parameterize geometric objects varying in families. It's said that now the stacks are outgrowing their basic function as a convenient framework and language for studying moduli and are becoming an essential tool in everyday research. However, it's not an easy concept to digest and it will take a lot of time to set up those "spaces" in one's mind. As for me, I like the notion because although I don't have much geometric intuition, learning stacks can help me study the very interesting subject-- the moduli space of curves. (It gives me rigorous proofs!)

If you are not interested in the moduli problems, you don't need worry since just basic topology and algebra is assumed and you may get some new ideas about the meaning of "space" after the talk.

January 31:
Speaker: Bob Palais
Title: DNA and SU(2)
A bit of applied math (genotyping and mutation scanning using high-resolution DNA melting analysis) and a bit of pure math (the unit quaternions, SU(2)) and and maybe even a bit of overlap between these two topics.

February 7:
Speaker: Jason Preszler
Title: Modular Forms and Number Theory
We will define modular forms as functions possessing nice symmetry under the action of fractional linear transformations on the complex plane. After seeing several examples, the relationships between these functions and other objects (Galois representations, elliptic curves, etc.) will be explored. Some highlights of these relationships are the Eichler-Shimura construction, Shimura-Taniyama Modularity theorem (including Fermat's last theorem), and conjectures of Serre and Fontaine-Mazur.

February 14:
Speaker: Jingyi Zhu
Title: Markov Chain and Credit Rating Transition
The recent downgrade of Ford's credit rating by S&P was followed by the announcement of massive restructuring (layoffs). What most of the public did not realize is that large amounts of money changed hands among investors as a consequence of the downgrade. As this example illustrates, modeling and understanding the dynamics of credit rating are critical for the financial market and the economy in general. The mathematical model for rating transition is based on Markov chains, and we in particular develop a partial integro-differential equation (PIDE) formulation for the general Levy process, which includes both Brownian motion and Poisson jumps, with variable coefficients in regions separated by time-dependent barriers. Each region corresponds to a particular rating and an upgrade/downgrade of the rating is triggered by barrier crossings. Model calibration requires solutions of an inverse problem and we provide results generated from historical and market data, with a glimpse of the generator that gives a better insight into the dynamics of the transition.

February 21:
Speaker: Nessy Tania and Liz Doman
Title: Propagation of Calcium Waves in Cardiac Cells and Dynamics of the Electric Field Mechanism
Calcium plays an important role in muscle contraction. Tight regulation of calcium release is particularly necessary in cardiac muscle in order for blood to be pumped effectively. Recent evidence suggests that many disturbances in heart rhythm, cardiac arrhythmias, are due to abnormal calcium handling in cardiac cell. In particular, calcium waves in cardiac ventricle cell is found to be pathological. In this talk, we will explore the nature of excitation-contraction coupling in healthy cells through the calcium induced calcium release mechanism. Following that, we will study the nature of wave propagation in deterministic and stochastic models of cardiac cell strand with discrete release sites.

Cardiac cells are excitable, which means that their membranes can undergo action potentials. As excitation spreads from cell to cell, over the myocardium, the muscle contracts and the heart pumps blood. In this talk, I will discuss the mechanisms by which action potentials are able to spread from one cell to the next. In particular, propagation is possible due to gap junction channels which connect the intracellular space of neighboring cells. These channels act as linear resistors to the spread of intracellular current, which is mathematically convenient. However, I will also discuss the possibility of a secondary mechanism for propagation. The "Electric Field Effect" was first proposed in the 50s by Nicholas Sperelakis, who suggested that propagation can occur via an electric field which forms in the narrow junctional cleft between neighboring cells. To better understand this mechanism, we can make a quasi-steady state approximation and study the dynamics of the system.

February 28:
Speaker: Christopher Hacon
Title: Rational and Unirational varieties
In this talk I will illustrate several ideas from birational algebraic geometry. I will then show how some of the geometric techniques can be used to study interesting problems in Algebra.

March 7:
No Talk

March 14:
No Talk

March 21:
Speaker: Sarah Kitchen
Title: Hilbert's 3rd Problem
Any plane polygon can be cut into smaller polygons and rearranged into any other plane polygon of the same area. In Hilbert's 1900 address, he posed the question as to whether the same relationship exists between polytopes (3-dim analogue to polygons) of the same volume. We will see Dehn's proof that it does not, and how related conjectures may provide some insight to understanding hyperbolic 3-manifolds.

March 27:
Note: Special Room and Time--JTB 120, 12:55 PM
Recruiting Weekend Grab Bag
Speaker: Jason Preszler
Title: Lifting a Representation
We will lift an explicit mod 3 representation to a 3-adic representation. We will then see how this example is part of a much larger problem and the more general theory of deformations of Galois representations.

Speaker: Erin Chamberlain
Title: Intersection Multiplicities and the Vanishing Theorem

Speaker: Russ Richins
Title: The Calculus of Variations in L^\infty
The calculus of variations is a very useful way to relate a PDE to the minimization of an integral functional, and vice versa. Usually this minimization is done over the space W^{1,p} for 1 < p < \infty. Some recent work has been done in the area of minimizing L^\infty functionals, which instead of integrals involve suprema. We will look at the simple example of the \infty-Laplace equation.

March 28:
Speaker: Russ Richins
Title: Viscosity Solutions to Second Order PDE
It often happens that we cannot expect a C^2 solution to a given PDE, and so we are forced to admit solutions in some weak sense. One of the most familliar methods for obtainig weak solutions is the use of the Lax-Milgram theorem. We will discuss another idea of weak solution, called viscosity solution, that is based on comparison with C^2 functions and will work for a large class of second order PDE, which includes some fully nonlinear PDE.

April 4:
Speaker: Karin Leiderman
Title: From Coulomb to Stokes: Potential Theory Applied to Fluid Mechanics
The application of integral equtions to formulate the fundamental boundary -value problems of potential theory dates back to the days of Fredholm in the early 1900's. In this talk I will show how to formulate an integral representation of the solution to Laplace's equation in a bounded domain. I will use this formulation to derive a boundary integral equation for the unknown values on the boundary. This boundary integral equation will be a Fredholm integral equation of the first or second kind depending on a Dirichlet or Neumann problem. Once the boundary values are found, the solution at any point in the domain is easily produced. This method effectively reduces the dimension of the solution space. From there I will show that this method can be applied to find the solution to Stokes equations by deriving a boundary integral equation from an integral representation analogous to Green's third identity for harmonic functions.

April 11:
Speaker: Yael Algom Kfir
Title: Rubik's Cube
The Magic Cube was invented in 1974 by Ern Rubik, a Hungarian sculptor and professor of architecture with an interest in geometry and the study of three dimensional forms. The first test batches of the product were produced in late 1977, and released to Budapest toy shops. Soon, over one hundred million cubes were sold all over the world. The beauty in the puzzle lies in the contrast between the sheer impossibility of solving it by chance (the cube has 43,252,003,274,489,856,000 different states), and the exsitence of simple and elegant algorithms to solve it.

Rotating any face of the cube changes the position and orientation of the little pieces comprising it. These rotations generate a group we shall call "Rubik's Group", where two actions are equivalent if their effects on the solved cube are identical. Our goal is to analyse the structure of this group.

As a warmup, what is wrong with this link?

April 18:
Speaker: Ken Chu
Title: The Good and Evil of the Axiom of Choice
The Axiom of Choice (AC) is undeniably the most (well, the only) controversial axiom of Set Theory, the foundation of modern mathematics. AC has two goods and one evil:

Good #1) AC is intuitively appealing (a lot of people will disagree).
Good #2) The use of AC is ubiquitous in mathematics.

Evil #1) AC has certain "absurd" consequences for which one might almost want to reject AC as an axiom.

You will see the proofs of a number of very elementary facts with the use of AC explicitly spelled out. You will see how many foundational results in areas ranging from abstract algebra to point-set topology to algebraic geometry to partial differential equations all rely crucially on AC. And yet, AC also has some very disturbing consequences, one of which being the famous Banach-Tarski Paradox, which Dr. Savin discussed in the Undergraduate Colloquium on April 4.

Come and hear about this fascinating axiom!

April 25:
No Talk--End of Year Wrapup

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