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Graduate Colloquia
Spring 2005
Tuesdays, 4:00 - 5:00 pm in JWB 335
Math 6960-010 (credit hours available!)

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 4:00pm in JWB 335, unless otherwise noted.

March 1:

Speaker: Josh Thompson
Title: Points in the State-Space of Topological Surfaces

In 1872, Felix Klein defined geometry as the study of the properties of a space which are left invariant under a group of transformations. With this is in mind, we define a notion of a "geometric structure" on a surface. If we define a "state" of a surface to be a particular "geometric structure" on that surface, then we are led to the notion of a "state-space" of a surface (commonly referred to as a "space of geometric structures"). In this talk, we will explore several such "states" of a surface, and then shed some light on the corresponding "state-space".

March 8:

Speaker: Will Nesse
Title: Contraction and Partial Contraction Analysis with Applications to Diffusion-Like Coupled Non-Linear Oscillators

Contraction analysis is a relatively new method to assess global stability of attractors. Rather than identifying the putative attractor and developing a Liapunov function, contraction analysis examines the small displacements in initial conditions and gives conditions on when trajectories resulting from the displacement will contract together. In this way we do not need to identify the attractor per se but we are able to say that there exists one that all trajectories go toward exponentially. Partial contraction analysis adapts the aforementioned method to study synchronization of coupled non-linear oscillators with diffusion-like coupling. Coupled oscillators, if they synchronize, do not necessarily contract as different initial conditions may force the oscillators to the same limit cycle but with a different common phase. After explaining the basic theory I will show some applications.

March 22:

Speaker: Matt Clay
Title: A Journey into Outer Space

Outer Space is a good geometric model space for the group of outer automorphisms of a free group. We will define what this means and then build Outer Space.

March 29:

Speaker: Stefanos Folias
Title: Breathers in Neural Networks

When a neural network model responds to a localized input, it responds with a localized region of activity (repetitive firing of neurons) which either remains stationary or periodically oscillates. I will explain why it happens and show you what it looks like.

April 4:

Special GSAC Smorgasboard By

Joro Todorov
Title: Parametrizations of algebraic sets

We will discus the definition, talk about some applications and some problems related to the parametrizations of algebraic sets.

Liz Doman
Title: Consequences of Spatial Organization of Cellular Connections on Action Potential Propagation

Cells in the ventricular myocardium are excitable, enabling the propagation of action potentials. This causes the cells to contract, which is how the heart pumps blood. On the cellular level, propagation from one cell to another is not smooth, but rather jumpy with a time delay on the order of $\mu$ seconds. Tissue architecture on the cellular level plays an important role in producing the reliable smooth wave fronts observed on the tissue level. The spatial organization of gap junctional connections is one characteristic of the tissue architecture. In particular, ventricular myocardial cells are each coupled to 11.3 $\pm$ 2.2 neighboring cells via gap junction channels. Therefore, wave fronts of excitation have many opportunities to propagate through connections in all directions. So, how does the spatial organization of cellular connections via gap junction channels affect propagation on the macroscopic level? In particular, what are the benefits of being coupled to $\sim$ 11 other cells? Does this spatial organization make propagation failure less likely? The model presented in the talk is a first approach to answering some of these questions.

Jason Preszler
Title: A brief introduction to $L$-functions

We will define several types of $L$-functions and illustrate some of their number theoretic applications including the Riemann hypothesis, Dirichlet's theorem, and class field theory.

April 5:

Speaker: An Le
Title: Mathematics is fun

I will talk about several interesting mathematics problems.

April 12:

Speaker: Fumitoshi Sato
Title: Introduction to Gromov-Witten Invariant

Kontsevich found a marvelous way to compute numbers of degree d rational curves which go through 3d-1 points. We will discuss his method.

April 19:

Speaker: Aaron McDonald
Title: The Role of R_0 in Epidemiology

Epidemiological modeling has been used extensively to understand the dynamics of an infectious disease within a population of hosts. Most epidemiological models suggest that the introduction of an infectious disease to a naive population via an infected individual will create an epidemic on average if the basic reproductive ratio, R_0, is greater than 1. Last semester, Amber introduced us to this concept for various ODE models. In this talk, I plan to formalize this idea for ODE, PDE, and Stochastic versions of the SIS Model. I also plan to demonstrate the importance of R_0 in Evolutionary Epidemiology, a field of Epidemiology that focuses on the adaptive dynamics of parasites.

April 26:

Title: TBA

Past Graduate Colloquia

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Department of Mathematics
University of Utah
155 South 1400 East, JWB 233 
Salt Lake City, Utah 84112-0090 
Tel: 801 581 6851, Fax: 801 581 4148