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Graduate Colloquia
Fall 2004
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.

August 31:

Speaker: Tom Robbins
Title: A mechanistic model for seed dispersal

In this talk, I will present a mechanistic model to describe wind dispersed plant seeds. I will discuss the general modeling approach and the assumptions that go into the model. In addition, I will talk about external constraints on the model, simplicity verses realism, and how the model compares with real data.

September 7:

Speaker: Dan Margalit
Title: Braids

While simple to define, braid groups display a surprising number of interesting features. After getting a feel for the basic structure of the group, we will explore connections to mapping class groups, geometric group theory, and representation theory.

September 14:

Speaker: Bori Mazzag
Title: A Mathematical Model of Pattern Formation by Aerotactic Bacteria

Conventional bacterial chemotaxis depends critically on the fast decrease and slow adaptation of the cell's turning frequency in a favorable environment. This mechanism is relatively ineffective: the aggregation of the bacterial colony toward chemoattractant is slow and incomplete. Aerotaxis is a particular form of chemotaxis in which oxygen plays the role of both attractant (at moderate concentrations) and repellent (at high concentrations). It has been experimentally demonstrated that this taxis does not require slow adaptation, making aerotaxis a very effective mechanism for complete aggregation toward the most favorable oxygen concentration. Our model consists of a system of hyperbolic differential equations, each describing a group of bacteria moving in the same direction, coupled to a diffusion equation describing the dynamics of oxygen. Simulations allow comparisons of theoretical and experimental results and help in making a judgment on the validity of various biochemical mechanisms. Analytical results give estimates for parameters that are otherwise difficult to obtain experimentally.

September 21:

Speaker: Ken Chu
Title: The First Things about ``Moduli''

The idea of ``moduli'' is to express a given family of geometric objects belonging to a certain category as an object of the same category (or at least a related, slightly generalized category).

For example, the so-called ``elliptic curves'' are complex manifolds. Then we ask: Does the set of all elliptic curves itself admit a natural structure of a complex manifold?

In complex algebraic and analytic geometry, there are plenty of non-trivial moduli problems. The main examples of this talk will be the elliptic curves. I will show that the moduli space (in the category of analytic spaces) of elliptic curves is a non-trivial quotient (analytic orbifold), whose underlying space is in fact simply the complex plane.

I will discuss the delicacy of constructing reasonable ``moduli spaces'' via forming quotients. Time permitting, I will also discuss deformation, singularities, and compactification of moduli spaces.

September 28:

Speaker: Robert Bell
Title: Algorithms and Undecidability

We will formalize the intuitive notion of an algorithm. In particular, we will prove the Markov-Post Theorem which states that there is a finitely presented semi-group (strings of letters on a finite alphabet together with a finite number of substitution rules) for which there does not exist an algorithm which decides whether or not two given strings are equivalent.

October 5:

Speaker: Jason Preszler
Title: p-Adic Numbers

The p-adic numbers will be constructed using the methods of 1) p-adic expansion, 2) completion, and 3) inverse limits. The result of Ostrowski concerning absolute values on the rationals will be discussed as well as the local-global principal and Hasse-Minkowski theorem. If time allows, the structure of the multiplicative group of $\mathbb{Q}_{p}$ will be explored.

October 12:

Speaker: Young-Seon Lee
Title: From Heart Failure to Intracellular Ca2+ cycling

Heart failure is a serious condition which causes about 300,000 deaths each year in the United States. Mechanical alternans,a phenomenon of alternating strong and weak beats at a constant heart rate,is the hallmark of heart failure. Recent experimental studies show that abnormal intracellular Ca2+ cycling is the key mechanism for mechanical alternans. In this talk, I will introduce mathematical models to understand the mechanism of mechanical alternans.

October 19:

Speaker:MunJu Kim
Title: Low Reynolds Number Flow

October 26:

Speaker: David Levin
Title: Exact Sampling

Many probability distribution with large combinatorial state spaces, such as the Ising model, are hard to simulate directly. The idea of Markov chain Monte Carlo is to create a Markov chain whose equilibrium distribution is the desired distribution, and simulate the chain for "a long period of time" with the hope that after this time, the chain is close to stationarity. There are two potential difficulties with the method: in most cases, it will be impossible to obtain exactly the equilibrium distribution in finite time, and moreover it is in general a difficult problem to bound the time required before the chain is close to its equilibrium distribution. In 1996, Propp and Wilson introduced an algorithm, "coupling from the past", which allows sampling exactly from the equilibrium distribution and does not require computation of convergence rates. I will explain the ideas of Propp and Wilson, and give some examples from statistical mechanics.

November 2:

Speaker: Kazuma Shimomoto
Title: Twin prime numbers

The study of prime numbers has a long history. Recently, it was announced that "Twin prime conjecture" was solved, but the proof, however, still contains a gap. While his proof is still incomplete, he found a nice approach to the conjecture. In this talk, I will present some basic ideas about this problem including historical aspects.

November 9:

Speaker: Amber Smith
Title: Population Invasions: Models, Ro, and NGOs for Infectious Diseases

The basic reproductive number (Ro) and the next generation operator (NGO) give us some mathematical tools to analyze infectious disease dynamics. This includes finding out whether a disease population will survive, if it can invade another, and if two populations can coexist. We'll develop these tools as well as apply them to some models including the flu, foot and mouth disease, STDs, malaria, and Strep pneumoniae.

November 16:

Speaker: Renzo Cavalieri
Title: Counting Bitangents With Stable Maps

Let's start from a pseudo high school problem: given a generic plane curve of degree d, count the number of lines that are tangent twice to this curve. If you try it at home, it's quite hard. A neat classical solution is a consequence of the Plucker formulas. However, the problem can be approached from a completely different angle, using moduli spaces and intersection theory. Goal of the talk is to sketch the strategy of this technique.

November 23:

Speaker: Dali Zhang
Title: Two-scale electromagnetic imaging problems with a microstructured medium

Many engineering problems contain materials which are heterogeneous on a length scale that is small (microscopic) compared to the typical length scale (macroscopic) of the problem. Electromagnetic materials are particularly important to study in this regard. We will discuss large-scale inverse problems based on Maxwell's equations in time or freqency domain. We will also discuss an inverse homogenization approach for microscale inversion to recover information about the fine structure of the random medium. We will show computational results regarding this approach and present the inverse problems related to this topic for future work.

November 30:

Speaker: Nessy Tania
Title: Restitution---how to study cardiac dynamics using one dimensional maps.

In this talk, we will learn how one-dimensional maps can be used to study the dynamics of electrical activities in the cardiac system. The map, known as the restitution relation, specifies an action potential duration based on the previous diastolic interval. We will look at two models in particular - one derived from a biological experiment, and one obtained by taking asymptotics limit of a Hodgkin-Huxley like system. Bifurcation structures of the latter will be explored in some details.

December 7:

Speaker: Elijah Newren
Title: Numerically Solving the Coupled Motion of a Fluid and an Elastic Material

There are a vast range of phenomena in biological fluid dynamics that involve the interaction of a viscous incompressible fluid with an elastic material--including the beating heart, insect flight, platelet aggregation, swimming, and many other applications. In this talk I will give a brief overview of a pair of methods used to solve such problems, namely the Immersed Boundary and Immersed Interface Methods.

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