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 inrtoductory 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.

Tentative speakers for this semester (not yet scheduled):
Aaron Fogelson
Jim Carlson

Brad
Frank

Jan 16 Eric Cytrynbaum

The Poincare Index Theorem and a swinging pendulum
Abstract: The Poincare Index Theorem gives some concrete constraints on the structure of vector fields, particularly, the nature of its critical points. In this talk, I will sketch a proof for two dimensional manifolds and use some examples from physics to illustrate the theorem including a discussion of several bifurcations for a swinging pendulum.

Jan 24 Angelo Vistoli

Two open questions in the theory of finite dimensional division algebras
Abstract: A division algebra is a nonzero associative ring with identity, in which every nonzero element has an inverse; of course commutative division algebras are known as fields. The theory of division algebras is one of the most attractive parts of algebra, and it has deep connections with algebraic geometry and number theory.

The algebra of quaternions is the only noncommutative division algebra of which the average mathematician is aware; in fact there are very many examples. The basic problem in the theory can be stated as follows: is it possible to describe all division algebras? Of course the answer is no, because even describing all fields is a hopeless task. So we will reformulate it: if we knew all fields, could we describe all division algebras? This is a much more reasonable mathematical question.

I am going to explain some of the basic results on finite dimensional division algebras, and two of the outstanding open questions. Most of the talk should be accessible to anyone who is comfortable with the definition of a field; to understand one of the two problems you will also need to know what a Galois extension is.

Jan 31 Misha Kapovich

How to straighten a carpenter's ruler
Abstract: Let L be an embedded finite polygonal chain in the the plane (the "carpenter's ruler"). Can one deform L through embedded polygonal chains (preserving the lengths of the edges) to a polygonal chain contained in a straight line? This question known as the "Carpenter's Ruler Problem" was first asked by Stephen Schanuel in the 1960-s, Kirby eventually included it in his list of problems in low-dimensional topology. I will explain a recent solution of this problem by Connelly, Demaine and Rote.

Feb 7

Title
Abstract:

Feb 14 Emina Alibegovic

The Banach-Tarski Paradox
Abstract: Stated popularly, the Banach-Tarski paradox says that it is possible to cut a pea into finitely many pieces which can then be rearranged to form a ball of the size of the sun. We will look at this paradox from a point of view of group theory. The talk will be accessible to all first year graduate students.

Feb 21 Kai-Uwe Bux

Dead Languages, Killing Fish, and Exhausted Kangaroos
Problems and Metaphors in Combinatorial Optimization

Abstract: Last millennium, we heard about genetic algorithms. However, it is always good to have many weapons at hand, Thus, I will present two other methods of combinatorial optimization based on more geometric metaphors.

The optimization problem I will focus on arose in comparative linguistics: given a list of cognates in two related languages (in out case: avestan and vedic, which died long ago), find a measure for the quality of this list. As typical for problems coming from outside mathematics, it is awfully vague. I will discuss one way to turn it into a well defined question to which discrete optimization can be applied.

Feb 28 Renzo Cavalieri

Approximating Jacobians...maybe...
Abstract: Jacobians are a very classical and fascinating topic in the study of algebraic curves. You can associate to any Riemann Surface its Jacobian, which is a complex torus; the amazing thing is that given a Jacobian and a little extra information you can pin down exactly the curve you started with. In the first part of the talk we'll get acquainted with this construction. Then, I'd like to present an idea by G.B. Shabat and V. Voevodsky (~1990), on how to use the combinatorial data of a dessin d'enfant on an algebraic curve over Qbar to create a sequence of abelian varieties supposedly converging to the Jacobian variety of the curve. I'd also like to point out some aspects of this construction that to me are still quite obscure.

March 7 Paul Bressloff

Synchronization in networks of coupled oscillators
Abstract: The Kuramoto model describes a large population of coupled limit-cycle oscillators with a prescribed distribution of natural frequencies. It has been used to study the phenomenon of collective synchronization in a variety of biological systems (neural oscillators, flashing fireflies, circadian pacemaker cells, cardiac pacemaker cells) and physical systems (laser arrays, superconducting Josephson junction arrays). We survey some of the mathematical approaches to understanding the onset of synchronization, which occurs when the coupling strength exceeds a critical threshold. This draws from a broad spectrum of ideas in mathematical biology, statistical physics, kinetic theory, bifurcation theory and plasma physics.

March 14 No colloquium (spring break)

Title
Abstract:

March 21 Annual Graduate Student Meeting

Title
Abstract:

March 28 Martin Deraux

Hyperbolic Tessellations
Abstract: I will give a basic introduction to the hyperbolic plane, and consider some questions related to its tessellations by triangles. For instance, it is an easy exercise to determine which Euclidean triangles tessellate the Euclidean plane; I will discuss the corresponding question in hyperbolic geometry.

April 4 Nick Cogan

The Belousov-Zhabotinskii reaction
Abstract:

April 11 Andrew Oster

Population Projection Via Computer Simulation
Abstract: Over the last two years while attending Cal Poly, I worked on a project for the reintroduction of the California Condors back into the wild. We developed a computer simulation program to help accomplish this task. I will discuss the biology behind our model and the mechanics of the program. Also, I will give some insight on the implications of the output.

April 18 Cord Erdenberg

Evolutionary Algorithms
Abstract: Evolutionary Algorithms are a class of search algorithms which is inspired by evolutionary change in populations of individuals. Although the first origins can be found in the early 1960s, the area of Evolutionary Algorithms is still quite young. They have been applied successfully to a broad variety of problems from many different fields. This led to a strongly practical oriented interest. Thus, today, theory is far behind "experimental knowledge". A solid theory has to be built... We will analyze the behavior of a simple Evolutionary Algorithm, the so called (1+1)EA, on linear and on unimodal functions. Several examples using plateaus of constant fitness will illustrate how small changes to this algorithm can influence its behavior significantly.

April 25 Tom Robbins

How in the hell can you say anything about that!
Abstract: There have been numerous developments in the modeling of systems that seem to obey the age old phenomenon in which "What goes up, must come down". In comparison, very little is known about systems in which stochastic factors often govern the evolution of the system. In this talk, I will outline several deterministic models and recent advances stochastic models that have been used to model the spread of biological systems. In addition, I will outline a hybrid mathematical-numerical model to describe spread of terrestrial plant species. To test this model, I will propose several ways in which the model can be used to describe the reintroduction of vegetation at Mount St. Helens in Washington state. At the end of the talk, I will "Drop my pants" and ask the question, "What the hell can you say about that!". [I believe this is supposed to be read as "I will expose my ignorance to the careful proding of well thought out questions".]