Graduate Student Advisory Committee (GSAC) Colloquium Schedule:

Welcome back.

This is an organizational meeting in which we'll discuss plans for upcoming events and colloquia.

Zero sets, cohomology, and p-torsion.

How many polynomials are needed to describe geometrical objects? Local cohomology helps to unswer this question. To try to better understand how big local cohomology modules are we study their associated primes. This talk is intended for a general audience. We will define local cohomology, and study torsion.

Dehn functions on different spaces

The isoperimetric inequality is a relationship between the length of a curve in the plane and the maximum area that curve can bound. We can generalize this question to spaces other than the plane. It turns out to be a geometrical invariant and an interesting approach to studying groups. This discussion will mostly be centered around examples.

Applied Statistics

Is R^n really just like R^2? Of course not, but we nonetheless tend to think of high dimensional spaces as being akin to R^2 and R^3. This talk is meant to shift your intuition, as it will examine some of the peculiarities that arise when working in high dimensional spaces. Not surprisingly, some knowledge of probability theory will be helpful.

Graphs and Free Groups

Using only basic topological properties of finite graphs and maps between them, we will prove several classical results about the structure of free groups, including the fact that the intersection of two finitely generated subgroups of a free group is free. I will follow the wonderfully pleasant paper of Stallings, "Topology of Finite Graphs."

Tolerance vs Immunity

When we have a viral or bacterial infection, the symptoms we feel (fever, runny nose, etc) are caused by our own immune response and not directly by the pathogen itself. The hope is that feeling the need to lie down and eat chicken soup is far better than the alternative (death). The fact that our own bodies contain so much self-destructive potential is somewhat troubling. An overactive immune system can cause anything from food allergies to diabetes to arthritis to transplant rejection.

In this talk, I will discuss some of the ways in which the immune system walks the tightrope between tolerance and immunity. I will show how math biologists turn biological questions into mathematical equations and how those equations can give important insights into how the immune system processes information.

Talk canceled.

There will be no colloquium this week.

Cayley Graphs, Reflections, and Coxeter Groups

Coxeter groups are a class of groups that arise naturally from symmetries of solids and also have a straightforward algebraic description. The Cayley graph of a Coxeter group is distinguishable by its many reflections under the group action. This talk will explore the nice properties of these reflections and characterize the Cayley graphs of Coxeter groups. The talk should be accessible to anyone with some background in group theory.

Keizer's Paradox

It should be intuitively obvious to the most casual observer that a chemical concentration in a solution is not a continuously changing quantity. An integer-valued number of molecules generates a discrete state space. However, chemists have been using the law of mass action to describe the evolution of chemical concentrations in reactions as continuous variables since the 1800's. How accurate are these approximations? In 1987 Joel Keizer proposed a particular type of reaction for which the continuous deterministic model and the discrete state stochastic model disagree. I will talk about why this paradox occurs and how it might be resolved.

Spring break!

There will be no colloquium this week.

Talk canceled.

There will be no colloquium this week.

Detecting Changes in Panel Data

We observe N panels and each panel is observed over time T. It is assumed that the panels are independent of each other but each panel might be time series. We are interested in testing if the mean of some of the panels have changed at the same but unknown time agains the alternative that the model is stable, i.e. the paramaters have not changed during the observation period.

We use a testing procedure which takes the average of the CUSUM statistics of each of the panels. Since N is large the average CUSUM will be approximately normal after centralizing and normalizing. However, when T is realtively small the norming and centering might not be close to the corresponding asymptotic values. We use simulation studies to reduce the bias when the normal approximation is used. We investigate what norming and centering should be used in case of small samples, i.e. when T is small. We consider the cases when the panels are based on independent observations and when they are based on autoregressive processes. The second method is based on the likelihood ratio test when instead of running CUSUM on the panels, we derive directly the likelihood ratio for all the panels at the same time, i.e. the likelihood is based on NT observations directly. With this approach we only need that NT is large. If the number of the panels is large we can analyze time series of short length.

Hecke Algebra

Hecke algebras are ubiquitous objects in modern representation theory, and critical to my own research, since they allow us to reduce much of the representation theory of a complicated algebraic group to the representation theory of a more manageable associative algebra with a finite presentation. In this talk I will try to make the concept of a Hecke algebra accessible, and then explain how they are useful in the representation theory of p-adic groups.

Representations of the Symmetric Group

This talk will focus on how the representations of S_n can be found and their connection to the object known as Young tableaux. As time allows, the analogous theory will be discussed for the generalized symmetric group.

End of the Year Party

As I see it there are three primary purposes to this meeting:
1. I'll take feedback (of really any variety I guess)
2. I will reorganize the various sub committees, and call for volunteers.
3. We'll try to agree on a date for the picnic.

Hopefully this will be relatively painless and extremely pizzaful.