In this talk, I am going to discuss some recent results concerning the derivation, from many body quantum mechanics, of a cubic nonlinear Schroedinger equation, known as the Gross-Pitaevskii equation, for the dynamics of Bose-Einstein condensation. This is a joint work with L. Erdos and H.-T. Yau.

Recently, Wang, Chakrabarti, Wang and Faloutsos have shown that the spectral radius of a graph plays an important role in modeling virus propagation in networks. This led Van Dam and Kooij to consider the following problem: which connected graph on n nodes and diameter D has minimal spectral radius ? Van Dam and Kooij answered this question for $D=n-1,n-2,n-3,n/2,2,1$ and provided a conjecture for the case $D=n-e$, when e is fixed. In this talk, I give an overview of their work and I will outline a proof of their conjecture for $e=4$ and
possible extensions for $e>4$.

This is joint work in progress with Edwin Van Dam (University of Tilburg,
The Netherlands).

This will be an expository talk. I'll begin by introducing the concepts of reduite (reduced function) and balayage (swept measure) in classical potential theory and their interpretations in terms of Brownian motion. I'll then discuss the extension of these ideas to Markov processes as in Hunt's fundamental memoir. After introducing h-transforms I'll be able to define the interior reduite and discuss some of its properties following Walsh. If time permits I'll give some indications of recent work in this area by Fitzsimmons and myself.

As any good TA knows, adding fractions can be tricky. But being able to work with fractions can enrich your mathematical life in many ways. Unfortunately, if you happen to be a ring, this luxury is not inherent. However, you could still hope to embed into some larger ring containing inverses--even to the extent that you become a field or division ring.

We will look first at the theory of localization for commutative rings where everything is fairly straightforward. Then, we will see what happens when we try to generalize to noncommutative rings and look at some examples of good and bad cases.

And the quote... Thoreau.

We will discuss a recent proof of the local Langlands conjecture for GSp(4). This is joint work with Shuichiro Takeda.