Georgia Institute of Technology College of Sciences

In this talk I will discuss random matrices that are matricial analogs of the well known binomial, Poisson, and negative binomial random variables. The common thread is the conditional variance of X given S = X+X', which is a quadratic polynomial in S and in the univariate case describes the family of six Meixner laws that will be described in the talk. The Laplace transform of a general n by n Meixner matrix ensemble satisfies a system of PDEs which is explicitly solvable for n = 2. The solutions lead to a family of six non-trivial 2 by 2 Meixner matrix ensembles. Constructions for the "elliptic cases" generalize to n by n matrices. The talk is based on joint work with Gerard Letac.

This will be a survey talk on some aspects of the geometry and topology of moduli spaces of representations of surface groups into Lie groups. I will discuss recent generalizations of the techniques of Atiyah and Bott on equivariant Morse theory. These extend results on stable bundles to Higgs bundles and associated moduli spaces, which correspond to representation varieties into noncompact Lie groups

We consider boundedness of singular integrals in the two weight setting. The problem consists in characterizing non-negative weights v and w for which H: L^{p}(v)\mapsto L^{p}(w) for 1

It has long been postulated and somewhat confirmed with limited biological experiment, that RNA structure affects the propensity of HIV-1 reverse transcriptase to undergo strand transfer, a prerequisite for recombination. Our goal was to use the large resource of in vivo recombinants isolated from patients and stored in the HIV database to determine whether there were signals in the HIV-1 genetic sequence, such as propensity to form RNA secondary structure, that promote recombination. Starting from 65,000 HIV-1 sequences at least 400 nucleotides long, we identified 2,360 recombinants involving exactly two distinct subtypes. Since we were interested in mechanistic causes, rather than selective causes, we reduced the number of recombinants to 544 verifiably unique events. We then fit a Gaussian Markov Random Field model with covariates in the mean to assess the impact of genetic features on recombination. We found SHAPE reactivities to be most strongly and negatively correlated with recombination rates, which agrees with the observation that pairing probabilities had an opposite, strong relationship with recombination. Less strongly associated, but still significant, we found G-rich stretches positively correlated, thermal stability negatively correlated, and GC content positively correlated with recombination. Interestingly, known in vitro hotspots did not explain much of the in vivo recombination.