Georgia Institute of Technology College of Sciences

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.

Classical Hardy, Sobolev, and Hardy-Sobolev-Maz'ya inequalities are well known results that have been studied for awhile. In recent years, these results have been been generalized to fractional integrals. This Dissertation proves a new Hardy inequality on general domains, an improved Hardy inequality on bounded convex domains, and that the sharp constant for any convex domain is the same as that known for the upper halfspace. We also prove, using a new type of rearrangement on the upper halfspace, based in part on Carlen and Loss' concept of competing symmetries, the existence of the fractional Hardy-Sobolev-Maz'ya inequality in the case p = 2, as well as proving the existence of minimizers, at least in limited cases.

I'll describe a new combinatorial method for computing the delta-graded knot Floer homology of a link in S^3. Our construction comes from iterating an unoriented skein exact triangle discovered by Manolescu, and yields a chain complex for knot Floer homology which is reminiscent of that of Khovanov homology, but is generated (roughly) by spanning trees of the black graph of the link. This is joint work with Adam Levine.

The theorem of Birman and Hilden relates the mapping class group of a surface and its image under a covering map. I'll explore when we can extend the original theorem and possible applications for further work.