Georgia Institute of Technology College of Sciences

Lots of attention and research activity has been devoted to partially hyperbolic dynamical systems and their perturbations in the past few decades; however, the main emphasis has been on features such as stable ergodicity and accessibility rather than stronger statistical properties such as existence of SRB measures and exponential decay of correlations. In fact, these properties have been previously proved under some specific conditions (e.g. Anosov flows, skew products) which, in particular, do not persist under perturbations. In this talk, we will construct an open (and thus stable for perturbations) class of partially hyperbolic smooth local diffeomorphisms of the two-torus which admit a unique SRB measure satisfying exponential decay of correlations for Hölder observables. This is joint work with C. Liverani

A 1986 article with this title, written by M. Zuker and published by the AMS, outlined several major challenges in the area. Stating the folding problem is simple; given an RNA sequence, predict the set of (canonical, nested) base pairs found in the native structure. Yet, despite significant advances over the past 25 years, it remains largely unsolved. A fundamental problem identified by Zuker was, and still is, the "ill-conditioning" of discrete optimization solution approaches. We revisit some of the questions this raises, and present recent advances in considering multiple (sub)optimal structures, in incorporating auxiliary experimental data into the optimization, and in understanding alternative models of RNA folding.

In the Landau-de Gennes theory to describe nematic liquid crystals, there exists a cubic term in the elastic energy, which is unusual but is used to recover the corresponding part of the classical Oseen-Frank energy. And the cost is that with its appearance the current elastic energy becomes unbounded from below. One way to deal with this unboundedness problem is to replace the bulk potential defined as in with a potential that is finite if and only if $Q$ is physical such that its eigenvalues are between -1/3 and 2/3. The main aim of our talk is to understand what can be preserved out of the physical relevance of the energy if one does not use a somewhat ad-hoc potential, but keeps the more common potential. In this case one cannot expect to obtain anything meaningful in a static theory, but one can attempt to see what a dynamical theory can predict.

I will discuss two problems in phylogenetics where a geometric perspective provides substantial insight. The first is the identifiability problem for phylogenetic mixture models, where the main problem is to determine which circumstances make it possible to recover the model parameters (e.g. the tree) from data. Here tools from algebraic geometry prove useful for deriving the current best results on the identifiability of these models. The second problem concerns the performance of distance-based phylogenetic algorithms, which take approximations to distances between species and attempt to reconstruct a tree. A classical result of Atteson gives guarantees on the reconstruction, if the data is not too far from a tree metric, all of whose edge lengths are bounded away from zero. But what happens when the true tree metric is very near a polytomy? Polyhedral geometry provides tools for addressing this question with some surprising answers.