Georgia Institute of Technology College of Sciences

Boros and Furedi (for d=2) and Barany (for arbitrary d) proved that there exists a constant c_d>0 such that for every set P of n points in R^d in general position, there exists a point of R^d contained in at least c_d n!/(d+1)!(n-d-1)! (d+1)-simplices with vertices at the points of P. Gromov [Geom. Funct. Anal. 20 (2010), 416-526] improved the lower bound on c_d by topological means. Using methods from extremal combinatorics, we improve one of the quantities appearing in Gromov's approach and thereby provide a new stronger lower bound on c_d for arbitrary d. In particular, we improve the lower bound on c_3 from 0.06332 due to Matousek and Wagner to more than 0.07509 (the known upper bound on c_3 is 0.09375). Joint work with Lukas Mach and Jean-Sebastien Sereni.

The Potts antiferromagnet on a random graph is a model problem from disordered systems, statistical mechanics with random Hamiltonians. Bayati, Gamarnik and Tetali showed that the free energy exists in the thermodynamic limit, and demonstrated the applicability of an interpolation method similar to one used by Guerra and Toninelli, and Franz and Leone for spin glasses. With Contucci, Dommers and Giardina, we applied interpolation to find one-sided bounds for the free energy using the physicists' ``replica symmetric ansatz.'' We also showed that for sufficiently high temperatures, this ansatz is correct. I will describe these results and some open questions which may also be susceptible to the interpolation method.

Starting in the 30's Physicists were concerned with the problem of motion of dislocations or the problem of deposition of materials over a periodic structure. This leads naturally to a variational problem (minimizing the energy). One wants to know very delicate properties of the minimizers, which was a problem that Morse was studying at the same time. The systematic mathematical study of these problems started in the 80's with the work of Aubry and Mather who developed the basis to deal with very subtle problems. The mathematics that have become useful include dynamical systems, partial differential equations, calculus of variations and numerical analysis. Physical intuition also helps. I plan to explain some of the basic questions and, perhaps illustrate some of the results.

We consider particle dynamics in the (unfolded) Ehrenfest Windtree Model and theflow along straight lines on a certain folded complex plane. Fixing some parameters,it turns out that both doubly periodic models cover one and the same L-shaped surface.We look at the case for which that L-shaped surface has a (certain kind of) structure preservingpseudo-Anosov. The dynamics in the eigendirection(s) of the pseudo-Anosovon both periodic covers is very different:The orbit diverges on the Ehrenfest model, but is dense on the folded complex plane.We show relations between the two models and present constructions of folded complex planes.If there is time we sketch some of the arguments needed to show escaping & density of orbits.There will be some figures showing the trajectories in different settings.