Georgia Institute of Technology College of Sciences

Let $K$ be a n dimensional convex body with of volume $1$. and barycenter of $K$ is the origin.  It is known that $|K \cap -K|>2^{-n}$.  Via thin shell estimate by Lee-Vempala (earlier versions were done by Guedon-Milman, Fleury, Klartag), we improve the bound by a sub-exponential factor.  Furthermore, we can improve  the Hadwiger’s Conjecture in the non-symmetric case by a sub-exponential factor.  This is a joint work with Boaz A. Slomka, Tomasz Tkocz, and Beatrice-Helen Vritsiou. 

The Caffarelli contraction theorem states that the Brenier map sending the
Gaussian measure onto a uniformly log-concave probability measure is
lipschitz. In this talk, I will present a new proof, using entropic
regularization and a variational characterization of lipschitz transport
maps. Based on joint work with Nathael Gozlan and Maxime Prod'homme.

One of the outstanding open problems of statistical mechanics is about the hard-core model which is a popular topic in mathematical physics and has applications in a number of other disciplines. Namely, do non-overlapping hard disks of the same diameter in the plane admit a unique Gibbs measure at high density? It seems natural to approach this question by requiring the centers to lie in a fine lattice; equivalently, we may fix the lattice, but let the Euclidean diameter D of the hard disks tend to infinity. In two dimensions, it can be a unit triangular lattice A_2 or a unit square lattice Z^2. The randomness is generated by Gibbs/DLR measures with a large value of fugacity which corresponds to a high density. We analyze the structure of high-density hard-core Gibbs measures via the Pirogov-Sinai theory. The first step is to identify periodic ground states, i.e., maximal-density disk configurations which cannot be locally `improved'. A key finding is that only certain `dominant' ground states, which we determine, generate nearby Gibbs measures. Another important ingredient is the Peierls bound separating ground states from other admissible configurations. In particular, number-theoretic properties of the exclusion diameter D turn out to be important. Answers are provided in terms of Eisenstein primes for A_2 and norm equations in the cyclotomic ring Z[ζ] for Z^2, where ζ is the primitive 12th root of unity. Unlike most models in statistical physics, we find non-universality: the number of high-density hard-core Gibbs measures grows indefinitely with D but
non-monotonically. In Z^2 we also analyze the phenomenon of 'sliding' and show it is rare.
This is a joint work with A. Mazel and Y. Suhov.

The Ehrhard-Borell inequality stands at the top of the pyramid of Gaussian inequalities. It is a powerful and delicate statement about the convexity of the Gaussian measure. In this talk I will discuss the inequality and its beautiful proof by Borell. The delicate nature of the inequality however makes the characterization of the equality cases difficult and they were left unknown. I will explain how we solved this problem. Joint work with Ramon van Handel.