Georgia Institute of Technology College of Sciences

To any finite real sequence, we can associate a permutation $\pi$, via: $\pi(k)$ is the index of the $k$th smallest element of the sequence. This association was introduced in a 1987 paper of Alavi, Malde, Schwenk and Erd\H{o}s, where they used it to study the possible patterns of rises and falls that can occur in the matching sequence of a graph (the sequence whose $k$th term is the number of matchings of size $k$), and in the independent set sequence. The main result of their paper was that {\em every} permutation can arise as the ``independent set permutation'' of some graph. They left open the following extremal question: for each $n$, what is the smallest order $m$ such that every permutation of $[n]$ can be realized as the independent set permutation of some graph of order at most $m$? We answer this question. We also improve Alavi et al.'s upper bound on the number of permutations that can be realized as the matching permutation of some graph. There are still many open questions in this area. This is joint work with T. Ball, K. Hyry and K. Weingartner, all at Notre Dame.

We discuss the asymptotic value of the maximal perimeter of a convex set in an n-dimensional space with respect to certain classes of measures. Firstly, we derive a lower bound for this quantity for a large class of probability distributions; the lower bound depends on the moments only. This lower bound is sharp in the case of the Gaussian measure (as was shown by Nazarov in 2001), and, more generally, in the case of rotation invariant log-concave measures (as was shown by myself in 2014). We discuss another class of measures for which this bound is sharp. For isotropic log-concave measures, the value of the lower bound is at least n^{1/8}.

In addition, we show a uniform upper bound of Cn||f||^{1/n}_{\infty} for all log-concave measures in a special position, which is attained for the uniform distribution on the cube. We further estimate the maximal perimeter of isotropic log-concave measures by n^2. 

I will briefly present our Stochastics Group and its main interests, and will continue with some of the problems I have worked on in recent years.

I this talk I will summerize some of our contributions in the analysis of parabolic elliptic Keller-Segel system, a typical model in chemotaxis. For the case of linear diffusion, after introducing the critical mass in two dimension, I will show our result for blow-up conditions for higher dimension. The second part of the talk is concentrated in the critical exponent for Keller-Segel system with porus media type diffusion. In the end, motivated from the result on nonlocal Fisher-KPP equation, we show that the nonlocal reaction will also help in preventing the blow-up of the solutions.