Georgia Institute of Technology College of Sciences

In this talk we will discuss properties of some random polytopes. In particular, we first propose a deviation inequality for the convex hull of i.i.d. random points, uniformly distributed in a convex body. We then discuss statistical properties of this random polytope, in particular, its optimality, when one aims to estimate the support of the corresponding uniform distribution, if it is unknown.We also define a notion of multidimensional quantiles, related to the convex floating bodies, or Tukey depth level sets, for probability measures in a Euclidean space. When i.i.d. random points are available, these multidimensional quantiles can be estimated using their empirical version, similarly to the one-dimensional case, where order statistics estimate the usual quantiles.

We express weight enumerator of each binary linear code as a product. An analogous result was obtain by R. Feynman in the beginning of 60's for the speacial case of the cycle space of the planar graphs.

Among n-dimensional regions with fixed volume, which one hasthe least boundary? This question is known as an isoperimetricproblem; its nature depends on what is meant by a "region". I willdiscuss variations of an isoperimetric problem known as thegeneralized Cartan-Hadamard conjecture: If Ω is a region in acomplete, simply connected n-manifold with curvature bounded above byκ ≤ 0, then does it have the least boundary when the curvature equalsκ and Ω is round? This conjecture was proven when n = 2 by Weil andBol; when n = 3 by Kleiner, and when n = 4 and κ = 0 by Croke. Injoint work with Benoit Kloeckner, we generalize Croke's result to mostof the case κ < 0, and we establish a theorem for κ > 0. It was originally inspired by the problem of finding the optimal shape of aplanet to maximize gravity at a single point, such as the place wherethe Little Prince stands on his own small planet.

The sharp A2 weighted bound for martingale transforms can be proved by a new elementary method. With additional work, it can be extended to the euclidean setting. Other generalizations should be possible.

The stochastic block model is a random graph model that was originally introduced 30 years ago to model community structure in networks. To generate a random graph from this model, begin with two classes of vertices and then connect each pair of vertices independently at random, with probability p if they are in the same class and probability q otherwise. Some questions come to mind: can we reconstruct the classes if we only observe the graph? What if we only want to partially reconstruct the classes? How different is this model from an Erdos-Renyi graph anyway? The answers to these questions depend on p and q, and we will say exactly how.

In this talk we will survey some recent development on statistical properties of matchings of very large and infinite graphs. The main goal of the talk is to describe a few applications of a new concept called matching measure. These applications include new results on the number of (perfect) matchings in large girth graphs as well as simple new proofs of certain statistical physical theorems. In particular, we will sketch the proof of Friedland's Lower Matching Conjecture, and a new proof of Schrijver's and Gurvits's theorems. This talk is based on joint papers with various subsets of Miklos Abert, Peter E. Frenkel, Tamas Hubai and Gabor Kun.

I will continue introduction to Nonnegative and PSD ranks of matrices. I will explain the connection between these ranks and constructing linear and semidefinite lifts of polytopes. I will give several different interpretations of these ranks and continue discussing some of their elementary properties.

In this the second of three talks about Stability of Matter I'll give a proof of the Lieb-Thirring inequality and then continue discussing an estimate on what is called the indirect term of the Coulomb energy of N electrons.