A graph G on n vertices is called an alpha-expander if the external neighborhood of every vertex subset U of size |U|<=n/2 in G has size at least alpha*|U|.

Extending and improving the results of Plotkin, Rao and Smith, and of Kleinberg and Rubinfeld from the 90s, we prove that for every alpha>0, an alpha-expander G on n vertices contains every graph H with at most cn/log n vertices and edges as a minor, for c=c(alpha)>0.

Alternatively, every n-vertex graph G without sublinear separators contains all graphs with cn/logn vertices and edges as minors.

Consequently, a supercritical random graph G(n,(1+epsilon)/n) is typically minor-universal for the class of graphs with cn/log n vertices and edges.

The order of magnitude n/log n in the above results is optimal.

A joint work with Rajko Nenadov.

We present recent results on the geometry of centrally-symmetric random polytopes generated by N independent copies of a random vector X. We show that under minimal assumptions on X, for N>Cn, and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector - namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body. Moreover, by identifying the floating bodies associated with various random vectors we recover the estimates that have been obtained previously, and thanks to the minimal assumptions on X we derive estimates in cases that had been out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when X is q-stable or when X has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing - noise blind sparse recovery. This is joint work with the speaker's PhD student Christian Kümmerle (now at Johns Hopkins University) as well as Olivier Guédon (University of Paris-Est Marne La Vallée), Shahar Mendelson (Sorbonne University Paris), and Holger Rauhut (RWTH Aachen).

I will talk about the beta orbital corners process, a natural interpolation (on the dimension of the base field) of orbital measures from random matrix theory. The new result is the convergence in probability of the orbital corners process to a Gaussian process. Our approach is based on the study of asymptotics of the multivariate Bessel functions via explicit formulas. I will also discuss some variations of the problem by means of multivariate special functions.

Free probability provides a unifying framework for studying random multi-matrix models in the large $N$ limit. Typically, the purview of these techniques is limited to invariant or mean-field ensembles. Nevertheless, we show that random band matrices fit quite naturally into this framework. Our considerations extend to the infinitesimal level, where finer results can be stated for the $\frac{1}{N}$ correction. As an application, we consider the question of outliers for finite-rank perturbations of our model. In particular, we find outliers at the classical positions from the deformed Wigner ensemble. No background in free probability is assumed.

Let s_n(M) denote the smallest singular value of an n x n random (possibly complex) matrix M. We will discuss a novel combinatorial approach (in particular, not using either inverse Littlewood--Offord theory or net arguments) for obtaining upper bounds on the probability that s_n(M) is smaller than a given number for quite general random matrix models. Such estimates are a fundamental part of the non-asymptotic theory of random matrices and have applications to the strong circular law, numerical linear algebra etc. In several cases of interest, our approach provides stronger bounds than those obtained by Tao and Vu using inverse Littlewood-Offord theory.