We study various structural information of a large network $G$ by randomly embedding a small motif $F$ of choice. We propose two randomized algorithms to effectively sample such a random embedding by a Markov chain Monte Carlo method. Time averages of various functionals of these chains give structural information on $G$ via conditional homomorphism densities and density profiles of its filtration. We show such observables are stable with respect to various notions of network distance. Our efficient sampling algorithm and stability inequalities allow us to use our techniques for hypothesis testing on and hierarchical clustering of large networks. We demonstrate this by analyzing both synthetic and real world network data.  Join with Facundo Memoli and David Sivakoff.

The Hamming cube of dimension n  consists of vectors of length n with coordinates +1 or -1.  Real-valued functions on the Hamming cube equipped with uniform counting measure can be expressed as Fourier--Walsh series, i.e., multivariate polynomials of n variables +1 or -1. The degree of the function is called the corresponding degree of its multivariate polynomial representation.  Pick a function whose Lp norm is 1. How large the Lp norm of the discrete (graph) Laplacian of the function can be if its degree is at most d, i.e., it lives on ``low frequencies''? Or how small it can be if the function lives on high frequencies, i.e., say all low degree terms (lower than d) are zero? I will sketch some proofs based on joint works (some in progress) with Alexandros Eskenazis.

In 2007, Virág and Válko introduced the Brownian carousel, a dynamical system that describes the eigenvalues of a canonical class of random matrices. This dynamical system can be reduced to a diffusion, the stochastic sine equation, a beautiful probabilistic object requiring no random matrix theory to understand. Many features of the limiting eigenvalue point process, the Sine--beta process, can then be studied via this stochastic process. We will sketch how this stochastic process is connected to eigenvalues of a random matrix and sketch an approach to two questions about the stochastic sine equation: deviations for the counting Sine--beta counting function and a functional central limit theorem.

Based on joint works with Diane Holcomb, Gaultier Lambert, Bálint Vet\H{o}, and Bálint Virág.

In this talk we discuss questions related to invertibility of random matrices, and the estimates for the smallest singular value. We outline the main results: an optimal small-ball behavior for the smallest singular value of square matrices under mild assumptions, and an essentially optimal small ball behavior for heavy-tailed rectangular random matrices. We make several remarks and outline some examples. We then explain the relation between such estimates and net constructions, and outline our main result in regards to existence of a net around the sphere with good properties. If time permits, we outline two more implications of this result.