Northwestern University

I will discuss statistical inference problems on edge-correlated stochastic block models. We determine the information-theoretic threshold for exact recovery of the latent vertex correspondence between two correlated block models, a task known as graph matching. As an application, we show how one can exactly recover the latent communities using multiple correlated graphs in parameter regimes where it is information-theoretically impossible to do so using just a single graph. Furthermore, we obtain the precise threshold for exact community recovery using multiple correlated graphs, which captures the interplay between the community recovery and graph matching tasks. This is based on joint work with Julia Gaudio and Anirudh Sridhar.

Chimera states have attracted significant attention as symmetry-broken states exhibiting the coexistence of coherence and incoherence. Despite the valuable insights gained by analyzing specific systems, the understanding of the physical mechanism underlying the emergence of chimeras has been incomplete. In this presentation, I will argue that an important class of stable chimeras arise because coherence in part of the system is sustained by incoherence in the rest of the system. This mechanism may be regarded as a deterministic analog of noise-induced synchronization and is shown to underlie the emergence of so-called strong chimeras. These are chimera states whose coherent domain is formed by identically synchronized oscillators. The link between coherence and incoherence revealed by this mechanism also offers insights into the dynamics of a broader class of states, including switching chimera states and incoherence-mediated remote synchronization.

In recent years, our understanding of the asymptotic behavior of characteristic polynomials of random matrices has seen much progression. A key paradigm in this area is that the asymptotic behavior is often captured by an appropriate family of Gaussian multiplicative chaos (GMC) measures (defined heuristically as the normalized exponential of log-correlated random fields). Indeed, such results have been shown for Harr distributed matrices for U(N), O(N), and Sp(2N), as well as for one-cut Hermitian invariant ensembles (and in particular, GUE(N)). In this talk we explain an extension of these results to GOE(2N) and GSE(N). The key tool is a new asymptotic relation between the moments of the characteristic polynomials of all three classical ensembles. 

The Yamabe problem asks whether, given a closed Riemannian manifold, one can find a conformal metric of constant scalar curvature (CSC). An affirmative answer was given by Schoen in 1984, following contributions from Yamabe, Trudinger, and Aubin, by establishing the existence of a function that minimizes the so-called Yamabe energy functional; the minimizing function corresponds to the conformal factor of the CSC metric.

We address the quantitative stability of minimizing Yamabe metrics. On any closed Riemannian manifold we show—in a quantitative sense—that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close to a CSC metric. Generically, this closeness is controlled quadratically by the Yamabe energy deficit. However, we construct an example demonstrating that this quadratic estimate is false in the general. This is joint work with Max Engelstein and Luca Spolaor.

Optimization methods for large-scale machine learning must confront a number of challenges that are unique to this discipline. In addition to being scalable, parallelizable and capable of handling nonlinearity (even non-convexity), they must also be good learning algorithms. These challenges have spurred a great amount of research that I will review, paying particular attention to variance reduction methods. I will propose a new algorithm of this kind and illustrate its performance on text and image classification problems.

The operation of sub-cellular processes in living organisms often require the transfer of biopolymers across impermeable lipid membranes. The emergence of new experimental techniques for manipulation of single molecules at nanometer scales have made possible in vitro experiments that can directly probe such translocation processes in cells as well as in synthetic systems. Some of these ideas have spawned novel bio-technologies with many more likely to emerge in the near future. In this talk I would review some of these experiments and attempt to provide a quantitative understanding of the data in terms of physical laws, primarily mechanics and electrostatics.

Finally we will discuss some recent developments in attacking the shock reflection-diffraction problems, including the existence, stability, and regularity of global regular configurations of shock reflection-diffraction by wedges. The approach includes techniques to handle free boundary problems, degenerate elliptic equations, and corner singularities, which is highly motivated by experimental, computational, and asymptotic results. Further trends and open problems in this direction will be also addressed. This talk will be mainly based on joint work with M. Feldman.