We study quasi-maximum likelihood (QML) breakpoint estimation for
 covariance regime switches in multivariate time series. We move beyond
 the classical framework and show that regime switches can be detected
 as soon as the signal-to-noise ratio is high enough. We identify a
 quantitative global recovery threshold that compares signal separation
 between regimes to signal fluctuations within regimes, and show its
 sharpness via an explicit counterexample.  We also discuss further
 developments in breakpoint detection in high dimensions.

 Joint work with Hubeyb Gurdogan (UCLA)

We consider the problem of recovering an unknown planted matching between a set of $n$ randomly placed points in \mathbb{R}^d and random perturbations of these points. Some recent works have established results for the error rates for this problem under Gaussian assumptions for both initial positions and noise at different scaling regimes of sample size and dimension. I will discuss some recent progress on establishing results which hold under general distributional assumption for both the the initial positions and noise. More precisely, I will show a general recipe to establish lower bounds via showing the existence of large matchings in random geometric graphs, which leads to simplified and generalized proofs of previous results. Time allowing I will also make some remarks regarding sufficient conditions for perfect recovery in high-dimensions for models where the noise is not isotropic. This is joint work with Roberto Oliveira.

In recent years, significant progress has been made in our understanding of the quantitative behavior of random matrices. One research direction of continued interest has been the estimation of the smallest singular value. A measurement of matrix’s ``invertibility’’, quantitative bounds on the smallest singular value are important for a variety of tasks including establishing a circular law for a non-Hermitian random matrix and for proving stability of numerical methods. In view of the universality phenomena of random matrices, one tries to prove these estimates for more general matrix ensembles satisfying weaker assumptions.

In the geometric approach to proving smallest singular value estimates a key ingredient is the use of a 'distance theorem', which is a small ball estimate for the distance between a random vector and subspace. In this talk we will discuss a new distance theorem and its application to proving smallest singular value estimates for inhomogeneous random rectangular matrix with independent entries. We will also discuss how the recent resolution of the Slicing Conjecture, due to Klartag, Lehec, and Guan, implies smallest singular values estimates for a number of log-concave random matrix ensembles. In some cases, independent entries are no longer necessary!

Random matrix statistics appear across diverse fields and are now recognized as exhibiting universal behavior. However, even within random matrix theory itself, the mechanism underlying the universality of local eigenvalue statistics beyond the mean-field setting remains poorly understood.

In this talk, we consider symmetric and Hermitian random matrices whose entries are independent random variables with an arbitrary variance pattern. Under a novel Short-to-Long Mixing condition—sharp in the sense that it excludes any deterministic correction at the spectral edge—we establish GOE/GUE edge universality for such inhomogeneous random matrices, which may be sparse or far from the classical mean-field regime. This condition reduces the universality problem to verifying the mixing properties of Markov chains defined by the variance profile matrix. This talk is based on joint work with Dang-Zheng Liu.