I will discuss the important phenomenon in mathematics that the solution to a question may reach out to concepts of complexity significantly higher than concepts needed for the formulation of the question.
 

Convex-geometric methods, involving random projection operators and coverings, have been successfully used in the study of the largest and smallest singular values, delocalization of eigenvectors, and, among further applications, in establishing the limiting spectral distribution for certain random matrix models. Conversely, random linear operators play a very important role in high-dimensional convex geometry, as a tool in constructing special classes of convex sets and studying sections and projections of convex bodies. In this talk, I will discuss some recent results (by my collaborators and myself) on the borderline between convex geometry and the theory of random matrices, focusing on invertibility of square non-Hermitian random matrices (with applications to the study of the limiting spectral distribution), edges of the spectrum of sample covariance matrices, as well as some applications of random operators to questions in high-dimensional convex geometry.

We will use the techniques of harmonic analysis to establish optimal geometric inequalities. These include the sharp Hardy-Adams inequalities on hyperbolic balls and Hardy-Sobolev-Mazya inequalities on upper half spaces or hyperbolic balls. Using the Fourier analysis on hyperbolic spaces, we will be able to establish sharper inequalities than those known classical inequalities in the literature.

The problem of detecting a deformation in a symmetric Gaussian random tensor is concerned about whether there exists a statistical hypothesis test that can reliably distinguish a low-rank random spike from the noise. Recently Lesieur et al. (2017) proved that there exists a critical threshold so that when the signal-to-noise ratio exceeds this critical value, one can distinguish the spiked and unspiked tensors and weakly recover the spike via the minimal mean-square-error method. In this talk, we will show that in the case of the rank-one spike with Rademacher prior, this critical value strictly separates the distinguishability and indistinguishability of the two tensors under the total variation distance. Our approach is based on a subtle analysis of the high temperature behavior of the pure p-spin model, arising initially from the field of spin glasses. In particular, the signal-to-noise criticality is identified as the critical temperature, distinguishing the high and low temperature behavior, of the pure p-spin model.

Probability theorey is now being inspired and transformed by challenges of big data. This decade is marked by a fascinating convergence of mathematics, statistics, computer science and electrical engineering. I will describe some data-driven advances in high-dimensional probability and high-dimensional inference.