Abstract: Conventional superconductivity emerges for weakly interacting Fermi gases, and its emergence has been studied in mathematical physics. Such conventional superconductors, however, have a very low critical temperature, making them very expensive in applications. Unconventional superconductors, such as cuperates, on the other hand exhibit a very high critical temperature, but we have very little understanding of the comprising mechanism. Surprisingly, a relatively simple material, namely twisted bilayer graphene (TBG) at a relative twist of 1.1°, the magic angle, has been shown to exhibit superconductivity at a very high temperature, compared with the electron density. Understanding superconductivity in TBG could provide us with insights into more general unconventional superconductors. One element that makes conventional superconductivity possible is the modification of the electron-electron interaction to become attractive due to the inclusion of scattering with an ionic background. In my talk, we will study how to rigorously access this scattering. In particular, we develop a mathematical framework that allows us to study phonons in incommensurate 2D materials.

Abstract: In this talk we will discuss a collection of convolution inequalities for real valued functions on the hypercube, motivated by combinatorial applications.

Abstract:  Multiple financial assets’ time-series data is stored in a matrix upon which we perform principal component analysis to find predominant factors in the market. Random matrix theory helps us to identify the number of factors present in the data, with the top eigenvalue-eigenvector pair bearing a strong resemblance to the market’s capitalization-weighted portfolio. This resemblance is consistent with fundamental concepts from portfolio theory, and can be extended to tensors of implied volatilities for which factors can be constructed using open interest as the analogue for capitalization. In our analyses we initially rely on the support of the Marchenko-Pastur distribution to serve as a cutoff for identification of outlying eigenvalues, but improved criteria can be developed using free probability.

Abstract: We consider the random Cayley graphs of a sequence of finite nilpotent groups of diverging sizes G = G(n), whose ranks and nilpotency classes are uniformly bounded.  For some k = k(n) such that 1 << log k << log |G|, we pick a random set of generators S = S(n) by sampling k elements Z_1, ..., Z_k from G uniformly at random with replacement, and set S = { Z_j^{\pm 1}: 1 <= j <= k }.  We show that the simple random walk on Cay(G,S) exhibits cutoff with high probability.  Some of our results apply to a general set of generators.  Namely, we show that there is a constant c > 0, depending only on the rank and the nilpotency class of G, such that for all symmetric sets of generators S of size at most c log |G| / log log |G|, the spectral gap and the mixing time of the simple random walk X = (X_t) on Cay(G,S) are asymptotically the same as those of the projection of X to the abelianization of G, given by [G,G]X_t.  In particular, X exhibits cutoff if and only if its projection does.  Based on joint work with Jonathan Hermon.

Abstract:  A number of continuous interacting particle systems can be described as collections of Brownian particles on the real line whose collision dynamics are mediated by the local times associated with the gaps between adjacent particles.  Examples of systems in this class include the ordered particle dynamics of rank-based diffusions and certain eigenvalue processes arising in random matrix theory.  We are interested in systems of this sort in which the number of particles is countably infinite.  In this talk, we will present new results on the strong existence and pathwise uniqueness of solutions to the SDEs describing the evolution of such systems.  The main challenge in this setting is establishing uniqueness, which requires controlling effects “coming from infinity.”  Our approach uses a novel technique based on some concentration bounds related to last-passage percolation.  We will also discuss new results on the structure of the set of stationary measures for infinite systems of competing Brownian particles.  This talk is based on joint work with Sayan Banerjee and Amarjit Budhiraja.

Abstract: The trace reconstruction problem asks to reconstruct an unknown n-bit string x given independent random "traces" of x, where a random trace is obtained by first deleting each bit of x independently with some probability (say 0.5), and then outputting the concatenation of the remaining bits of x. A basic question is to determine the number of traces required to reconstruct x (with high probability).  Despite many efforts, this question remains largely open.  In particular, the best known upper bound is exp(~O(n^{1/5})), while the best known lower bound is barely superlinear.

In this talk, I will survey several recent results on this problem and its variants, and highlight its connections to the study of the extremal properties of Littlewood polynomials on complex disks.

Abstract: Given a set K in R^2 with Hausdorff dimension t \in [0, 2], what can we say about a typical orthogonal projection of K? Marstrand (1954) proved that for Lebesgue almost all unit vectors \theta \in S^1, the dimension of the projection \pi_\theta (K) to \theta is min{t, 1}. To refine the question, we can replace Lebesgue measure with an s-dimensional Frostman measure \nu for s \in (0, 1], and ask for the dimension of typical projections \pi_\theta (K) for \theta in the support of \nu. We give an essentially sharp answer to this question, drawing on deep connections to harmonic analysis and geometric measure theory. Joint works with Yuqiu Fu and Hong Wang.

Abstract: The six-vertex model, also known as the square-ice model, is one of the central and most studied systems of 2d statistical mechanics. It offers various combinatorial interpretations. One of them involves molecules of water on the square grid; another one deals with non-intersecting lattice paths, which can be also viewed as level lines of an integer-valued height function. Despite many efforts since the 1960s, the limit shapes for the height function are still unknown in general situations. However, we recently found ways to compute them in a degeneration, which leads to a low density of corners of paths (or, equivalently, of horizontal/vertical molecules of water). I will report on the progress in this direction emphasizing various unusual features: appearance of hyperbolic PDEs; discontinuities in densities; connections to random permutations.

Abstract: The past decade has witnessed a remarkable surge in breakthroughs in artificial intelligence (AI), with the potential to profoundly impact various aspects of our lives. However, the fundamental mathematical principles underlying the success of deep learning, the core technology behind these breakthroughs, is still far from well-understood. In this presentation, I will share some interesting connections between the theory of deep learning and harmonic analysis. The first half provides a gentle introduction to machine learning and deep learning. The second half focuses on two technical topics: An uncertainty principle between space and frequency and its significance in overcoming the curse of dimensionality. The multi-scale Marchenko-Pastur law and its interplay with the multiple-descent learning curve phenomenon.

Abstract: A Bohr set in an abelian group G is a subset of the form

B(K, \epsilon) = {g \in G: |\chi(g) - 1| < \epsilon for all \chi \in K }

where K is a finite subset of the dual group \widehat{G}. A classical theorem of Bogolyubov says that if A \subset \mathbf{Z} has positive upper density \delta, then A+A-A-A contains a Bohr set B(K, \epsilon) where |K| and \epsilon depend only on \delta. While the same statement for A-A is not true (a result of Kříž), Bergelson and Ruzsa proved that if r + s + t = 0, then rA + sA + tA contains a Bohr set (here rA = { ra: a \in A }).  I will discuss this phenomenon in compact abelian groups and countable discrete abelian groups, as well as analogous results for partitions. This talk is based on joint works with Anh Le and John Griesmer.