A class C of Banach spaces is said to be rigid in a certain nonlinear category (e.g. isometric, Lipschitz, coarse Lipschitz, coarse...), if we can assert that a Banach space X is a member of C whenever it nonlinearly embeds into a member of C. We will first recall classical rigidity results in the isometric and Lipschitz category and more recent coarse Lipschitz rigidity results. Then we will discuss a very recent joint work with P. Motakis, G. Lancien, and Th. Schlumprecht were it is shown that the class of reflexive asymptotic-c₀ Banach spaces is coarsely rigid. The rigidity result is obtained by metrically characterizing the class under scrutiny in terms of a concentration inequality for Lipschitz maps on the Hamming graphs. The talk will be made accessible to non-specialists.

Due to its compositional nature, the function computed by a deep neural net often suffers from the so-called exploding and vanishing gradient problem (EVGP), which occurs when the derivatives of the function computed by the neural net vary wildly depending on the parameter with respect to which one takes the derivative. The EVGP happens especially at the start of training when the weights and biases of the net are random. The purpose of this talk is to present several rigorous probabilistic results which answer the question of which neural net architectures fall prey the EVGP in the simple case fully connected ReLU nets.

In recent years a number of conjectures have appeared concerning the improvement of inequalities of Brunn-Minkowski type under the additional assumptions of symmetry; this includes the B-conjecture, the Gardner-Zvavitch conjecture of 2008, the Log-Brunn-Minkowski conjecture of 2012, and some variants. The conjecture of Gardner and Zvavitch, also known as the Brunn-Minkowski conjecture, states that even log-concave measures in ℝⁿ are in fact ¹⁄_n-concave with respect to the addition of symmetric convex sets. In this talk we shall prove that the standard Gaussian measure enjoys ¹⁄_2n-concavity with respect to centered convex sets. The improvements to the case of general log-concave measures shall be discussed as well: under a certain compactness assumption, we show that any symmetric log-concave measure is indeed better than log-concave with respect to the addition of symmetric convex sets. This is a joint work with A. Kolesnikov.

Binary, or one-bit, representations of data arise naturally in many applications, and are appealing in both hardware implementations and algorithm design. In this talk, we provide a brief background to sparsity and 1-bit measurements, and then present new results on the problem of data classification from binary data that proposes a framework with low computation and resource costs. We illustrate the utility of the proposed approach through stylized and realistic numerical experiments, provide a theoretical analysis for a simple case, and discuss extensions including a hierarchical approach.

In our work with R. Vershynin we have shown that as long as i.i.d. entries of an n x n random matrix A have zero mean and finite second moment, the operator norm of A can be regularized to the optimal order O(sqrt(n)) locally (i.e., by zeroing out a small fraction of the entries of A). A natural question arising from this result is how to find this small subset of the entries to be deleted. In this talk I will discuss challenges of the constructive local regularization of ∥A∥, some known strategies that work in special cases (e.g., for Bernoulli matrices), and a recent result providing a simple constructive way to regularize norm of a general matrix to the almost optimal order O(sqrt(log log n) n). All the results mentioned are non-asymptotic: they hold with high probability (that we find explicitly) for all matrices of large enough size n.

A concrete C*-algebra A of operators on a Hilbert space H is quasidiagonal if there are arbitrarily large finite-dimensional subspaces of H almost reducing A. Through a series of deep results starting with Blackadar and Kirchberg in the mid 90's, it is known that simple, nuclear, quasidiagonal C*-algebras have a very rigid structure and admit certain internal approximations relevant to the Elliott's classification programme for simple, nuclear C*-algebras. A long standing question of Blackadar and Kirchberg asks if all nuclear C*-algebras with a faithful trace are quasidiagonal. In 2015, substantial progress on the quasidiagonality problem was made by Tikuisis, White, and Winter, and combined with the work of Elliott, Gong, Lin, and Niu earlier that year, this partial solution to the quasidiagonality question led to a positive solution to Elliott's classification conjecture. I will discuss the quasidiagonality and classification problems and how they are related and will also discuss how a self-contained proof of the quasidiagonality theorem has led to new methods in the classification and structure theory for simple, nuclear C*-algebras. This is partly based on joint work with José Carrión, James Gabe, Aaron Tikuisis, and Stuart White.

We show that for any n>2 there is a symmetric convex body B in Rⁿ whose Banach-Mazur distance to cross-polytope P is bounded below by n^{(9/5)/polylog(n)}. This improves upon an earlier result of S.Szarek who showed that there exists B' in Rⁿ with d(P,B')>n^(1/2)log(n). Just as the result of S.Szarek, our proof is based on studying geometry of random Gaussian polytopes.

Secondary index theoretic invariants naturally arise in geometry and topology. A good understanding of these secondary invariants have important implications in geometry and topology. For example, in my joint work with Shmuel Weinberger and Guoliang Yu, we use secondary invariants to study the structure group of a topological manifold and the space of positive scalar curvature metrics on a given spin manifold. In this talk, I will consider two different notions of secondary invariants, one K-theoretic and one numerical. I will explain the connection between the two, and some consequences as a result of this. The talk is based on my joint work with Guoliang Yu and Jinmin Wang, and the work of Sheagan John.