Abstract:  In 1989, Bourgain proved the existence of maximal \Lambda(p)-subsets within the collection of mutual orthogonal functions. We shall explore the Euclidean analogue of \Lambda(p)-sets through localization. As a result, we construct maximal \Lambda(p)-subsets on a large class of curved manifolds, in an optimal range of Lebesgue exponents p. This is joint work with C. Demeter and D. Ryou.

Abstract:  Many physical phenomena involve the nonlinear, conservative transfer of energy from weakly damped degrees of freedom driven by an external force to other modes that are more strongly damped. For example, in hydrodynamic turbulence, energy enters the system primarily at large spatial scales, but at high Reynolds number, dissipative effects are only significant at very high frequencies. Nevertheless, empirical observations suggest that the nonlinearity transfers energy to small scales at a rate that allows statistically stationary solutions to have bounded energy in the infinite Reynolds number limit, with the energy input balanced by a nontrivial flux of energy through arbitrarily small length scales. This is an instance of a phenomenon typically referred to as anomalous dissipation and is one of the fundamental predictions of turbulence theory. A rigorous understanding of anomalous dissipation in realistic, infinite-dimensional systems seems largely out of reach and it is natural to start with simplified (e.g., finite-dimensional) models. 

In this talk, I will discuss the existence of invariant measures for a class of SDEs on R^d with a bilinear nonlinearity B(x,x) constrained to possess various properties common to finite-dimensional fluid models and a linear damping term -Ax with a nontrivial kernel. Since kerA is nontrivial, an invariant measure can exist only if the nonlinearity transfers energy from the undamped modes to the damped modes sufficiently fast. We develop a set of sufficient dynamical conditions on B that guarantees the existence of an invariant measure and prove that they hold “generically” within our constraint class of nonlinearities provided that at least 1/3 of the modes are damped and the stochastic forcing acts directly on at least two degrees of freedom. This is joint work with J. Bedrossian, A. Blumenthal, and K. Callis.

Abstract:  We consider two related questions about the extremal statistics of Wigner matrices (random symmetric matrices with independent entries). First, how much can their eigenvalues fluctuate? It is known that the eigenvalues of such matrices display repulsive interactions, which confine them near deterministic locations. We provide optimal estimates for this “rigidity” phenomenon. Second, what is the behavior of the maximum of the characteristic polynomial? This question is motivated by a conjecture of Fyodorov–Hiary–Keating on the maxima of logarithmically correlated fields, and we will present the first results for Wigner matrices. This talk is based on joint work with Paul Bourgade and Ofer Zeitouni.

Abstract: Let p be a prime. Bounding short Dirichlet character sums is a classical problem in analytic number theory, and the celebrated work of Burgess provides nontrivial bounds for sums as short as p^{1/4+epsilon} for all epsilon>0. In this talk, we will first survey known bounds in the original and generalized settings. Then we discuss the so-called "Burgess method" and present new results that rely on bounds on the multiplicative energy of certain sets in products of finite fields.

Abstract: First-passage percolation on the square lattice is a random growth model in which each edge of Z^2 is assigned an i.i.d. nonnegative weight.  The passage time between two points is the smallest total weight of a nearest-neighbor path connecting them, and a path achieving this minimum is called a geodesic.  Typically, the number of edges in a geodesic is comparable to the Euclidean distance between its endpoints.  However, when the edge-weights take the value 0 with probability 1/2, a strikingly different behavior occurs: geodesics travel primarily on critical clusters of zero-weight edges, whose internal graph distance scales superlinearly with Euclidean distance.  Determining the precise degree of this superlinear scaling is a challenging and ongoing endeavor.  I will discuss recent progress on this front (joint work with David Harper, Xiao Shen, and Evan Sorensen), along with complementary results on a dual problem, where we restrict path lengths and analyze passage times (joint with Jack Hanson and Daniel Slonim).

Abstract: In the 1970s, Stein raised the Fourier restriction conjecture, aiming to study the L^p behavior for Fourier transforms restricted to hypersurfaces. It turns out that the conjecture is closely related to many other branches of math, such as number theory and PDEs. In this talk, I will survey the conjecture and discuss some of its recent progress. The key tools are the decoupling inequality and a discretized (continuous) version of the Szemeredi-Trotter theorem. 

Abstract:  One way to understand the concentration of the norm of a random matrix X with Gaussian entries is to apply a standard concentration inequality, such as the one for Lipschitz functions of i.i.d. standard Gaussian variables, which yields subgaussian tail bounds on the norm of X. However, as was shown by Tracy and Widom in 1990s, when the entries of X are i.i.d. the norm of X exhibits even sharper concentration. The phenomenon of a function of many i.i.d. variables having strictly smaller tails than those predicted by classical concentration inequalities is sometimes referred to as «superconcentration», a term originally dubbed by Chatterjee. I will discuss novel results that can be interpreted as superconcentration inequalities for the norm of X, where X is a Gaussian random matrix with independent entries and an arbitrary variance profile. We can also view our results as a nonhomogeneous extension of Tracy-Widom-type upper tail estimates for the norm of X.

Abstract:  It has long been predicted that the ground state of the Edwards-Anderson spin glass model is highly sensitive to small perturbations in disorder, a phenomenon known as disorder chaos. In a recent breakthrough, Chatterjee rigorously established disorder chaos in the EA spin glass model using the Hermite spectral method.

In this talk, I will discuss generalizations of Chatterjee's results, following the same Hermite spectral approach, for two related spin glass models: the mixed even p-spin short-range model and the diluted mixed p-spin model. The key novelty of our proof lies in an elementary algebraic identity involving the Fourier-Hermite series coefficients of the two-spin correlation functions. This identity allows us to derive sufficient combinatorial conditions on the underlying interaction hypergraph to determine the contributing coefficients in the overlap function.

This talk is based on joint work with Wei-Kuo Chen and Heejune Kim.

Abstract:  In this talk, we consider a class of infinite horizon optimal control problems subject to semilinear parabolic equations.  First and second-order optimality conditions are obtained in the presence of constraints on the controls, which can be either pointwise in space-time or pointwise in time and L^2 in space.  These results rely on a new L^\infty estimate for nonlinear parabolic equations in an essential manner.  We also approximate the problem by finite horizon optimal control problems and derive error estimates for these approximations.  The proof of these estimates is based on the established second-order optimality conditions.

Abstract: Counting the number of real roots of a generic polynomial is a classical question that has been studied by many authors during the last century. In this talk, I would like to discuss a brief history of the subject and some recent progress, where ideas from Fourier analysis are helpful.

Abstract:  For a scalar conservation law with strictly convex flux classical Oleinik's estimates ensure that the total variation of a solution with bounded initial data decays like 1/t.  We show that a faster decay  rate of the total variation is achieved if the initial datum is taken in a proper class of  ''intermediate domains''.

A key ingredient of the analysis is a ''Fourier-type" decomposition into components which oscillate more and more rapidly for a function belonging to such a class of intermediate domains.

The result aims at extending the theory of fractional domains for analytic semigroups to an entirely nonlinear setting in order to establish well-posedness of solutions of hyperbolic systems of conservation laws.

Joint work with A. Bressan, E. Marconi and L. Talamini.