https://yale.zoom.us/j/95303636613 [15]

In 1957, Huang and Yang predicted an asymptotic formula for the ground state energy of a dilute Fermi gas in the thermodynamic limit. This formula highlights a remarkable universality, showing that the correlation energy depends solely on the interaction's scattering length. In this talk, I will present a rigorous proof of the Huang-Yang prediction, employing a bosonization approach that interprets suitable pairs of fermions as bosons.

The celebrated prime geodesic theorem gives the asymptotic formula for the number of primitive closed geodesics according to their length in a closed hyperbolic manifold, i.e., for uniform lattices in SO(n, 1). What’s more, the error term is power-saving. This was generalized with the error term for convex cocompact subgroups by Naud and Stoyanov. It is natural to wonder whether one can generalize this further to the higher rank setting such as SL(n, R). This was done without the error term for Anosov subgroups by Sambarino. In keeping with a principle from Margulis's thesis, we go further and establish exponential mixing of an appropriate dynamical system and use that to produce a power-saving error term.

I will discuss geometric properties of the regular representation of hyperbolic groups. Given a torsion free hyperbolic group G, one can build a natural quotient Q of a Hilbert sphere from the regular representation of G. I am interested in the geometry of this infinite dimensional Riemannian space Q, and also of its ultralimit Q_omega. This ultralimit encodes the asymptotic geometry of Q, for instance possible limits of the regular representation. I will mention an application: the spherical volume (a topological invariant defined by Besson-Courttois-Gallot) of a negatively curved manifold is realized by a minimal surface inside the corresponding Q_omega. This result is motivated by the problem of constructing a "minimal surface geometry" on closed manifolds.

The existence of multi black hole solutions in General Relativity is one of the expectations from the final state conjecture, the analogue of soliton resolution. In this talk, I will present preliminary works in this direction via a semilinear toy model in dimension 3. In particular, I show 1) an algorithm to construct approximate solutions to the energy critical wave equation that converge to a sum of solitons at an arbitrary polynomial rate in (t-r); 2) a robust method to solve the remaining error terms for the nonlinear equation. The methods apply to energy supercritical problems.

Hierarchically hyperbolic groups (HHGs) expand on work of Masur and Minsky on mapping class groups of compact surfaces to provide a geometric model for certain acylindrically hyperbolic groups Gromov hyperbolic groups, and subgroups of right-angled Artin groups.
This geometric frame work has been especially useful for verifying finer properties of acylindrically hyperbolic groups such as the Tits alternative and uniform exponential growth.
I will describe why quotients obtained from subgroups normally generated by finitely many independently distributed random walks preserve hierarchical hyperbolicity.
These ideas also let us understand the geometry of random quotients of hyperbolic, and relatively hyperbolic groups. This is joint work in progress with Abbott, Berlyne, Mangioni, and Rasmussen.

https://yale.zoom.us/j/95303636613 [15]
This talk is based on joint works with Hiroshi Tsuji (Saitama Japan).
The Blaschke—Santal\'{o} inequality describes a correlation between a convex body and its dual object (polar body). Motivated by the recent studies in convex geometry, optimal transportation theory, as well as information theory, a problem of extending the inequality to multiple convex bodies was proposed by Kolesnikov--Werner. Their formulation of the problem naturally involves some generalization of the (functional) Legendre duality. In this talk, we are going to establish a genuine Gaussian saturation principle for the generalized Blaschke--Santal\'{o}-type inequality, and in particular give an affirmative answer to the conjecture of Kolesnikov--Werner.

Our novel observation is a simple but crucial link between the above problem and the inverse form of the Brascamp--Lieb (multilinear) inequality (IBL inequality). The study of the IBL inequality was initiated by Chen--Dafnis--Paouris, and then later Barthe--Wolff developed its theory in more general framework, but under a certain non-degeneracy condition.
Our second main result is about the Gaussian saturation principle for the IBL inequality beyond the framework of Barthe—Wolff.
The above result on the generalized Blaschke—Santal\’{o}-type inequality is a consequence of this second result.
There are further fruitful consequences from our study of the IBL inequality, which we will present as long as time permits.

This talk is based on joint work with Pablo Lessa. We provide an example of a probability distribution on the group GL(2,R) with finite first moment such that the corresponding random product of i.i.d. matrices has two distinct Lyapunov exponents, but the angle between the Oseledets directions is not log-integrable. We prove that, on the other hand, if the second moment is finite, then this angle, if defined, is log-integrable. Next, we turn our attention to general GL(2,R)-cocycles over ergodic automorphisms, and ask ourselves if there is any criterion for log-integrability of the angle between the Oseledets directions in terms of a suitable integrability condition. The answer is negative. In fact, we show the following flexibility result: given any ergodic automorphism T of a non-atomic Lebesgue probability space, we can find a GL(2,R)-cocycle over T whose Lyapunov exponents and joint distribution of Oseledets spaces are prescribed a priori, and meeting any prescribed integrability condition.

https://yale.zoom.us/j/95303636613 [15]
Lieb--Thirring inequalities are functional inequalities that generalize Sobolev inequalities and that have proved to be powerful tools in several questions from mathematical physics, PDEs and functional analysis. Recently, in the spirit of Lieb--Thirring inequalities, certain inequalities in harmonic analysis were extended to the setting of orthonormal functions. In this talk we give a gentle introduction to classical aspects of the subject, some recent progress and some open problems.