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
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.