We study the growth rate of the number of maximal arithmetic subgroups of bounded covolumes in a semisimple Lie group using an extension of the method developed by Borel and Prasad

We study the $C^{*}$-algebras and von Neumann algebras associated with the universal discrete quantum groups. They give rise to full prime factors and simple exact $C^{*}$-algebras. The main tool in our work is the study of an amenable boundary action, yielding the Akemann-Ostrand property. Finally, this boundary can be identified with the Martin or the Poisson boundary of a quantum random walk

Let $P(S)$ be the space of projective structures on a closed surface $S$ of genus $g>1$, and let $Q(S)$ be the subset of $P(S)$ of projective structures with quasi-Fuchsian holonomy. It is known that $Q(S)$ consists of infinitely many connected components. In this article, we show that the closure of any exotic component of $Q(S)$ is not a topological manifold with boundary and that any two components of $Q(S)$ have intersecting closures

In this article, we make the first steps toward developing a theory of intersections of coisotropic submanifolds, similar to that for Lagrangian submanifolds.

For coisotropic submanifolds satisfying a certain stability requirement, we establish persistence of coisotropic intersections under Hamiltonian diffeomorphisms, akin to the Lagrangian intersection property. To be more specific, we prove that the displacement energy of a stable coisotropic submanifold is positive, provided that the ambient symplectic manifold meets some natural conditions. We also show that a displaceable, stable, coisotropic submanifold has nonzero Liouville class. This result further underlines the analogy between displacement properties of Lagrangian and coisotropic submanifolds

We establish global well-posedness and scattering for solutions to the defocusing mass-critical (pseudoconformal) nonlinear Schrödinger equation ${\mathrm{iu}}_t+\Delta u=|u{|}^{4/n}u$ for large, spherically symmetric, $L_x^2({\mathbb{R}}^n)$ initial data in dimensions $n\ge 3$. After using the concentration-compactness reductions in [32] to reduce to eliminating blow-up solutions that are almost periodic modulo scaling, we obtain a frequency-localized Morawetz estimate and exclude a mass evacuation scenario (somewhat analogously to [10], [23], [36]) in order to conclude the argument