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Noncommutative ergodic theory of higher rank lattices

Noncommutative ergodic theory of higher rank lattices is a current topic that has seen several exciting developments in the last five years. Among these recent advancements, in joint work with Rémi Boutonnet (2019), we proved a no... ver más

30/09/2029

ENS

2M€

Presupuesto del proyecto: 2M€

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TRL 4-5

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Financiación concedida El organismo HORIZON EUROPE notifico la concesión del proyecto el día 2024-10-01

ERC-2023-ADG: ERC ADVANCED GRANTS Scope:Objectives

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Duración del proyecto: 59 meses Fecha Inicio: 2024-10-01
Fecha Fin: 2029-09-30

Líder del proyecto

ECOLE NORMALE SUPERIEURE No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 2M€

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Descripción del proyecto Noncommutative ergodic theory of higher rank lattices is a current topic that has seen several exciting developments in the last five years. Among these recent advancements, in joint work with Rémi Boutonnet (2019), we proved a noncommutative analogue of Nevo-Zimmer's structure theorem for actions of higher rank lattices on von Neumann algebras. First of all, we derived several novel applications to ergodic theory, topological dynamics, unitary representation theory and operator algebras associated with higher rank lattices. Then we obtained a noncommutative analogue of Margulis' factor theorem that provides strong evidence towards Connes' rigidity conjecture for the group von Neumann algebra of higher rank lattices. These results revealed deep and unexpected interactions between the field of discrete subgroups of semisimple Lie groups and the field of operator algebras. In this research project, I plan to build upon these recent achievements to develop new directions in noncommutative ergodic theory of higher rank lattices. This research proposal is centered around two main interconnected themes.Firstly, I plan to work on several problems and conjectures around the dynamics of the space of positive definite functions and character rigidity for higher rank lattices. These include the classification of characters for higher rank lattices of product type, and more general lattices with dense projections in product groups as well as stiffness results for stationary positive definite functions.Secondly, drawing inspiration from Margulis' superrigidity theorem, I plan to tackle Connes' rigidity conjecture for higher rank lattices by developing a novel strategy combining techniques from boundary theory, C*-algebras and von Neumann algebras. These methods will lead to new rigidity phenomena for operator algebras arising from irreducible lattices in higher rank semisimple connected Lie groups.

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