Unravelling The Mysteries Of Homogeneous Dynamics

How do you study arithmetic objects like integer points using the theory of dynamical systems? The answer is homogenous dynamics, and this connection goes both ways. The GMODGAMMADYNAMICS project, funded by an ERC grant, took a broad approach towards studying this rich interplay.

Homogeneous dynamics is the study of the asymptotic properties of the action of Lie groups on homogenous spaces such as the space of lattices in n dimensional space. “This project looked at the connection these actions have to number theory, to the spectral theory of homogenous spaces and to arithmetic combinatorics,” says Prof. Elon Lindenstrauss. “It’s a beautiful area that interacts with a lot of mathematics.”

One of the mysteries that the project team addressed was the subtle rigidity properties of higher rank abelian actions on homogenous spaces. They also focused on the combinatorial properties of arithmetic groups, equidistribution, spectral gaps, random walks and quantum ergodicity. A key achievement of this work was the creation of a very general joining classification theorem for higher rank diagonalisable groups. “In some sense, this tool is a definitive result: one that has already found interesting applications,” says Prof. Lindenstrauss. For example, this joining result was used by other researchers to establish the joint distribution of integer vectors on Eucleadian spheres and the shape of the lattice orthogonal to this vector. Another important outcome was the solving of a 50-year-old conjecture on local limit theorem for random walks on the isometry group of Euclidean spaces by a team member, Dr. Peter Varju, which gave way to a very precise local-central limit theorem for such random walks. Despite these significant breakthroughs, Prof. Lindenstrauss says that this is just the beginning: “There are ideas developed in this project that we are eager to take even further forward. Furstenberg’s Conjecture on x2 x3 invariant measures and Littlewood’s Conjecture are both still open, so there’s still plenty more to think about!”

Elon Lindenstrauss obtained his PhD at the Hebrew University of Jerusalem (Israel) under the guidance of Benjamin Weiss in 1999. He held positions at the IAS in Princeton, Stanford University and Princeton University (USA) before returning to the Hebrew University in 2008 where he is currently Professor of Mathematics and since 2016 the Chair of the Einstein Institute of Mathematics. He received several prizes for his mathematical achievements, including the Fields Medal. He is a member of the Israel Academy of Sciences and Humanities and the Academia Europaea.


For more information: https://erc.europa.eu/projects-figures/stories/unravelling-mysteries-homogeneous-dynamics