Research Projects

The proposed project is pure mathematical in nature. It deals with problems at the forefront of the dynamics of group actions on homogeneous spaces. In addition to their fundamental value, solutions to these problems will open the door to long standing open problems in Diophantine approximation and the geometry of locally symmetric spaces. Recent years have seen spectacular advances at the interface of homogeneous dynamics, number theory and geometry. The work of Einsiedler, Katok and Lindenstrauss towards Littlewood's conjecture, the work of Benoist and Quint on random walks and stationary measures, and the work of Eskin and Lindenstrauss are some prominent examples. The proposed project aims to further these spectacular achievements and advance them in a "quantitative" aspect. Making ergodic theory quantitative or "effective" is a very fundamental challenge and one that is being pursued by the world's leading mathematicians at present. A famous, century old theorem of Levy and Khintchine provides a limiting growth rate for the convergents of continued fraction estimates of real numbers. However, no quantitative information, for example a speed of convergence is known. This problem can be translated to homogeneous dynamics where it leads to the problem of studying "how fast" the orbits of certain flows distribute on the modular surface. The PI aims to develop new ideas to study the latter problem, and this will then give a wealth of information on the Levy-Khintchine theorem.  A related, remarkable programme of research was suggested by Mahler. He asked about the Diophantine properties of the middle third Cantor set. This is a very difficult circle of problems whose study has only begun relatively recently with the advent of random walk techniques. In short, it is often possible to model Diophantine behaviour on the limit set of an iterated function system using an associated random walk on a homogeneous space. The PI plans to further develop the seminal works of Benoist and Quint and Eskin and Lindenstrauss and build a comprehensive set of tools which can be used to study these challenging questions. Much of the dynamics that will be studied involves flows on homogeneous spaces, including the far more challenging multi-parameter actions. These are naturally connected to geometric objects, namely the geodesic flow on a locally symmetric space. A part of the project will involve a detailed examination of the asymptotic behaviour of geodesics with a view towards establishing high rank analogues of the famous "logarithm laws" of Sullivan and of Kleinbock and Margulis. The high rank situation is far more challenging than the rank one situation as one loses hyperbolicity and with it, many nice aspects of Patterson-Sullivan theory. The PI has extensive experience and an established record of studying these problems at the highest levels.