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Homogeneous Dynamics and Number Theory

Published in Courses
Wednesday, 09-25-2019

Topic and importance

Homogeneous dynamics is the study of actions of subgroups of Lie groups on their quotients called homogeneous spaces. This has been a very active area of research for several decades. Developments in the area have also led to important breakthroughs in other areas, in particular number theory and mathematical physics. A few examples are the proof of the Oppenheim conjecture (Margulis, 1986), a proof of Quantum Unique Ergodicity (QUE) for particular arithmetic surfaces (Lindenstrauss, 2006), and a strong result towards the Littlewood conjecture (Einsiedler, Katok, Lindenstrauss, 2006). Another application is to describe the periodic Lorentz gas in the Boltzmann-Grad limit (Marklof, Strombergsson, 2011). Many of the results obtained in homogeneous dynamics are intrinsically ineffective, and there is currently much interest in the problem of proving effective versions of such results.


We will start by giving an introduction to the setting and problems in homogeneous dynamics, and we will discuss various examples in detail. We will also describe the basic set-up for how results from homogeneous dynamics are applied in the proof of the Oppenheim conjecture, the Littlewood conjecture and Arithmetic QUE. Finally, we will discuss by now classical applications of homogeneous dynamics to problems in Diophantine approximation and to the problem of asymptotically counting the number of integer points on varieties with large symmetry groups.


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