Homogeneous dynamics, unitary representations, and Diophantine exponents
Seminar Room 1, Newton Institute
AbstractWe will describe an explicit quantitative form of the duality principle in homogeneous dynamics. This allows the reduction of a diverse set of quantitative equidistribution problems on homogeneous spaces G/H to the problem of giving explicit spectral bounds for the restriction of automorphic representations of G to the stability subgroup H. We will demonstrate this approach by deriving bounds for Diophantine approximation exponent on homogeneous varieties, a problem raised by Serge Lang already in 1965, but which have seen little progress since then. The Diophantine exponents we derive are in many cases best possible, a remarkable fact that follows from an important and useful representation-theoretic phenomenon which we will highlight. Based on Joint work with Alex Gorodnik (Bristol University) and on joint work with Anish Ghosh (TIFR Mumbai) and Alex Gorodnik.
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