Abstract

We 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.