Abstract: The dynamics on homogeneous spaces has many interesting connections to number theory. One of the main problems here is to understand the distribution of closed orbits for subgroups H of the ambient Lie group G. In joint work with G.Margulis and A.Venkatesh we prove an error rate in the equidistribution for semisimple subgroups H acting on congruence quotients of G. This makes use of spectral gap in the form of property (tau). However, the proof of our theorem can also be used to prove all cases of property (tau) except for groups of type A_1. We will discuss the relationship between spectral gap, effective decay of matrix coefficients, and effective equidistribution, as well as the main ideas of our argument.

Notes: Dr. Einsiedler, Manfred is an invited speaker at the Congress of European Mathematical Society this summer where he will report on work presented in this colloquium lecture.