Title: “The geometry of eigenvalue extremal problems”
Abstract: When we choose a metric on a manifold we determine the spectrum of
the Laplace operator. Thus an eigenvalue may be considered as a functional
on the space of metrics. For example the first eigenvalue would be the fundamental
vibrational frequency. In some cases the normalized eigenvalues are bounded
independent of the metric. In such cases it makes sense to attempt to find
critical points in the space of metrics. In this talk we will survey two cases in
Abstract - Equidistribution problems, originating from the classical works of Kronecker, Hardy and Weyl about equidistribution of sequences mod 1, are of major interest in modern number theory.
We will discuss how some of those problems relate to unipotent flows and present a conjecture by Margulis, Sarnak and Shah regarding an analogue of those results for the case of the horocyclic flow over a Riemann surface. Moreover, we provide evidence towards this conjecture by bounding from above the Hausdorff dimension of the set of points which do not equidistribute.
Title: Counting points and counting representations
I will talk about the following questions:
1) Given a system of polynomial equations with integer coefficients, how many solutions does it have in the ring Z/N?
2) Given a polynomial map f:R^a-->R^b and a smooth, compactly supported measure m on R^a, does the push-forward of m by f have bounded density?
3) Given a lattice in a higher rank Lie group (say, SL(n,Z) for n>2). How many d-dimensional representations does it have?