Tuesday, November 17, 2020
11/17/2020 - 9:00am
The QUE conjecture of Rudnick and Sarnak asserts that high-frequency Laplace eigenfunctions on a negatively curved compact manifold become equidistributed. A well-studied variant of this problem, known as Arithmetic QUE, concerns the distribution of Hecke--Maass forms on locally symmetric spaces. In 2014 Brooks and Lindenstrauss proved AQUE for certain compact hyperbolic surfaces, for eigenfunctions of the Laplacian and only one Hecke operator. We generalize this result to higher rank spaces.
11/17/2020 - 4:00pm
In this talk we will discuss quasi-Hitchin representations in Sp(4,C). In the same way that Hitchin representations correspond to a natural higher-rank generalization of Fuchsian representations, quasi-Hitchin representations correspond to a higher-rank generalization of quasi-Fuchsian representations. Unfortunately, in the higher rank setting, an interpretation in term of geometric structures is much less obvious than in the classical case, and this is what we will discuss in this talk in the specific case of quasi-Hitchin representations in Sp(4,C). Our result will follow from a parametrization of the space Lag(C^4) of complex lagrangian grassmanian subspaces of C^4 as the space of regular ideal hyperbolic tetrahedra and their degenerations. This theory generalizes the classical and important theory of quasi-Fuchsian representations and their action on the Riemann sphere CP^1 = Lag (C^2). (This is joint work with D. Alessandrini and A. Wienhard.)