Geometry and Spectra of Compact Riemann Surfaces

There are many geometric and topological invariants one can associate with homotopy classes of closed curves. These include algebraic and geometric self-intersection number, intersection with curves in a class of curves (for example, simple ones), mapping class group stabilizers of a curve, and many others. How these invariants interact and determine the curve type (mapping class group orbit) is an active area of research today. In this talk, we consider the relationship, in various settings, between length and self-intersection. One such setting we will discuss involves the so-called inf invariant (shortest length among all metrics) for a closed curve, and its relationship with the geometric self-intersection number and the optimal (designer) metric that is tailored to produce the minimum length.

A finitely generated subgroup of a finitely generated group admits two natural word (Cayley graph) metrics: that of the subgroup and that of the ambient group. The subgroup is said to be undistorted if these two metrics are equivalent (i.e., the ratio between the two is bounded away from 0 and infinity), and distorted otherwise. I will first survey the notion of distortion in groups, with an emphasis on cyclic subgroups. Then I will address the question of finding distorted abelian subgroups of minimal rank in semidirect products ℤ^d ⋊ ℤ. This question turns out to have a number-theoretic flavor.

For a divergent sequence of Schottky groups in the hyperbolic space ℍ^N, the Hausdorff dimension of their limit sets tends to 0. By embedding these groups into the group of isometries of infinite-dimensional hyperbolic space, we determine the rate of convergence. (Joint work with Antonin Guilloux.)

In this talk, we review classical results on how geometry influences the small eigenvalues of the Laplace-Beltrami operator on surfaces. Then we discuss recent work extending this picture to Steklov eigenvalues on hyperbolic surfaces.

We know from work of Thurston that every surface S of negative Euler characteristic admits plenty of pseudo-Anosov homeomorphisms, and that their mapping torus is always a hyperbolic 3-manifold. It is natural to ask whether such a picture has some higher-dimensional analogues. We describe one case where it does: we propose a higher-dimensional generalization of the notion of pseudo-Anosov homeomorphism, and build one 4-dimensional case whose mapping torus is a hyperbolic 5-manifold.

The systole of a hyperbolic surface is the length of the shortest closed geodesic on that surface. How large the systole of a closed hyperbolic surface of a fixed genus can be is a classical question that is still mostly open. I will speak about joint work with Mingkun Liu on how random constructions can be used to build surfaces with large systoles.

Given a hyperbolic manifold, the spectrum of its Laplacian contains a great deal of information about its geometric structure. In this talk, I will present joint work with Will Hide, Bram Petri and Joe Thomas in which we study the spectral gap -the first nonzero eigenvalue- of two models of random hyperbolic 3-orbifolds related to the Apollonian group. If time permits, I will also mention how this can be used to investigate the bass note spectrum of the set of hyperbolic 3-orbifolds.

I will talk about joint work with my students (Max Blom, Henrik Nordell, Oliver Thim, Jack Vahnberg) and more recently with Gustav Mårdby and Fabio Francesconi, in which we explore connections between the regularity of Laplace eigenfunctions, the geometry of polytopes, and algebraic structures.