Schedule for: 16w5010 - Flat Surfaces and Dynamics of Moduli Space

A welcome drink will be served at the hotel.

We explain that the SL(2,R) orbit closure of a typical periodic orbit for the Teichmuelller flow on a stratum of abelian differentials in genus at least three equals the entire stratum. Similar ideas also yield finiteness of algebraically primitive Teichmüller curves in genus at least three.

In this talk, I will introduce a compactification of the strata of abelian differentials inspired by the Deligne-Mumford compactification of the moduli space of abelian differentials. The main result will be the explicit description of the elements of this compactification for every stratum. Moreover, I will emphasize the flat geometric point of view of this construction. This is joint work with M. Bainbridge, D. Chen, S. Grushevsky and M. M\"oller.

We will present an algebraic surface in the moduli space of triply punctured tori with the remarkable property that it is totally geodesic for Teichmuller metric. Our example reveals a surprising connection between Teichmuller theory and the geometry of cubic curves and is the first of its kind not arising from a covering construction. This is joint work with C. McMullen and A. Wright.

A k-differential is a section of the k-th power of the cotangent bundle on a Riemann surface. The space of k-differentials is stratified by prescribing the number and multiplicities of their zeros and poles. The cases of abelian and quadratic differentials, corresponding to k=1 and k=2 respectively, exhibit fascinating geometry that arises from the associated flat structures, which provide us a good understanding of the dimension, local coordinates, connected components, and compactification for the strata. In this talk I will report some recent work in progress that studies these aspects of the strata of k-differentials for general k. It is part of a joint project with Bainbridge, Gendron, Grushevsky, and Moeller.

I will describe recent joint work with Matt Bainbridge and Barak Weiss on horocycle flows in for 3 dimensional GL(2,R) invariant loci in genus 2. This work depends on applying methods developed for the analysis of the case of higher dimensional tori to higher genus surfaces. I will examine the extent to which these methods succeed in this setting.

For any flat surface, the closure of its orbit under the horocycle flow in almost any direction is equal to its $SL(2, \mathbb R)$ orbit closure. I will sketch the proof of this result as well as present a characterization of lattice surfaces in terms of minimal sets for the horocycle flow. These results are joint work with Jon Chaika.

Computing the distribution of the gaps between slopes of saddle connections is a question that was studied first by Athreya and Cheung in the case of the torus, motivated by the connection with Farey fractions, and then in the case of the golden L by Athreya, Chaika, and Lelievre. Their strategy involved translating the question of gaps between slopes of saddle connections into return times under horocycle flow on the space of translation surfaces to a specific transversal. We show how to use this strategy to explicitly compute the distribution in the case of the octagon, the first case where the Veech group has multiple cusps, how to generalize the construction of the transversal to the general Veech case (both joint work with Caglar Uyanik), and how to parametrize the transversal in the case of a generic surface in $\mathcal{H}(2)$.

Given a holomorphic family of Riemann surfaces is it possible to associate a holomorphically varying finite collection of points to each Riemann surface in the family? Hubbard showed that when the family is the entire moduli space of genus g Riemann surfaces this is possible only when g = 2 and the marked points are fixed points of the hyperelliptic involution. We will pose and resolve analogous questions for strata of translation surfaces with marked points. We will draw connections between GL(2,R)-invariant families of marked points on affine invariant submanifolds and holomorphically varying collections of points on closed totally geodesic families of Riemann surfaces. Finally we will discuss applications to billiard problems, specifically the finite blocking and illumination problems.

Motivated by some number theoretic considerations, Peter Sarnak has conjectured that the Möbius function of number theory is disjoint from any zero entropy dynamical system. We prove this conjecture for interval exchanges of three intervals under a no loss of mass assumption. This is joint work with Jon Chaika.

Teichmueller geodesics in moduli space are typically dense in this non-compact space. It is natural to ask how long it takes the typical geodesic to leave compact sets for the first time. In particular, we can exhaust moduli space by compact sets given by surfaces with no closed geodesics with length strictly less than c and ask how much time it takes a Teichmueller geodesic to leave such a set. Masur addressed this question by proving a logarithm law. Translation surfaces give Teichmueller geodesics and it is natural to ask if a Teichmuller geodesic satisfying Masur's logarithm law has that the vertical flow on the surface it arises from is uniquely ergodic. We show that it is for the flat systole but not necessarily for the extremal length systole (which coarsely gives distance in Teichmuller space). This is joint work with Rodrigo Trevino.

Among all complex two-dimensional manifolds, K3 surfaces are distinguished for having a wealth of extra structures. They admit dynamically interesting automorphisms, have Ricci-flat metrics (by Yau's solution of the Calabi conjecture) and at the same time can be studied using algebraic geometry. Moreover, their moduli spaces are locally symmetric varieties and many questions about the geometry of K3s reduce to Lie-theoretic ones. In this talk, I will discuss the analogue on K3 surfaces of the following asymptotic question in billiards - How many periodic billiard trajectories of length at most L are there in a given polygon? The analogue of periodic trajectories will be special Lagrangian tori on a K3 surface. Just like for billiards, such tori come in families and give torus fibrations on the K3. I will provide the necessary background on K3 surfaces.

The "rel surgery" (also referred to as kernel foliation, Schiffer variation, absolute period foliation) give rises to an almost-everywhere defined flow which commute with the horocycle flow. In a recent work with Matt Bainbridge and John Smillie, the rel surgery was used in order to construct and classify horocycle invariant measures in the eigenform locus. In another recent work with Pat Hooper, the horocycle flow was used to classify closures of rel leaves. While the results are different, there are certain common aspects to the arguments and certain common difficulties to address.

For every discriminant D, McMullen constructed the Prym-Teichmüller curves in the moduli space of genus 3 and 4 curves. In joint work with David Torres Teigell, which is partially still in progress, we determine the number and orders of orbifold points occurring on these curves. This involves the explicit analysis of the Jacobians of certain families of curves with automorphisms.

I will describe a topology on the space of all translation structures on surfaces with a basepoint (including surfaces of arbitrary topological types). The topology is built in two steps. First, the space of translation structures on the open disk and the natural disk bundle over this space are given second-countable locally-compact metrizable topologies. The observation that the universal cover of a surface with a translation structure is a topological disk allows us to place a topology on the space of all translation structures. I will attempt to concentrate on the practical use of the topologies such as proving a sequence converges in this topology and consequences of such convergence.

The mapping class group of surfaces is a very rich object, directly related to the geometry of the moduli space. Pseudo-Anosov homeomorphisms give a way to understand this geometry. We propose a general framework for studying these homeomorphisms on (half-)translation surfaces. As an application, among other consequences, this allows us to determine the systole of the hyperelliptic connected components (for the Teichmueller metric) for any genus. This is joint work with Corentin Boissy.