Counting meanders and square-tilings on surfaces- Eduard Duryev (Institut de mathématiques de Jussieu in Paris)
Abstract: Meander is a homotopy class of a pair of transversal simple closed curves on a sphere. They appear, in particular, as natural enumeration of polymer foldings. Statistics of meanders as the number of intersection grows is one question about meanders people have been interested in. We will show how meanders are related to a particular class of square tilings on surfaces and explain an approach to this more general counting problem that uses ribbon graphs and intersection theory on moduli space.
Extremal length systole of the Bolza surface- Didac Martinez-Granado (Indiana University Bloomington)
Abstract: In Bers-Teichmueller theory there is a rich interaction between the hyperbolic geometry and the conformal geometry, as a result of the uniformization theorem. If the length notion in the hyperbolic side of this dichotomy is the hyperbolic length, on the conformal side the counterpart is extremal length, a conformal invariant notion of homotopy classes of curves. For a choice of hyperbolic structure, there is an essential curve of minimal extremal length and one of minimal hyperbolic length. We call both the curve and its length, respectively, the extremal length and hyperbolic length systoles. These are functions on moduli space. Bounds for maximal values of the hyperbolic length systole for some geni are known, and for genus 2, its absolute maximum is the Bolza surface, a triangle surface. In this talk we show that the Bolza surface realizes a local maximum of the extremal length systole, and compute its value: square root of 2. This is work in progress with Maxime Fortier Bourque and Franco Vargas Pallete.
Gaps of saddle connection directions for some branched covers of tori- Anthony Sanchez (University of Washington Seattle)
Abstract: Translation surfaces given by gluing two identical tori along a slit have genus two and two cone-type singularities of angle $4\pi$. There is a distinguished set of trajectories called saddle connections that are the straight lines trajectories between cone points. We can associate a holonomy vector in the plane to each saddle connection whose components are the horizontal and vertical displacement of the saddle connection. How random is the planar set of holonomy of saddle connections? We study this question by computing the gap distribution for slopes of saddle connections for these and other related classes of translation surfaces.
The topology and geometry of random square-tiled surfaces- Sunrose Thapa Shrestha (Tufts University)
Abstract: A square-tiled surface (STS) is a branched cover of the standard square torus with branching over exactly one point. They are concrete examples of translation surfaces which are an important class of singular flat metrics on 2-manifolds with applications in Teichmüller theory and polygonal billiards. In this talk, we will consider a randomizing model for STSs based on permutation pairs and see how to use it to compute the genus distribution. We will also look at holonomy vectors (Euclidean displacement vectors between cone points) on a random STS. Holonomy vectors of translation surfaces provide coordinates on the space of translation surfaces and their enumeration up to a fixed length has been studied by various authors such as Eskin and Masur. We will obtain finer information about the set of holonomy vectors, Hol(S), of a random STS. In particular, we will see how often Hol(S) contains the set of primitive integer vectors and find how often these sets are exactly equal.