Counting simple closed multi-geodesics on hyperbolic surfaces with respect to the lengths of individual components- Francisco Arana-Herrera (Stanford)
Abstract: In her thesis, Mirzakhani showed that on any closed hyperbolic surface of genus g, the number of simple closed geodesics of length at most L is asymptotic to a polynomial in L of degree 6g-6. Wolpert conjectured that analogous results should hold for more general countings of multi-geodesics that keep track of the lengths of individual components. In this talk we will present a proof of this conjecture which combines techniques and results of Mirzakhani as well as ideas introduced by Margulis in his thesis.
Cutting and pasting along measured laminations- Aaron Calderon (Yale)
Abstract: One of the fundamental techniques of low-dimensional topology is cutting and pasting along embedded codimension 1 submanifolds. In surface theory these submanifolds are just simple closed curves, and cutting and pasting gives rise to Fenchel-Nielsen coordinates for Teichmüller space, normal forms for simple closed curves, and many other foundational results. The set of simple closed curves completes to the space of “measured laminations,” and in my first talk, I will summarize how to build coordinates for Teichmüller space by cutting and pasting along any lamination, generalizing the shear coordinates of Bonahon and Thurston. In my second talk, I will explain how these coordinates lead to an extension of Mirzakhani’s conjugacy between the earthquake and horocycle flows, two notions of unipotent flow coming from hyperbolic, respectively flat, geometry. This represents joint work with James Farre.
Volumes and filling collections of multicurves- Andrew Yarmola (Princeton)
Abstract: Consider a link L in a Seifert-fibered space N over a surface S of negative Euler characteristic. If the fiber-wise projection of L to S is a collection C of closed curves in minimal position, then N \ L is hyperbolic if and only if C is filling and N \ L is acylindrical. The behavior of vol(N \ L) in terms of the topology and geometry of C have been studied in recent years, but effective lower bounds have been elusive. In this talk we will focus on the case where C is a collection of simple closed curves. In the special case where N = PT(S) is the projectivized tangent bundle and L is the canonical lift of a pair of filling multicurves, we show that vol(N \ L) is quasi-isometric to the Weil-Petersson distance between the corresponding strata in Teichmuller space. In the more general setting, we show that vol(N \ L) is quasi-isometric to expressions involving distances in the pants graph whenever L is a stratified link. This is joint work with T. Cremaschi and J. A. Rodriguez-Migueles.