|Friday May 18|
|8:30 - 9:20||Registration||4th floor Math Department|
|9:20 - 9:30||Welcome||380C||9:30 - 10:30||Jenya Sapir||380C||10:30 - 11:00||Break||11:00 - 12:00||Ben Dozier||380C||12:00 - 1:30||Lunch||1:30 - 2:30||Jon Chaika||380C||2:30 - 3:00||Break||3:00 - 4:00||Amir Mohammadi||380C||4:15 - 5:15||Elon Lindenstrauss||380C||5:30||Reception||4th floor Math Department|
|Saturday May 19||9:30 - 10:30||Hee Oh||380C||10:30 - 11:00||Break||11:00 - 12:00||Francois Labourie||380C||12:00 - 2:00||Lunch||2:00 - 3:00||Alex Eskin||380C||3:00 - 3:30||Break||3:30 - 4:30||Anton Zorich||380C||4:45 - 5:45||Curtis McMullen||380C||6:30||Banquet||Faculty Club|
|Sunday May 20||9:15 - 10:15||David Dumas||380C||10:30 - 11:30||Amie Wilkinson||380C||11:30 - 12:30||Catered brunch/lunch||4th floor Math Department||12:30 - 1:30||Kasra Rafi||380C||1:45 - 2:45||Alex Wright||380C|
Title: Horocycle orbits in strata of translation surfaces
Abstract: Ratner, Margulis, Dani and others, showed that the horocycle
flow on homogeneous spaces has strong measure theoretic and topological
rigidity properties. Eskin-Mirzakhani and Eskin-Mirzakhani-Mohommadi,
showed that the action of SL(2,R) and the upper triangular subgroup of
SL(2,R) on strata of translation surfaces have similar rigidity
properties. We will describe how some of these results fail for the
horocycle flow on strata of translation surfaces. In particular,
1) There exist horocycle orbit closures with fractional Hausdorff dimension.
2) There exist points which do not equidistribute under the horocycle flow with respect to any measure.
3) There exist points which equidistribute distribute under the horocycle flow to a measure, but they are not in the topological support of that measure.
This is joint work with John Smillie and Barak Weiss.
Title: Equidistribution of saddle connections on translation surfaces
Abstract: I will show that on any translation surface, the collection of saddle connections of length at most R becomes equidistributed on the surface as R tends to infinity. This result has analogs in the world of hyperbolic and negatively curved surfaces, but the proof in the flat setting is completely different. A key ingredient in the proof is a result on counting saddle connections whose angle lies in a prescribed interval, which generalizes Masur's original quadratic upper bound on saddle connections.
Title: Asymptotics of Hitchin's metric on Teichmüller space
Abstract: In 1987, Hitchin constructed a metric on the Teichmüller space of a compact surface that depends on the choice of a base point in this space. This metric arises naturally from the construction of the moduli space of Higgs bundles, and it is the restriction of a hyperkähler metric on the SL(2,C) character variety, however it is not invariant under the action of the mapping class group.
I will discuss recent work with Andrew Neitzke where we show that as one goes to infinity in a generic direction in Teichmüller space, Hitchin's metric is exponentially asymptotic to another metric that has a simple, explicit formula (the "semiflat approximation"). This behavior was predicted in a conjecture of Gaiotto-Moore-Neitzke, who also proposed a more general asymptotic formula for the hyperkähler metric on the complex character variety.
Title: On stationary measure rigidity and orbit closures for actions of non-abelian groups
Abstract: I will describe joint work in progress with Aaron Brown, Federico Rodriguez-Hertz and Simion Filip. Our aim is to find some analogue, in the context of smooth dynamics, of Ratner's theorems on unipotent flows. This would be a (partial) generalization of the results of Benoist-Quint and my work with Elon Lindenstrauss in the homogeneous setting, the results of Brown and Rodriguez-Hertz in dimension 2, and my results with Maryam Mirzakhani in the setting of Teichmuller dynamics.
Title: The probabilistic nature of McShane--Mirzakhani's identities
Abstract: I will explain a joint work with Ser Peow Tan giving a probabilistic approach to McShane--Mirzakhani's identities. I will then discuss a few possible applications.
Title: On stationary and invariant measures.
Abstract: Eskin and Mirzakhani famously classified the measures on moduli space of quadratic differentials invariant under the Borel subgroup of SL_2(R) --- a 2 parameter group that includes a diagonalizable action and a unipotent one. In their proof they employed ideas that came from the Benoist-Quint work on classifying stationary measures on homogeneous spaces.
In my talk I will discuss the relation between classifying stationary measures and classifying measures invariant under a diagonalizable group, and present new results - one joint with Eskin on stationary measures and one joint with Einsiedler on diagonalizable actions.
Title: Polygons, periods, and Teichmüller theory
Abstract: We will discuss a unified approach, based on quadrilaterals, to a suite of new totally geodesic curves and surfaces in moduli space, new SL_2(R) invariant varieties in the space of 1-forms, and new billiard tables with optimal dynamics. Based on joint works with Eskin, Mukamel and Wright.
Title: Effective results in homogeneous dynamics
Abstract: Rigidity phenomena in homogeneous dynamics have been extensively studied over the past few decades with several striking results and applications. In this talk we will give an overview of the more recent activities which aim at presenting quantitative versions of some of these strong rigidity results.
Title: Orbit closures of the SL(2,R) action on Kleinian manifolds
Abstract: We will discuss the action of SL(2,R) on the quotient space X=Gamma\SL(2,C) for a discrete subgroup Gamma < SL(2,C). More precisely, let Gamma < SL(2,C) be a convex cocompact acylindrical Kleinian group, and let F be the minimal open SL(2,R)-invariant subset of X above the interior of the convex core of the hyperbolic manifold Gamma\H^3. We classify all possible closures of SL(2, R) orbits in F. An immediate consequence is the classification of all possible closures of geodesic planes in the interior of the core of Gamma\H^3.
By Mostow rigidity, there are only countably many lattices in SL(2,C) up to conjugation and in those cases, these results were proved by Ratner and Shah independently almost 30 years ago. Our results present the first quasi-isometry invariant family of uncountably many Kleinian manifolds for which a strong topological rigidity of geodesic planes is established.
This talk is based on joint work with McMullen and Mohammadi.
Title: Local maxima of the systole function
Abstract: We construct infinite families of closed hyperbolic surfaces that are local maxima for the systole function on their respective moduli spaces. The systole takes values along a linearly divergent sequence L_n at these local maxima. The only surface corresponding to L_1 is the Bolza surface in genus 2, but for large g and n > 2, the number of local maxima of the systole in M_g at height L_n grows super-exponentially in g. Most of the surfaces we construct have no non-trivial orientation-preserving automorphisms, and are the first examples of local maxima with this property. This is a joint work with Maxime Fortier Bourque.
Title: Tessellations from long geodesics on surfaces
Abstract: I will talk about a recent result of Athreya, Lalley, Wroten and myself. Given a hyperbolic surface S, a typical long geodesic arc will divide the surface into many polygons. We give statistics for the geometry of a typical tessellation. Along the way, we look at how very long geodesic arcs behave in very small balls on the surface.
Title: Partial hyperbolicity, exponents and (super)rigidity
Abstract: A superrigidity phenomenon, in its loosest formulation, occurs when one can reconstruct the action of a Lie group from that of a discrete subgroup. In Margulis's superrigidity theorem, the Lie group is large (higher rank semisimple) and the subgroup is a lattice. In this talk, I will indicate how one of the core ideas behind Margulis's superrigidity also lies behind a superrigidity phenomenon in partially hyperbolic dynamics: under relatively mild hypotheses, one can reconstruct a flow from the action of a diffeomorphism. Here the Lie group is quite small (the real numbers), but hidden in the wings is the action of a much larger group.
Title: Is there a rank 3?
Abstract: We will discuss open problems regarding the classification of GL(2,R) invariant subvarieties of the Hodge bundle. It may be too optimistic to hope for a complete classification in the foreseeable future, and in any case it is helpful to have intermediate goals that would clarify the mystery: Are there many new invariant subvarieties out there? We will present more than one such intermediate goal, but focus special attention on the wide open problem of whether there is a non-trivial orbit closure of rank at least 3. These terms will all be defined, and we will outline the different tools available for attacking these problems.
Title: Equidistribution of square-tiled surfaces, meanders, and Masur-Veech volumes
Abstract: We show how recent results of the authors on equidistribution of square-tiled surfaces of given combinatorial type allow to compute approximate values of Masur-Veech volumes of the strata in the moduli spaces of Abelian and quadratic differentials by Monte Carlo method.
We also show how similar approach allows to count asymptotical number of meanders of fixed combinatorial type in various settings in all genera. Our formulae are particularly efficient for classical meanders in genus zero.
We construct a bridge between flat and hyperbolic worlds giving a formula for the Masur-Veech volume of the moduli space of quadratic differentials in terms of intersection numbers (in the spirit of Mirzakhani's formula for Weil-Peterson volume of the moduli space of pointed curves).
Joint work with V. Delecroix, E. Goujard, P. Zograf