Videos
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Videos on high rank
These videos describe work with Alex Wright to show that if an invariant subvariety has rank at least one plus half genus, then it is either a stratum or a locus of double covers.
In depth (five part lecture series delivered with Alex Wright at the Pacific Dynamics Seminar in January-February 2021):
lecture 1,
lecture 2,
lecture 3 (pdf),
lecture 4 (pdf),
lecture 5.
Moderate detail (60 minute lecture delivered at BIRS CMO Oaxaca May 2019): If an orbit closure is big then it is trivial!
Quick synopsis (25 minute lecture delivered at the Nearly Carbon Neutral Geometric Topology Conference May 2020): Orbits in the moduli space of translation surfaces
Videos on other research projects
Marked points in genus two and beyond (Institut Fourier June 2018)(abstract)
In the principal stratum in genus two, McMullen observed that something odd happens - there is only one nonarithmetic Teichmuller curve - the one generated by the decagon. This strange phenomenon begets another - a primitive translation surface in genus two admits a periodic point that is not a Weierstrass point or zero only if it belongs to the golden eigenform locus. In this talk, we will explain how to leverage results of Mirzakhani-Wright to study the orbit closures of translation surfaces with marked points and sketch a proof of the previously mentioned result in genus two. We will also explain how the result in genus two proves another uniqueness results - that there is at most one nonarithmetic rank two orbit closure in the minimal stratum in genus four - the one discovered by Eskin-McMullen-Mukamel-Wright.
(talk by Howard Masur on joint work) Teichmuller geodesics that diverge on average in moduli space (Warwick EPSRC Symposium on Geometry, Topology and Dynamics in Low Dimensions March 2018) (abstract)
Given any quadratic differential the Hausdorff dimension of the divergent on average directions is 1/2.
Shouting across the void - Low dimensional orbit closures in hyperelliptic components of strata (Warwick EPSRC Symposium on Geometry, Topology and Dynamics in Low Dimensions March 2018) (abstract)
Every orbit in a hyperelliptic component of a stratum of Abelian differentials is closed, dense, or contained in a locus of branched covers.
Marked points, Hubbard and Earle-Kra, and illumination (CMO BIRS Oaxaca 2016) (abstract)
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.
Expository videos
The counting formula of Eskin and McMullen (University of Utah Seminar on Ergodic Theory June 2020) (abstract)
Given a lattice acting on the hyperbolic plane, how many orbits of a point intersect the ball the radius of r as r gets big? Similarly, given a hyperbolic surface with a geodesic gamma, how many lifts of gamma to the hyperbolic plane intersect the ball of radius r? Using the mixing of geodesic flow on hyperbolic surfaces, Eskin and McMullen found a short beautiful argument to find the asymptotics for these counting questions (and more general ones on affine symmetric spaces). In this talk, we explain the main ideas behind this formula.