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Pillowcases and quasimodular forms
Alex Eskin, Andrei Okounkov
We prove that natural generating functions for enumeration of branched coverings of the pillowcase orbifold are level 2 quasimodular forms. This gives a way to compute the volumes…
Unipotent flows on the space of branched covers of Veech surfaces
Alex Eskin, Jens Marklof, Dave Witte Morris
There is a natural action of SL(2,R) on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup $U = {\begin{pmatrix} 1 & * 0 & 1 \end{pmatrix…
Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems and the Siegel--Veech Constants
Alex Eskin, Howard Masur, Anton Zorich
A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found…
Billiards in rectangles with barriers
Alex Eskin, Howard Masur, Martin Schmoll
We use Ratner's theorem to compute the asymptotics of the number of (cylinders of) periodic trajectories in a rectangle with a barrier, assuming that the location p/q of the barrie…