Title: Counting for Right-Angled Billiards
Abstract: In joint work with A. Eskin and A. Zorich, we compute (weak) quadratic asymptotics for counting special trajectories for billiards in polygons whose angles are integer multiples of 90 degrees. A key tool is computing volumes of moduli spaces of meromorphic quadratic differentials on CP, via enumerating pillowcase covers.
Abstract: Teichmueller space is topologically an open ball which has numerous compactifications. In joint work with H. Masur and M. Wolf we show that there are abelian differentials with minimal but not uniquely ergodic vertical foliations so that their limit set in Thurston's compactification, PMF,
a) is a unique point
b) is a line segment
c) an ergodic (but not uniquely ergodic) minimal abelian differential
which has a line segment as its limit set.
d) an ergodic (but not uniquely ergodic) minimal abelian differential which has a unique point as its limit set.
These examples arise from Veech's example of minimal and not uniquely ergodic Z_2 skew products of rotations which are related to two tori glued along a slit. Masur proved that the geodesic defined by a quadratic differential with uniquely ergodic vertical foliation has a (unique) limit in PMF and that it was what one would expect. Lenzhen constructed an example of a non-minimal quadratic differential that did not have a limit in PMF (the limit set was a line segment).
Abstract: SL(2,R) invariant subspaces of the homology of a surface which are also measurable with respect to an SL(2,R) invariant measure are one of the most important properties of the measure. They appear in several contexts, such as generalizations of the wind-tree model or the classification of Teichmuller curves. I will state the definitions, survey what is known, and state some conjectures.
Abstract: In the most classic example of a translation surface there is a close connection between dynamics and algebra: A billiard trajectory on the torus is periodic if and only if its slope lies in the field of rational numbers. In this talk I want to discuss work of a number of people which relates translation surfaces to number fields. In particular I will relate new work of Wright on fields of definition to old joint work with Calta.