Spring 2020:
Feb 7: Barak Weiss (Tel Aviv University)
Spaces of Cut and Project Quasicrystals: Classification and Statistics
Cut and project sets are wellstudied models of almostperiodic discrete subsets of ${\mathbb R}^d$. In
2014 Marklof and Strombergsson introduced a natural class of random processes which
generate cut and project sets in a way which is invariant under the group $ASL(d, {\mathbb R})$.
Using Ratner’s theorem and the theory of algebraic groups we classify all these measures.
Using the classification we obtain results analogous to those of Siegel, Rogers, and
Schmidt in geometry of numbers: summation formulas and counting points in large sets
for typical cut and project sets. Joint work with Rene Ruehr and Yotam Smilansky.
Feb 14: Alba Málaga (ICERM)
Generic Dynamics of Staircase Surfaces
A staircase surface is an infinite measure, infinite genus translation surface obtained by gluing rectangles in a
way similar to steps of a staircase. In this talk I will introduce parameterized families of staircases, then I'll
explain why for a $G_\delta$ dense set of parameters the corresponding staircase has nice dynamics (minimality, ergodicity, etc).
This is joint work with Serge Troubetzkoy.
Feb 21: Dragomir Saric (Queens College and Graduate Center of CUNY)
The Heights Theorem for Infinite Riemann Surfaces
Marden and Strebel established the Heights Theorem for integrable holomorphic quadratic differentials
on parabolic Riemann surfaces. We extends the validity of the Heights Theorem to all surfaces whose
fundamental group is of the first kind. In fact, we establish a more general result: the horizontal
map which assigns to each integrable holomorphic quadratic differential a measured lamination obtained
by straightening the horizontal trajectories of the quadratic differential is injective for an arbitrary
Riemann surface with a conformal hyperbolic metric. This was established by Strebel in the case of the unit disk.
When a hyperbolic surface has a bounded geodesic pants decomposition, the horizontal map assigns a bounded measured
lamination to each integrable holomorphic quadratic differential. When surface has a sequence of closed geodesics whose
lengths go to zero, then there exists an integrable holomorphic quadratic differential whose horizontal measured
lamination is not bounded. We also give a sufficient condition for the nonintegrable holomorphic quadratic
differential to give rise to bounded measured laminations.
Feb 28: Benjamin Dozier (Stony Brook University)
Translation Surfaces with Multiple Short Saddle Connections
Questions about billiards on rational polygons can be converted into questions about the straightline flow on translation surfaces. These in turn can be converted (via renormalization) into questions about the dynamics of the $SL_2({\mathbb R})$ action on strata of translation surfaces. By the pioneering work of EskinMirzakhani, to understand dynamics on strata, one is led to study "affine" measures.
It is natural to ask about the relation between measures of certain subsets of surfaces and the geometric properties of the surfaces. I will discuss a proof of a bound on the volume, with respect to any affine measure, of the locus of surfaces that have multiple independent short saddle connections. A key tool is the new compactification of strata due to BainbridgeChenGendronGrushevskyMoller, which gives a good picture of how a translation surface can degenerate.
The remaining scheduled talks of our seminar have been canceled until further notice.
