Complex Analysis and Dynamics Seminar

Spring 2020 Schedule

Feb 7: Barak Weiss (Tel Aviv University)
Spaces of Cut and Project Quasicrystals: Classification and Statistics

Cut and project sets are well-studied models of almost-periodic 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 non-integrable 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 straight-line 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 Eskin-Mirzakhani, 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 Bainbridge-Chen-Gendron-Grushevsky-Moller, which gives a good picture of how a translation surface can degenerate.

The remaining scheduled talks of the seminar were canceled due to Covid-19 pandemic.