# Informal Geometry and Dynamics Seminar

## Spring 2020

### Upcoming talks

 No more talks this semester. Please check the following pages for other seminars throughout the summer: BiSTRO, ADIOS.

### Past talks

 Feb 5: Open Neighborhood Seminar: Negatively curved crystals Curtis McMullen, Harvard University 4:30 - 5:30 pm, Science Center 507 *Different time and venue* This is part of the Open Neighborhood Seminar. [abstract] Imagine the universe is a periodic crystal. If gravity makes space negatively curved, the thin walls of the crystalline structure might trace out a pattern of circles in the sky, visible at night. In this talk we will describe how to generate pictures of these patterns and how to think like a hyperbolic astronomer. We also touch on the connection to knots and links and arithmetic groups. The lecture is accompanied by an exhibit of prints in the Science Center lobby. (This talk will be accessible to members of the department at all levels.) Feb 19: Spaces of knots in exotic 4-manifolds Alexander Kupers, Harvard University [abstract] Can you use the homotopy type of the space of knots in a simply-connected 4-manifold to distinguish smooth structures? The answer is no, using embedding calculus. I will also give some examples which show that embedding calculus does distinguish smooth structures in high dimensions. This is joint with Ben Knudsen. Mar 4: Pseudo-Anosov mapping classes with large dilatation Eriko Hironaka, Harvard University [abstract] I'll talk about some subclasses of pseudo-Anosov mapping classes whose dilatations are bounded away from 1. Mar 11: Billiards, heights and non-arithmetic groups Curtis McMullen, Harvard University Mar 25: Absolute period leaves and the Arnoux—Yoccoz example in genus 3 Karl Winsor, Harvard University April 1: Heights Curtis McMullen, Harvard University [abstract][slides] We will describe how the problem of finding periodic trajectories in a regular pentagon can be solved using a new height on P^1 coming from real multiplication. April 8: Effective density for values of generic quadratic forms Dubi Kelmer, Boston College [abstract] The Oppenheim Conjecture, proved by Margulis, states that any irrational quadratic form, has values (at integer coordinates) that are dense on the real line. However, obtaining effective estimates for any given form is a very difficult problem. In this talk I will discuss recent results, where such effective estimates are obtained for generic forms using a combination of methods from dynamics and analytic number theory. I will also discuss some results on analogous problems for inhomogenous forms and more general higher degree polynomials. Followed by a 30-min conclusion to the talk from last Wednesday (April 1): Heights Curtis McMullen, Harvard University [abstract][slides] We will describe how the problem of finding periodic trajectories in a regular pentagon can be solved using a new height on P^1 coming from real multiplication. April 15: Framed mapping class groups and strata of abelian differentials Nick Salter, Columbia University [abstract][slides] Strata of abelian differentials have long been of interest for their dynamical and algebro-geometric properties, but relatively little is understood about their topology. I will describe a project aimed at understanding the (orbifold) fundamental groups of non-hyperelliptic stratum components. The centerpiece of this is the monodromy representation valued in the mapping class group of the surface relative to the zeroes of the differential. For g \ge 5, we give a complete description of this as the stabilizer of the framing of the (punctured) surface arising from the flat structure associated to the differential. This is joint work with Aaron Calderon. April 22: Pseudo-Anosov maps and toral automorphisms Rick Kenyon, Yale University [abstract][slides] We give a construction of a pseudo-Anosov map of a surface starting from (and almost isomorphic to) a hyperbolic automorphism of an n-torus. The construction arises from a peano curve based on an invariant space-filling tree. This construction allows to confirm (for degree 3) a conjecture of Fried regarding stretch factors of pseudo-Anosov maps. April 29: Large genus bounds for the distribution of triangulated surfaces in moduli space Sahana Vasudevan, MIT [abstract][slides] Triangulated surfaces are compact (hyperbolic) Riemann surfaces that admit a conformal triangulation by equilateral triangles. Brooks and Makover started the study of the geometry of random large genus triangulated surfaces. Mirzakhani later proved analogous results for random hyperbolic surfaces. These results, along with many others, suggest that the geometry of triangulated surfaces mirrors the geometry of arbitrary hyperbolic surfaces especially in the case of large genus asymptotics. In this talk, I will describe an approach to show that triangulated surfaces are asymptotically well-distributed in moduli space. May 6: Coarse density of subsets of moduli space Benjamin Dozier, Stony Brook [abstract][arxiv paper] I will discuss coarse geometric properties of algebraic subvarieties of the moduli space of Riemann surfaces. In joint work with Jenya Sapir, we prove that such a subvariety is coarsely dense, with respect to either the Teichmuller or Thurston metric, iff it has full dimension in the moduli space. This work was motivated by an attempt to understand the geometry of the image of the projection map from a stratum of abelian or quadratic differentials to the moduli space of Riemann surfaces. As a corollary of our theorem, we characterize when this image is coarsely dense. A key part of the proof of the theorem involves comparing analytic plumbing coordinates at the Deligne-Mumford boundary to hyperbolic/extremal lengths of curves on nearby smooth surfaces. May 13: In the moduli space of Abelian differentials, big invariant subvarieties come from topology Paul Apisa, Yale University [abstract][slides] It is a beautiful fact that any holomorphic one-form on a genus g Riemann surface can be presented as a collection of polygons in the plane with sides identified by translation. Since GL(2, R) acts on the plane (and polygons in it), it follows that there is an action of GL(2, R) on the collection of holomorphic one-forms on Riemann surfaces. This GL(2, R) action can also be described as the group action generated by scalar multiplication and Teichmuller geodesic flow. By work of McMullen in genus two, and Eskin, Mirzakhani, and Mohammadi in general, given any holomorphic one-form, the closure of its GL(2, R) orbit is an algebraic variety. While McMullen classified these orbit closures in genus two, little is known in higher genus. In the first part of the talk, I will describe the Mirzakhani-Wright boundary of an invariant subvariety (using mostly pictures) and a new result about reconstructing an orbit closure from its boundary. In the second part of the talk, I will define the rank of an invariant subvariety - a measure of size related to dimension - and explain why invariant subvarieties of rank greater than g/2 are loci of branched covers of lower genus Riemann surfaces. This will address a question of Mirzakhani. No background on Teichmuller theory or dynamics will be assumed. This material is work in progress with Alex Wright. June 3: Slices of Thurston's Master Teapot Kathryn Lindsey, Boston College [abstract][slides] Thurston's Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a "restricted iterated function system." An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.

Harvard Math Department events

Organized by Tina Torkaman, Karl Winsor and Yongquan Zhang