Complex Analysis and Dynamics Seminar
Fridays 2:00 - 3:00 pm
Room 6417
Organizers: David Aulicino, Ara Basmajian, Patrick Hooper, Jun Hu
New location this semester: Room 6417.
Spring 2024: Upcoming Seminars
May 3: Sam Freedman (Brown University)
Periodic points of Veech surfaces
We will consider the dynamics of automorphisms acting on highly-symmetric flat surfaces called Veech surfaces. Specifically, we’ll examine the points of the surface that are periodic, i.e., have a finite orbit under the whole automorphism group. While this set is known to be finite for primitive Veech surfaces, for applications it is desirable to determine the periodic points exactly. In this talk we will classify periodic points for the case of minimal Prym eigenforms, certain primitive Veech surfaces in genera 2, 3, and 4.
May 10: Dragomir Saric (Queens College and Graduate Center of CUNY)
The finite-area holomorphic quadratic differentials and the geodesic flow on infinite Riemann surface
Let $X$ be an infinite Riemann surface with a conformally hyperbolic metric. The Hopf-Tsuji-Sullivan theorem states that the geodesic flow is ergodic iff the Poincare series is divergent iff the Brownian motion is recurrent, and many other equivalent conditions are given in the literature. We added an equivalent condition: the Brownian motion on $X$ is recurrent iff almost every horizontal leaf of every finite-area holomorphic quadratic differential is recurrent.
A finite-area holomorphic quadratic differential on $X$ is uniquely determined by the homotopy class of its horizontal foliation, uniquely represented by a measured geodesic lamination on $X$. Most measured geodesic laminations do not come from the horizontal foliations of finite-area differentials. The problem of intrinsically deciding which measured laminations are induced by finite-area differentials is highly transcendental. From now on, assume that $X$ is equipped with a geodesic pants decomposition whose cuffs are bounded. The space of finite-area holomorphic quadratic differentials on $X$ is in a one-to-one correspondence with the measured geodesic laminations on $X$ whose intersection numbers with the cuffs (and “adjoint cuffs”) are square summable. Using this parametrization, we establish that the Brownian motion on $X$ is recurrent iff the simple random walk on the graph dual to the pants decomposition is recurrent.
Spring 2024: Previous Seminars
February 2: Jenya Sapir (Binghamton University)
Geodesics on high genus surfaces
In joint work with Ben Dozier, we study the behavior of closed geodesics on hyperbolic surfaces of large genus. In her celebrated thesis, Mirzakhani showed that the number of simple closed geodesics of length at most $L$ grows asymptotically like the polynomial $L^{6g-6}$, where $g$ is the genus. On the other hand, the number of all closed geodesics grows exponentially, like $e^L/L$ (Delsart, Huber, Selberg). For large $L$, therefore, most curves are non-simple. However, when $L$ is small relative to the genus, $g$, this is no longer the case. In this talk, we will explore what "not-too-long" closed geodesics on large genus hyperbolic surfaces look like.
February 16: Jun Hu (Brooklyn College and The Graduate Center)
Parameter space for cubic polynomials with a period-2 Siegel disk of bounded type
In the talk, we explore the dynamics of cubic polynomials with a period-2 Siegel disk of bounded type
and study different loci on the parameter plane of such cubic polynomials. We show the locus for both two critical points on the boundaries of the two Siegel disks on the 2-cycle consists of a Jordan curve and two arcs. This is a joint work with Yuming Fu and Oleg Muzician.
February 23: Ara Basmajian (Hunter College and The Graduate Center)
From Collars in Riemann surfaces to Tubes in Complex Hyperbolic Manifolds
The celebrated Keen collar lemma guarantees that a simple closed geodesic on a hyperbolic Riemann surface has a collar (tubular neighborhood) whose width only depends on its length. Viewing a Riemann surface as the quotient of the unit ball in the complex plane, a natural generalization is to ball quotients in higher dimensions where the
Poincaré metric is replaced by the Bergman metric (also known as the complex hyperbolic metric). Such ball quotients are called
complex hyperbolic manifolds. The focus of this talk will be on embedded complex (totally) geodesic surfaces in complex hyperbolic manifolds. Note that such a surface has real codimension 2. We prove a tubular neighborhood theorem for such surfaces where the width of the tube depends only on the Euler characteristic of the embedded surface. We give an explicit estimate for this width.
After giving a short history of the collar lemma generalizations and discussing the basics on complex hyperbolic geometry, we will discuss the ideas leading to the proof of this tubular neighborhood theorem. Time permitting we'll supply applications and discuss the sharpness of the width as a function of the Euler characteristic. This is joint work with Youngju Kim.
March 1: Kathryn Lindsey (Boston College)
Topological Entropy, Thurston Sets, and the Mandelbrot Set
Given a family of dynamical systems, which numbers are realized as the topological entropy of some member of the family, and how does the answer to this question inform our understanding of the family? A way to get at this question is to study the Thurston Set -- a subset of the complex plane defined in terms of the collection of all topological entropies realized by the family. I and collaborators proved a variety of results about the geometrical and topological structures of Thurston sets for families of quadratic polynomials. In particular, I will discuss results (joint with G. Tiozzo and C. Wu) about how core entropy varies over the Mandelbrot set. Specifically, we proved that the collection of all Galois conjugates of norm at least 1 of the exponential of core entropy, together with the unit circle, varies continuously in the Hausdorff topology with external angle for the Mandelbrot set. On the other hand, the Galois conjugates with norm < 1 exhibit "Persistence" along principal veins in the Mandelbrot set.
March 8: Rodrigo Treviño (University of Maryland)
The Aperiodic Lorentz Gas
I will talk about joint work with A. Zelerowicz on the statistical properties of the Lorentz gas with aperiodic scatterer configurations. No prior knowledge of Lorentz gases or aperiodic tilings will be assumed.
March 15: Zachary Smith (UCLA)
Thurston’s theorem for four marked points
A (marked) Thurston map $f: (S^2, A)$ to $(S^2, A)$ is a branched covering map of the 2-sphere which is not a homeomorphism, has finite postcritical set $P_f$, and has a forward-invariant marking set $A$ containing $P_f$. Thurston’s famous characterization theorem gives necessary and sufficient conditions for when such a map is realized by a rational map in a suitable sense. The proof amounts to showing that the induced pullback map on Teichmüller space $\sigma_f: T_A$ to $T_A$ has a fixed point. In this talk I will present a new proof of this result in the special case where $|A|=4$ using only classical complex analysis theorems and the fact that $\sigma_f$ satisfies a certain family of functional equations. Time permitting, I will discuss how this perspective yields new insights into related dynamical questions.
March 22: David Aulicino (Brooklyn College and The Graduate Center)
The Forni Subspace: A Survey of Recent Results
The Forni subspace, defined by Avila-Eskin-Moeller (in the context of translation surfaces), is a mechanism for producing Lyapunov exponents equal to zero in the Kontsevich-Zorich cocycle. In this talk we will introduce the basic objects and survey various classification results. We will explain some of the techniques and highlight possible future directions. This will include joint work with Frederik Benirschke and Chaya Norton.
April 12: Hongming Nie (Institute for Mathematical Sciences -- Stony Brook University)
Bounded hyperbolic components of bicritical rational maps
In this talk, I will show that the hyperbolic components of bicritical rational maps having two distinct attracting cycles each of period at least two are bounded in the moduli space of bicritical rational maps. The arguments rely on arithmetic methods. This is a joint work with Kevin Pilgrim.
Previous Semesters:
- Fall 2006
- Spring 2007
- Fall 2007
- Spring 2008
- Fall 2008
- Spring 2009
- Fall 2009
- Spring 2010
- Fall 2010
- Spring 2011
- Fall 2011
- Spring 2012
- Fall 2012
- Spring 2013
- Fall 2013
- Spring 2014
- Fall 2014
- Spring 2015
- Fall 2015
- Spring 2016
- Fall 2016
- Spring 2017
- Fall 2017
- Spring 2018
- Fall 2018
- Spring 2019
- Fall 2019
- Spring 2020
- Fall 2021
- Spring 2022
- Fall 2022
- Spring 2023
- Fall 2023
For information about the history of our seminar, please visit: History.