Complex Analysis and Dynamics Seminar
Fall 2025 Schedule
September 5: Xinlong Dong (Kingsborough Community College)
On the derivatives of the Liouville currents
The Liouville map, introduced by Bonahon, assigns to each point in the Teichmüller space a natural Radon measure on the space of geodesics of the base surface. The Liouville map is real analytic and it even extends to a holomorphic map of a neighborhood of the Teichmüller space in the Quasi-Fuchsian space of an arbitrary conformally hyperbolic Riemann surface. The earthquake paths and by their extension quake-bends, introduced by Thurston, are particularly nice real-analytic and holomorphic paths in the Teichmüller and the Quasi-Fuchsian space, respectively. We find a geometric expression for the derivative of the Liouville map along earthquake paths. This is joint work with Dragomir Saric and Zhe Wang.
September 12: Howard Masur (University of Chicago)
Counting Saddle Connections on Translation Surfaces
A translation surface can be thought of as a polygon(s) in the plane with pairs of sides that are parallel and are of the same length that are glued by translations. The Euclidean metric in the plane gives rise to a metric on the surface which is locally Euclidean away from the glued vertices which become cone points of the metric. A saddle connection is a geodesic joining cone points with no cone point in its interior. One problem in the subject is to find the growth rate for the number of saddle connections in terms of their length. I will give some background to this problem. I will then address a new question where one fixes a saddle connection, and then tries to find the growth rate for the number of saddle connections that are disjoint from the fixed saddle connection. I will motivate this question and then give some recent results. This is joint with David Aulicino, Huiping Pan, and Weixu Su.
September 19: Juliet Aygun (Cornell University)
Counting geodesics on prime-order k-differentials
It has been of popular interest over the last several decades to count geodesics of a specified type with respect to their length on flat surfaces. Asymptotics of these counting functions for generic translation surfaces, which are Riemann surfaces with a holomorphic one form, have been determined by the pioneering work of Eskin-Masur-Zorich. There is a more general type of flat surface called a $(1/k)$-translation surface, which is a Riemann surface with a $k$-differential. Equivalently, a $(1/k)$-translation surface is a collection of polygons in the complex plane with sides identified pairwise by translation and possible rotations of $2\pi/k$. In this talk, we will determine a weaker asymptotic of these counting functions on generic $(1/k)$-translation surfaces when k is prime and genus is more than two. The main tools I will discuss are $\text{GL}_+(2,R)$-orbit closures and a result of Eskin-Mirzakhani-Mohammadi which relates asymptotics to $\text{GL}_+(2,R)$-orbit closures.
October 3: Sudeb Mitra (Graduate Center and Queens College)
Teichmüller space of a closed set in the sphere — Part I
Associated to each closed set $E$ in the Riemann sphere, there is a Teichmüller space $T(E)$, which is the universal parameter space for holomorphic motions of that set. In this talk, we will discuss some properties of $T(E)$. In particular, we will introduce the concept of a maximal holomorphic motion and give an explicit example of such a motion over a simply connected parameter space.
October 10: Tao Chen (Laguardia Community College and the Graduate Center)
Accessibility of some points on bifurcation locus
Escaping parameters in the family of exponential maps have been well investigated. It is known that they are in the bifurcation locus, approached by hyperbolic components and not on the boundary of any hyperbolic components. In this talk, we will show that they are on the boundary of hyperbolic components in meromorphic maps, and this could be extended to more general escaping parameters. Moreover, we will also discuss some application of this theorem.
October 17: Jon Chaika (University of Utah, Rice University and the Institute of Advanced Study)
Billiards in polygons
Consider a point mass traveling in a polygon. It travels in a straight line, with constant speed, until it hits a side, at which point it obeys the rules of elastic collision. What can we say about this? When all the angles of the polygon are rational multiples of $\pi$, the unit tangent bundle is foliated by invariant surfaces and we know a lot about it. In the case when at least one of the angles is irrational, it is much less understood, though from approximating with the rational case we know a couple of things. Kerckhoff, Masur and Smillie proved that there exists a billiard in an irrational polygon where the billiard flow is ergodic with respect to the natural measure. This talk will present two results both concerning a strengthening of ergodicity called weak mixing:
1) A strengthening of Kerckhoff, Masur and Smillie's result: There exists a polygon where billiard flow is weakly mixing with respect to the natural volume on the unit tangent bundle.
2) A classification of the rational polygons where the billiard flow is weakly mixing with respect to the natural area on the invariant surfaces that foliate the unit tangent bundle.
Open questions will be presented and no previous knowledge of billiards will be assumed. This is joint work with Giovanni Forni and Francisco Arana-Herrera.
October 31: Linda Keen (Graduate Center and Lehman College)
Meromorphic Functions with $2$ Asymptotic Values and No Critical Points
Just as understanding the dynamical behavior and the parameter spaces rational maps of degree $2$ gives insight into the theory for rational maps of degree $d > 2$, understanding of meromorphic maps whose singular set contains two asymptotic values gives insight into the more general theory for meromorphic maps with finitely many singular values. In this talk, I will present a number of examples of 1 complex dimensional "slices" of the two complex space. Each illustrates the new phenomena that arise because, unlike rational functions, these functions are infinite to one coverings of the sphere. Much of this work is joint with others, or based on work with others, including Tao Chen, Bob Devaney, Nuria Fagella, Lisa Goldberg, Yunping Jiang and Janina Kotus,
November 7: Michael Yampolsky (University of Toronto)
Theoretical challenges of computing dynamical systems
Asking and answering the right questions on what can and cannot be modeled numerically leads to a beautiful synthesis between dynamics and theoretical computer science. I will try to give a survey of some of the directions of study and the principal challenges.
November 21: Omri Solan (Princeton University and the Institute of Advanced Study)
Critical exponent gap in $SL_2(R)$ orbits of translation surfaces
We discuss the $SL_2(R)$ action on the space of translation surfaces (with genus g). McMullen and independently Hubert and Schmidt points with weird stabilizers: the stabilizers can be Zariski dense in $SL_2(R)$ and infinitely generated (and in particular, not lattices).
In the talk I will introduce the constructions, and show that Zariski dense stabilizers are quantitatively far from being lattices. Explicitly, the critical exponent - the exponential growth rate - is uniformly bounded away from the growth rate of the entire $SL_2(R)$.
December 5: Yunping Jiang (Graduate Center and Queens College)
Tameness Conditions in Complex Dynamics and Quasiconformal Motions
Extending mathematical results from finite settings to infinite ones presents significant challenges. In this survey, I will discuss our efforts to address these challenges through two illustrative examples. First, in complex dynamical systems, we study how Thurston's theorem for post-critically finite branched coverings can be extended to geometrically finite cases. Second, in the setting of quasiconformal motions, we investigate how the full extension theorem—originally valid for motions of finite subsets of the Riemann sphere over simply connected Hausdorff parameter spaces—may be generalized to motions of infinite subsets. While elegant theorems hold under finite assumptions, these results generally fail in infinite frameworks. However, by introducing suitable "tameness" conditions in each context, we are able to recover the essential structure of these theorems and preserve much of their elegance even in the infinite setting. Our project in complex dynamical systems involves Guizhen Cui, Gaofei Zhang, Tao Chen, Linda Keen, and Lei Tan and me. Our project on holomorphic and quasiconformal motions involves Sudeb Mitra, Hiroshige Shiga, Zhe Wang, Mike Beck, Fred Gardiner, and me.
December 12: Sudeb Mitra (Graduate Center and Queens College)
Some generalized Teichmüller spaces and applications
Let $E$ be a closed set in the Riemann sphere $\hat C$, such that the points $0, 1, \infty$ belong to $E$. The Teichmüller space $T(E)$, first studied by G. S. Lieb, in his 1990 Cornell University doctoral dissertation, is a contractible complex Banach manifold, which is a universal parameter space for holomorphic and tame quasiconformal motions of the set $E$. More generally, if $E$ is a closed set in a hyperbolic Riemann surface $X$, the Teichmüller space of $X$ rel $E$, denoted by $T(X,E)$, was first studied by A. L. Epstein in his 1993 CUNY doctoral dissertation. The space $T(E)$ has been extensively studied, whereas the space $T(X,E)$ has not been well explored.
In this talk, we will discuss a conformal naturality property of $T(E)$; we expect this to have applications in the study of $T(X,E)$. We will also discuss two applications - one for "holomorphic axiom of choice", as introduced by Sullivan and Thurston, and the other for holomorphic families of Jordan curves, improving a result of Pommerenke and Rodin.
This talk is based on some joint works with Xinlong Dong, Arshiya Farhath. G, and Yunping Jiang.
