Complex Analysis and Dynamics Seminar at CUNY's GC

Complex Analysis and Dynamics Seminar

Department of Mathematics

Graduate Center of CUNY

Fridays 1:45 - 2:45 pm

Room 5417


Organizers: David Aulicino, Patrick Hooper, Jun Hu, Dragomir Saric

New time and location this semester: Room 5417 from 1:45 to 2:45 pm.

Fall 2024: Upcoming Seminars

December 6: Samuel Grushevsky (Stony Brook University)
Ends and compactifications of strata of differentials

The strata of differentials, that is the moduli spaces of Riemann surfaces together with a differential with prescribed multiplicities of zeroes (and poles) are the central object of study of Teichmuller space. We describe a nice algebro-geometric compactification of the strata of differentials and indicate how to quickly use the degeneration techniques, to the boundary of this compactification, to show that the strata of differentials have one topological end. Based on joint work with Ben Dozier.

Fall 2024: Previous Seminars

September 13: Michael Pandazis (John Jay College)
Parabolic and Non-Parabolic Surfaces with Small or Large End Spaces via Fenchel-Nielsen Parameters

We consider conditions on the Fenchel-Nielsen parameters of a Riemann surface X that determine whether or not a surface X is parabolic. Fix a geodesic pants decomposition of a surface and call the boundary geodesics in the decomposition cuffs. For a zero or half-twist flute surface, we prove that parabolicity is equivalent to the surface having a covering group of the first kind. Using that result, we give necessary and sufficient conditions on the Fenchel-Nielsen parameters of a half-twist flute surface X with increasing cuff lengths such that X is parabolic. As an application, we determine whether or not each half-twist flute surface in the Hakobyan slice is parabolic. Let X be a Cantor tree surface or a blooming Cantor tree surface. Basmajian, Hakobyan, and Saric proved that if the lengths of cuffs are rapidly converging to zero, then X is parabolic. More recently, Saric proved a slightly slower convergence of lengths of cuffs to zero implies X is not parabolic. We interpolate between the two rates of convergence of the cuffs to zero and find that these surfaces are not parabolic, thus completing the picture.

September 20: Ara Basmajian (Hunter College and The Graduate Center)
A Bers type classification of big mapping classes

For an infinite type surface $\Sigma$, we consider the space of marked (convex) hyperbolic structures on $\Sigma$, denoted $H(\Sigma)$, with the Fenchel-Nielsen topology. The (big) mapping class group acts faithfully on this space allowing us to investigate a number of mapping class group invariant subspaces of $H(\Sigma)$ which arise from various geometric properties (e.g. geodesic or metric completeness, ergodicity of the geodesic flow, lower systole bound, discrete length spectrum) of the hyperbolic structure. In particular, we show that the space of marked geodesically complete hyperbolic structures in $H(\Sigma)$ is locally path connected, connected and decomposes naturally into Teichmüller subspaces. The big mapping class group of $\Sigma$ acts faithfully on this space allowing us to classify mapping classes into three types (always quasiconformal, sometimes quasiconformal, and never quasiconformal) in terms of their dynamics on the Teichmüller subspaces. Moreover, each type contains infinitely many mapping classes, and the type is relative to the underlying subspace of $H(\Sigma)$ that is being considered.

This talk will have three parts starting with basics. The first part will be a quick introduction to hyperbolic geometry and quasiconformal homeomorphisms on finite type surfaces, and the second part on the geometry and topology of infinite type surfaces. Finally, the third part will be about the results mentioned above on big mapping class groups. This is joint work with Yassin Chandran.

September 27: Insung Park (Stony Brook University)
Julia sets having conformal dimension one

Conformal dimension of a metric space X is the infimal Hausdorff dimension of metric spaces that are quasisymmetric to X. Let f be a hyperbolic rational map, which is conjecturally a generic rational map. Suppose that the Julia set J(f) of f is connected. As a fractal embedded in the Riemann sphere, J(f) has conformal dimension between 1 and 2. J(f) has conformal dimension 2 if and only if J(f) is the whole sphere. On the other hand, the other extreme case, when the conformal dimension of J(f) is one, contains many interesting families of rational maps, such as postcritically finite polynomials, Newton maps, and critically fixed rational maps. In this talk, we show that J(f) has conformal dimension one if and only if there is a zero entropy f-invariant graph that contains all the critical values of f.

October 18: Anca Rădulescu (SUNY New Paltz)
Complex dynamics in networks, templates and mutated systems

We will continue a previous discussion around extending the traditional theory of complex quadratic iterations to coupled and time-dependent systems. These emerge as frameworks that better match natural complexities than single map iterations. We will focus in particular on defining their Mandelbrot set, and presenting recent results around its combinatorics and topology, with interpretations to applications, when possible.

Our first extension will be to networks of coupled quadratic nodes. We will describe how the asymptotic properties of the orbits in general, and critical orbit in particular, depend on the distribution and weights of the connections between nodes. We will present theoretical results for low dimensions (two and three nodes), and numerical simulations for larger networks.

For our second study case, we will consider non-autonomous iterations of two complex quadratic functions, applied according to an infinite binary symbolic sequence (which we call template). Here, the Mandelbrot set is a subset of the product between a two-dimensional complex space (of the two complex parameters) and the unit interval (the space of all binary templates). We discuss the set's slices in either subspace and present a Hausdorff convergence theorem.

Our final extension will explore the effects of inserting a local perturbation of the intact iteration function, that acts within a small disk around an epicenter and tapers off radially until the original function is recovered. We track how the Julia set is perturbed in response to errors of different size, with different radius and different location.

October 25: Runze Zhang (Stony Brook University)
Main hyperbolic component boundary for cubic polynomials

The geometry of the Julia set undergoes a radical change as a hyperbolic rational map approaches the hyperbolic component boundary. It is therefore important to understand the structure of the boundary and dynamics of maps on the boundary. This talk will focus on the family of cubic polynomials. The main hyperbolic component is defined by the hyperbolic component containing $z^3$, which is a topological real 4-ball. We prove that its boundary presents canonically copies of Julia sets of quadratic polynomials.

November 1: Sahana Vasudevan (Institute for Advanced Study and Princeton University)
Triangulated surfaces in moduli space

Triangulated surfaces are Riemann surfaces formed by gluing together equilateral triangles. They are also the Riemann surfaces defined over the algebraic numbers. Brooks, Makover, Mirzakhani and many others proved results about the geometric properties of random large genus triangulated surfaces, and similar results about the geometric properties of random large genus hyperbolic surfaces. These results motivated the question: how are triangulated surfaces distributed in the moduli space of Riemann surfaces, quantitatively? I will talk about results related to this question.

November 8: Sayantika Mondal (The Graduate Center)
Distinguishing filling curve types via special metrics

The study of extremal lengths of curves and their relations to intersection numbers has a very rich history. In this talk, I look at filling curves on hyperbolic surfaces and consider its length infima in the moduli space of the surface as a type invariant. Then explore the relations between this geometric invariant and a topological namely the self-intersection number of a curve. In particular, for all finite type surface, construct infinite families of filling curves that cannot be distinguished by self-intersection number but via length infimum. Time permitting, I will also discuss some coarse bounds on the metrics associated to these infimum lengths.

November 15: Xinlong Dong (Kingsborough Community College)
Divergent geodesics in the universal Teichmüller space

Thurston boundary of the universal Teichmüller space is identified with the space of projective bounded measured laminations. A geodesic ray in the universal Teichmüller space is of generalized Teichmüller type if it shrinks the vertical foliation of a holomorphic quadratic differential. In the previous work of Hakobyan and Saric, existence of limits at infinity was proved for many generalized Teichmüller rays in the universal Teichmüller space. In this paper we provide the first example of generalized Teichmüller rays which do not have a unique limit point in Thurston boundary and completely determine the limit sets of these rays in the space of projective bounded measured laminations. This is a joint work with Hrant Hakobyan.

November 22: Yassin Chandran (The Graduate Center)
Density results for modular groups of infinite-type surfaces

The mapping class group $MCG(\Sigma)$ of an infinite-type surface $\Sigma$ is a (non-discrete) topological group. We will discuss some density results for the group of quasiconformal mapping classes also known as the modular group $Mod(X) \subset MCG(X)$ of a hyperbolic surface $X$, such as the following. For any infinite-type surface $\Sigma$, there exists a hyperbolic structure $X$ carried by $\Sigma$ such that the the quasiconformal mapping class group $Mod(X)$ is dense in the pure mapping class group $PMCG(X)$. In fact, if a surface has at most countably many ends, then there is a hyperbolic structure $X$ such that $Mod(X)$ is dense in the full mapping class group. This is joint work with Tommaso Cremaschi.

Previous Semesters:

For information about the history of our seminar, please visit: History.