Fractal 1

Pacific Quasiworld

This is a virtual international seminar organized by Mario Bonk, Yunping Jiang, Zhiqiang Li, and Mitsuhiro Shishikura. The seminar focuses on quasiconformal geometry, complex dynamics, and analysis and geometry on metric spaces. We invite anyone interested to participate, and to volunteer talks.

The seminar runs monthly on Tuesdays (US time) / Wednesdays (Asia time) around mid of the month. We will have two speakers each month.

Dates for Fall 2023:
Dates for Spring 2024:

Date & Time:
  • Tue Oct 17, 6-8pm Pacific time /
  • Tue Oct 17, 9-11pm Eastern time /
  • Wed Oct 18, 9-11am China time /
  • Wed Oct 18, 10am-12pm Japan time

Speaker 1: Weixiao Shen (Fudan University, China)

Title: Prevalent periodic maximization over an expanding circle map

Abstract: I will discuss a joint work with Rui Gao (in progress), where we study the ergodic optimization problem over a real analytic expanding cicle map $T$. Let $M$ be the set of $T$-invariant Borel probability measures on the unit circle. For a continuous real valued function $f$ defined on the unit circle, a measure $\mu_0\in M$ is called maximal if $\mu\mapsto \int f d\mu$ takes its maximum at $\mu=\mu_0$. We shall show that for a typical (in both topological and measure-theoretical sense) real analytic performance function $f$, there is a unique maximizing measure and the measure is supported on a periodic orbit.

Speaker 2: Xin Sun (Peking University, China)

Title: Two dimensional percolation and Liouville quantum gravity

Abstract: Smirnov's proof of Cardy's formula for percolation on the triangular lattice leads to a discrete approximation of conformal maps, which we call the Cardy-Smirnov embedding. Under this embedding, Holden and I proved that the uniform triangulation converge to a continuum random geometry called pure Liouville quantum gravity. There is a variant of the Gaussian free field governing the random geometry, which is an important example of conformal field theory called Liouville CFT. A key motivation for understanding Liouville quantum gravity rigorously is its application to the evaluation of scaling exponents and dimensions for 2D critical systems such as percolation. Recently, with Nolin, Qian and Zhuang, we used this idea and the integrable structure of Liouville CFT to derive a scaling exponent for planar percolation called the backbone exponent, which was unknown for several decades.

Date & Time:
  • Tue Nov 21, 5-7pm Pacific time /
  • Tue Nov 21, 8-10pm Eastern time /
  • Wed Nov 22, 9-11am China time /
  • Wed Nov 22, 10am-12pm Japan time

Speaker 1: Gaven Martin (Massey University, New Zealand)

Title: From Teichmüller to Schoen-Yau: Extremal mappings between Riemann surfaces

Abstract: There are two now classical descriptions of the moduli space of a Riemann surface via the theory of extremal mappings. The first from Teichmüller in the 1940s (rigorously established by Ahlfors in 1953) and through the existence of extremal quasi- conformal mappings. The second is through Schoen-Yau's existence theory for unique harmonic diffeomorphisms in the 1970s, and developed into a theory of moduli by many, including Wolf, Tromba and Wolpert many years later. The important ingredient in both is the existence of a holomorphic quadratic differential, from the Beltrami coefficient of an extremal quasiconformal mapping (Teichmüller) or from the Hopf equation (Harmonic). These quadratic differentials define the cotangent space to the moduli space. Here we show that in fact both of these approaches are manifestations of the same theory (that of existence of diffeomorphic extremal mappings of finite distortion) in limiting regimes. We identify parameterised families of moduli spaces (Beltrami coefficients) interpolating between these two end cases defined by a parametrised family of degenerate elliptic nonlinear PDEs giving holomorphically parameterised homotopy between the extremal quasiconformal mapping [which is not a diffeomorphism] and the harmonic diffeomorphism.

Speaker 2: Zachary Smith (University of California, Los Angeles, USA)

Title: Curve attractors for marked rational maps

Abstract: A Thurston map f: (S^2, A) -> (S^2, A) with marking set A induces a pullback relation on isotopy classes of Jordan curves in (S^2, A). If every curve lands in a finite list of possible curve classes after iterating this pullback relation, then the pair (f,A) is said to have a finite global curve attractor. Conjecturally all rational maps that are not flexible Lattes maps have a finite global curve attractor. In this talk I present partial progress on this problem. Specifically, I will prove that if A has four points and the postcritical set (which is a subset of A) has two or three points, then (f,A) has a finite global curve attractor. Time permitting, I will also discuss how to extend this result to certain special cases where there are four postcritical points.

Date & Time:
  • Tue Dec 12, 5-7pm Pacific time /
  • Tue Dec 12, 8-10pm Eastern time /
  • Wed Dec 13, 9-11am China time /
  • Wed Dec 13, 10am-12pm Japan time

Speaker 1: Roland Roeder (Indiana University-Purdue University Indianapolis, USA)

Title: Questions about the holomorphic group action dynamics on a natural family of affine cubic surfaces

Abstract:I will describe the dynamics by the group of holomorphic automorphisms of the affine cubic surfaces \begin{align*} S_{A,B,C,D} = \{(x,y,z) \in \mathbb{C}^3 \, : \, x^2 + y^2 + z^2 +xyz = Ax + By+Cz+D\}, \end{align*} where $A,B,C,$ and $D$ are complex parameters. This group action describes the monodromy of the famous Painlevé 6 Equation as well as the natural dynamics of the mapping class group on the $SL(2,\mathbb{C})$ character varieties associated to the once punctured torus and the four times punctured sphere. For these reasons it has been studied from many perspectives by many people including Bowditch, Goldman, Cantat–Loray, Cantat, Tan–Wong–Zhang, Maloni–Palesi–Tan, and many others. In this talk I will describe my recent joint with Julio Rebelo and I will focus on several interesting open questions that arose while preparing our work "Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlevé 6" and during informal discussions with many people.

Speaker 2: Jinwoo Sung (University of Chicago, USA)

Title: Quasiconformal deformation of the chordal Loewner driving function

Abstract: The Loewner chain provides a method for encoding a simple planar curve by a family of uniformizing maps satisfying a differential equation driven by a real-valued function. For instance, choosing Brownian motion for the driving function gives Schramm–Loewner Evolution (SLE). Driving functions with finite Dirichlet energy encode the class of Weil–Petersson quasicircles, which was identified as the semiclassical limit of SLE in a series of works by Yilin Wang. In this talk, we consider the Loewner chain of a simple planar curve under infinitesimal quasiconformal deformations. We provide a variational formula of the Loewner driving function when the Beltrami coefficients are supported away from the curve. As an application, we obtain the first variation of the Loewner energy of a Jordan curve, defined as the Dirichlet energy of the driving function of the curve. This gives another explanation of the identity between the Loewner energy and the universal Liouville action introduced by Takhtajan and Teo. This is joint work with Yilin Wang.

Date & Time:
  • Tue Jan 23, 5-7pm Pacific time /
  • Tue Jan 23, 8-10pm Eastern time /
  • Wed Jan 24, 9-11am China time /
  • Wed Jan 24, 10am-12pm Japan time

Speaker 1: Tao Chen (City University of New York, USA)

Title: A family of meromorphic functions: ergodic or non-ergodic

Abstract: McMullen's dichotomy states that a rational map is ergodic (the Julia set is the whole sphere, and the map is ergodic) or attracting (almost every point on the sphere is attracted to the post-critical set). However, it may be challenging to determine which case applied to a given map, of which the Julia set is the sphere. In this talk, for a family of meromorphic maps with finite many asymptotic values and no critical points, called Nevanlinna functions in the literature, we give a criterion to determine whether it is ergodic or not. This is a joint work with Yunping Jiang and Linda Keen.

Speaker 2: Vanessa Matus de la Parra (University of Rochester, USA)

Title: Dynamics of covering correspondences

Abstract: In this talk, we will describe the dynamics of compositions of deleted covering correspondences. These are a particular case of holomorphic correspondences on the Riemann sphere which yield interesting families studied by Bullett, Penrose and Lomonaco. These families lie in the gap between modularity and weak modularity, and turn out to be “matings” of rational maps and Kleinian groups. The main goal of this talk is to use the mating structure to deduce equidistribution results, as well as finding measures of maximal entropy.

Date & Time:
  • Tue Feb 20, 5-7pm Pacific time /
  • Tue Feb 20, 8-10pm Eastern time /
  • Wed Feb 21, 9-11am China time /
  • Wed Feb 21, 10am-12pm Japan time

Speaker 1: Hideki Miyachi (Kanazawa University, Japan)

Title: Towards the function theory on Teichmüller spaces

Abstract: In this talk, I will give the recent progress of my research on the function theory on Teichmüller space. We first start with the background of the research and recall the characterization of the pluriharmonic measure on the Bers slice (in the sense of Demailly) in terms of the Thurston measure. Then, I will discuss versions of the Fatou theorem and the F. and M. Riesz theorem for the Teichmüller space. As an application of these results, I will show that the action of the Torelli group on the space of projective measured foliations is not ergodic.

Speaker 2: Yan Gao (Shenzhen University, China)

Title: Invariant graphs and decomposition of rational maps

Abstract: In this talk, I will show that for any post-critically finite rational map f, there exists a finite and connected graph G in the Julia set of f, such that f^n(G) ⊂ G for every large n, G contains all the post-critical points in the Julia set and every complementary component of G contains at most one post-critical point. The proof is based on the cluster-exact decomposition of post-critically finite rational maps.

Date & Time:
  • Tue Mar 12, 6-8pm Pacific time /
  • Tue Mar 12, 9-11pm Eastern time /
  • Wed Mar 13, 9-11am China time /
  • Wed Mar 13, 10am-12pm Japan time

Speaker 1: Hiroshige Shiga (Kyoto Sangyo Univercity, prof. emer. Tokyo Institute of Technology, Japan)

Title: Quasiconformal equivalence of Cantor sets

Abstract: Cantor sets appear in many fields of mathematics. In this talk, we are interested in Cantor sets in the complex plane $\mathbb C$. We recognize those sets have two aspects. One is that they are compact and totally disconnected perfect subset of $\mathbb C$ and another is that their complements are Riemann surfaces of infinite type of genus zero. We consider the quasiconformal (= qc) equivalence of Cantor sets from the two aspects. In the first part of the talk, we show qc-equivalence or non-qc-equivalemce of Cantors set which are obtained from some dynamical systems. In the second part, we focus on Cantor sets which are generalized ones of the middle one-third Cantor set and consider their qc-equivalence in terms of sequences of $(0, 1)^{\mathbb N}$ which give those Cantor sets. Finally, we exhibit some conjectures about qc-eqivalence of Cantor sets and their moduli spaces.

Speaker 2: Jinsong Liu (Chinese Academy of Sciences, China)

Title: Riesz conjugate function theorem for harmonic quasiconformal mappings

Abstract:In this talk we will introduce the Riesz conjugate functions theorem for planar harmonic K-quasiregular mappings (when 1 <p ≤2) and harmonic K-quasiconformal mappings (when 2 <p <∞) in the unit disk. Moreover, if K=1, then our constant coincides with the classical analytic case. For the n-dimensional case (n >2), we also obtain the Riesz conjugate functions theorem for invariant harmonic K-quasiregular mappings when 1 <p≤2. This is a joint work with Dr. Jianfeng Zhu.

Date & Time:
  • Wed Apr 17, 6-8pm Pacific time /
  • Wed Apr 17, 9-11pm Eastern time /
  • Thu Apr 18, 9-11am China time /
  • Thu Apr 18, 10am-12pm Japan time

Speaker 1: Mariusz Urbański (University of North Texas, USA)

Title: Hyperbolic dimension gaps for entire functions (joint with Volker Mayer)

Abstract: Polynomials and entire functions whose hyperbolic dimension is strictly smaller than the Hausdorff dimension of their Julia set are known to exist but in all known examples the latter dimension is maximal, i.e. equal to two. In this talk I will describe hyperbolic entire functions $f$ having Hausdorff dimension of the Julia set $HD (J _f)<2$ and hyperbolic dimension $HypDim (f) < HD (J _f)$.

Speaker 2: Junyi Xie (Peking University, China)

Title: The multiplier spectrum morphism is generically injective

Abstract:We consider the multiplier spectrum of periodic points, which is a natural morphism defined on the moduli space of rational maps on the projective line. A celebrated theorem of McMullen asserts that aside from the well-understood flexible Lattès family, the multiplier spectrum morphism is quasi-finite. In this paper, we strengthen McMullen's theorem by showing that the multiplier spectrum morphism is generically injective. This answers a question of McMullen and Poonen.

Date & Time:
  • Wed May 15, 6-8pm Pacific time /
  • Wed May 15, 9-11pm Eastern time /
  • Thu May 16, 9-11am China time /
  • Thu May 16, 10am-12pm Japan time

Speaker 1: Antoine Song (California Institute of Technology, USA)

Title: Geometry of the regular representation of hyperbolic groups

Abstract: Given a torsion free hyperbolic group G, one can build a natural quotient Q of a Hilbert sphere from the regular representation of G. We will talk about the geometry of this infinite dimensional Riemannian space Q, and also of its ultralimit. This discussion involves studying properties of limits of the regular representation. It has applications to the construction of minimal surfaces related to a topological invariant for manifolds, called the spherical volume. I will also mention some possible connections between this setup and Cannon's conjecture.

Speaker 2: Yusheng Luo (Cornell University, USA)

Title: Renormalization, uniformization and circle packings

Abstract:Circle packings have many applications in geometry, analysis and dynamics. The combinatorics of a circle packing is captured by the contact graph, called the nerve of the circle packing. It is natural and important to understand 1. Given a graph G, when is it isomorphic to the nerve of a circle packing? 2. Is the circle packing rigid? Or more generally, what is the moduli space of circle packings with nerve isomorphic to G? 3. How are different circle packings with isomorphic nerves related? For finite graphs, Kobe-Andreev-Thurston’s circle packing theorem give a complete answer to the above questions. The situation is more complicated for infinite graphs, and has been extensively studied for locally finite triangulations. In this talk, I will describe how to use renormalization theory to study these questions for infinite graphs. In particular, I will explain how it gives complete answers to the above questions for graphs with subdivision rules. I will also discuss some applications on quasiconformal geometries for dynamical gasket sets. This is based on some joint works with Y. Zhang and an ongoing work with D. Ntalampekos.

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The zoom invitation will be distributed through the email list. If you do not receive it, you can contact one organizer to give you the link and password. You may also share it within our community, but do not post it on a publicly viewable website. It is the same every week, so you do not need to receive a new one each week.

All participants will join automatically muted and with no video on entry. The format worked fine with lots of videos, so you can keep your video on, and if there is an issue, we will address it. You may, and are encouraged, to unmute yourself to ask questions. Keep yourself muted otherwise. You may also post questions in the chat, that the organizers and/or speaker will follow. At the end the host may choose some of these questions to ask from the speaker. This is a friendly and conversational seminar, so many questions are encouraged. The chat window will be monitored by hosts, speaker and/or possibly collaborators, and will answer questions in a live feed format (you can answer too if you know the answer).

We are looking for volunteer speakers. Contact one of the organizers to be added to the schedule. If you want and choose to use slides, you can send your slides to be added to this website.

Also, we may consider using the email list for announcements of more broad interest to our community. Contact one of the organizers if you wish to post such an announcement.

Fractal 1