Pacific Quasiworld

Speaker 1: Mikhail Hlushchanka (University of Amsterdam, Netherlands)

Title: The independence polynomial on recursive graph sequences - the dynamical perspective

Abstract: The distribution of the zeros of partition functions on graphs are intimately related to the analyticity of physical quantities and their phase transitions. In their pioneering work in the 1950s, Lee and Yang proved that the free energy per site of the cubic lattice is analytic at a given positive real parameter, provided that the complex zeros of the partition functions for a sequence of finite graphs converging to the lattice avoid a neighborhood of this parameter.

In the last decade, a special focus was put on graph sequences that do not converge to a regular lattice but are instead defined recursively. Examples of such recursive graph sequences include hierarchical lattices, Cayley trees, Sierpiจฝski gasket graphs, and various self-similar Schreier graphs. Significant progress has been made in studying the partition functions of the Ising, Potts, and Hard-Core models on these graphs (with the latter two corresponding to the chromatic and independence graph polynomials, respectively). One main advantage of working with recursive sequences of graphs is that the underlying recursion naturally induces an iterative system on the level of partition functions, often given in terms of rational maps in one or several (complex) variables. By analyzing the dynamical behavior of these iterative systems, we can gain insights into the properties of the respective graph polynomials, such as phase transitions and computational complexity.

In our current work with Han Peters (University of Amsterdam), we attempt to establish a unified framework for recursive graph sequences in the Hard-Core model setting. We start with an arbitrary graph with k marked vertices. At each recursive step, we construct a new graph by taking n copies of the previous graph and connecting these copies along the marked vertices (according to a specified rule). The dynamical systems that emerge from the respective independence polynomials are represented by homogeneous polynomials of degree n in 2^k variables. Somewhat surprisingly, it turns out that these systems can be successfully analyzed in this general context. In the talk, I will report on our results on the structure of the zero sets of the independence polynomials for these recursive graph sequences, particularly highlighting the absence of phase transitions.

Speaker 2: Enrique Pujals (City University of New York, US)

Title: Henon like Renormalization

Abstract: I will present the renormalization theory of non-perturbative dissipative H\'enon-like maps (meaning that they are not necessary close to 1-d dynamics) developed jointly with S. Crovisier, M. Lyubich and J. Yang, which enables us to establish a priori bounds and control the small-scale geometry of dynamics.

Speaker 1: Dragomir Saric (City University of New York, USA)

Title: Harmonic functions, finite-area holomorphic quadratic differentials and classification of infinite Riemann surface

Abstract: An infinite Riemann surface is closest to a compact Riemann surface if it does not support a Green’s function. This condition is equivalent to the ergodicity of the geodesic flow as well as the transience of the Brownian motion. We give another characterization of this class of Riemann surfaces using the space of holomorphic quadratic differentials and prove that the set of holomorphic quadratic differentials with single cylinders is dense among all holomorphic quadratic differentials. It is well known that the class of Riemann surfaces without Green’s function is invariant under quasiconformal mappings. A class of Riemann surfaces without non-constant harmonic functions with finite Dirichlet integral properly contains the class of surfaces without Green’s function. We prove that the larger class is also invariant under quasiconformal mappings. The classes of Riemann surfaces that satisfy the (strong) Liouville property are sandwiched between the above two classes, and it is known that the (strong) Liouville property is not a quasiconformal invariant.

Speaker 2: Vladimir Markovic (Tsinghua University, China)

Title: Teichmuller flow and Caratheodory metrics on Moduli Spaces

Abstract: In this talk I will explain my results that the Caratheodory and Teichmuller metrics do not agree on Teichmueller spaces, and the proof of the convexity conjecture of Siu. Moreover, I will illustrate how deep theorems in Teichmuller dynamics play an important role in classifying Teichmuller discs where the two metrics agree.

Speaker 1: Hao Wu (Tsinghua University, China)

Title: Connection probabilities for loop O(n) models and BPZ equations

Abstract: Critical loop O(n) models are conjectured to be conformally invariant in the scaling limit. In this talk, we focus on connection probabilities for loop O(n) models in polygons. Such probabilities can be predicted using two families of solutions to Belavin-Polyakov-Zamolodchikov (BPZ) equations: Coulomb gas integrals and SLE pure partition functions. The conjecture is proved to be true for the critical Ising model, FK-Ising model, percolation, and uniform spanning tree.

Speaker 2: Yuan Zhou (Beijing Normal University, China)

Title: Jacobian determinants for nonlinear gradient of planar p-harmonic and $\infty$-harmonic functions

Abstract: In dimension 2, we introduce a distributional Jacobian determinant for the nonlinear complex gradient $V_\beta(Dv)$ of a function $v\in W^{1,2 }_\loc$ with $\beta |Dv|^{1+\beta}\in W^{1,2}_\loc$, where $\beta>-1$. This is new when $\beta\ne0$. Given any planar $\infty$-harmonic function $u$, we show that such distributional Jacobian determinant $\det DV_\beta(Du)$ is a nonnegative Radon measure with some quantitative local lower and upper bounds. Denoting by $u_p$ the $p$-harmonic function having the same nonconstant boundary condition as $u$, we show that $\det DV_\beta(Du_p) \to \det DV_\beta(Du)$ as $p\to\infty$ in the weak-$\star$ sense in the space of Radon measure. Recall that $V_\beta(Du_p)$ is always quasiregular mappings, but $V_\beta(Du)$ is not in general.

Speaker 1: Alex Kapiamba (Harvard University, USA)

Title: A priori bounds for some near-parabolic primitive combinatorics

Abstract: The local connectivity of the Mandelbrot set (MLC) is a long outstanding conjecture in complex dynamics. Nearly twenty years ago, Kahn and Lyubich established MLC for all “definitely primitive” combinatorics. In this talk we will discuss MLC for some primitive combinatorics which accumulate on parabolic parameters in the Mandelbrot set. Based on joint work with Jeremy Kahn.

Speaker 2: Yan Mary He (Oklahoma University, USA)

Title: Geometry of polynomial shift locus and the space of trees

Abstract: The degree d \ge 2 shift locus S_d consists of affine conjugacy classes of complex polynomials whose critical points all escape to infinity under iterations of the polynomial. We relate generic points in the shift locus S_d to metric graphs. Using thermodynamic metrics on the space of metric graphs/Culler-Vogtmann outer space, we obtain a distance function on S_d. We show that S_d is incomplete if and only if d \ge 3 and the metric completion contains a subset homeomorphic to the space of trees PST_d^* introduced by DeMarco-Pilgrim. This is joint work with Hongming Nie.

Speaker 1: Gaofei Zhang (Nanjing University, China)

Title: Polynomial curve systems are exponentially decaying

Abstract: The finite global attractor problem for polynomial case was recently solved by using tree lifting algorithm. Since the global contraction rate of the simplicial map of the tree complex is in general unclear, it is not known how fast the pull backs of a curve converge to the attractors. In the work we will introduce the ideas of quick returns and barrier lakes to analyze the combinatorial models of curves. These allow us to prove that the complexity of a curve is exponentially decreased under the pull backs by the iteration of a polynomial.

Speaker 2: Takuya Murayama (Kyushu University, Japan)

Title: Additive processes on the real line and Loewner chains

Abstract: Loewner's differential equation, which describes the time-evolution of slit mappings, was initially introduced to attack an extremal problem for univalent functions (Bieberbach's conjecture) solved finally by de Branges in 1984. It was then used to indroduce the Schramm--Loewner evolution (SLE), which yielded a breakthrough in the analysis of two-dimensional critical models in statistical physics, in 2000. In this talk, I shall describe yet another, recent application of Loewner's method to one-dimensional, non-commutative stochastic processes. In non-commutative probability, we can consider several ``independences'' for algebraic random variables and corresponding stochastic processes with ``independent'' increments. Loewner chains are naturally associated with such processes if we consider ``monotone'' independence. If time permits, I shall also discuss how some function-theoretic (or potential-theoretic) properties are related to probabilistic ones. This talk is based on a joint work with Takahiro Hasebe (Hokkaido University) and Ikkei Hotta (Yamaguchi University).

Speaker 1: Dylan Thurston (Indiana University, Bloomington)

Title: Expansivity of hyperbolic dynamical systems

Abstract: A topologically expansive system X (one where the diagonal is repelling in X^2) has an expansive metric, where sufficiently close points are moved apart by a definite ratio. We define the expansivity to be the supremal ratio one can achieve by varying the metric. It is bounded above by the topological entropy. Finer upper bounds come from the topological entropy of invariant subgraphs, while lower bounds come from constructing metrics, which can also be made precise using techniques from graphs. In this way we find the expansivity exactly for many polynomials and rational maps. Closer investigation reveals a finer structure of degrees of expansion of the dynamical system, all bounded by the topological entropy.

Speaker 2: Fedor Manin (University of Toronto, Canada)

Title: Elliptic and quasiregularly elliptic manifolds

Abstract: A Lipschitz map from ℝ^n to an n-manifold M has positive asymptotic degree if, roughly speaking, it wraps efficiently around M. M is elliptic in the sense of Gromov if it admits such a map. Similarly, M is quasiregularly elliptic if it admits a quasiregular map from ℝ^n: a map with geometric properties similar to those of holomorphic functions. Gromov in his book Metric Structures suggested that there may be a connection or even equivalence between ellipticity and quasiregular ellipticity. This is supported by the fact that the known obstructions to ellipticity and quasiregular ellipticity for closed manifolds are exactly the same (for example, in both cases the fundamental group has to be abelian). On the other hand, we know much less about constructing quasiregular maps than Lipschitz maps of positive asymptotic degree. Moreover, for open manifolds, a host of simple examples shows that these two properties are quite different. I will discuss joint work with Berdnikov and Guth and with Prywes and highlight a number of open problems.

Speaker 1: Jun Kigami (Kyoto University, Japan)

Title: Sobolev spaces on metric spaces, recent developments

Abstract: The classical theory of Sobolev spaces on metric spaces has been developed consierably from 1990’s. The theory is based on the notion of “upper gradient” which is a generalization of the gradient of smooth functions on smooth spaces. However, recent results by Murugan-Kajino showed that this theory does not work for certain self-similar sets like the Sierpinski carpet, higher-dimensional Sierpinski gaskets and the Viscek set. Recently, a new theory covering such metric spaces has emerged. In this talk, we will review this new theory and examples of self-similar sets where this new theory can be applied.

Speaker 2: Weiwei Cui (Shandong University, China)

Title: Perturbations of non-recurrent exponential maps and applications

Abstract: We consider dynamics of the exponential family. A parameter in this family is non-recurrent if the only singular value is not recurrent. We show that the set of non-recurrent parameters has Lebesgue measure zero. Moreover, non-recurrent parameters can be approximated by hyperbolic ones. We also discuss applications to the measurable dynamics of exponential functions. The talk is based on joint works with Magnus Aspenberg (Lund University), Jun Wang (Fudan University) and Jiaxing Huang (Shenzhen University), respectively.

Speaker 1: Giulio Tiozzo (University of Toronto, Canada)

Title: Open Dynamical Systems: Stochastics, Geometry, and Thermodynamic Formalism

Abstract: An "open dynamical system” is obtained by cutting a “hole” in the phase space of a dynamical system, and considering the set of points that never falls into the hole under forward iteration. As a simple, but already nontrivial, example, one can consider the set of points that never hit a fixed interval under iteration of the doubling map. It is of interest to look at the Hausdorff dimension of the set of remaining points (sometimes called the survival set), and to consider natural measures, such as Gibbs measures, supported on it. If the hole varies in a family (for instance, if one removes the interval (0, t) as t varies), all the associated quantities vary with the parameter, and one looks at how fast they vary, looking for example at their Holder exponent. In joint work with T. Das, M. Urbanski, and A. Zdunik, we provide a general framework to deal with thermodynamic formalism of open dynamical systems in metric spaces, and establish formulas for escape rates in this general context.

Speaker 2: Fei Yang (Nanjing University, China)

Title: Rational maps with smooth degenerate Herman rings

Abstract: We prove the existence of rational maps having smooth degenerate Herman rings. This answers a question of Eremenko affirmatively. The proof is based on the construction of smooth Siegel disks by Avila, Buff and Chéritat as well as the classical Siegel-to-Herman quasiconformal surgery. A crucial ingredient in the proof is the surgery's continuity, which relies on the control of the loss of the area of quadratic filled-in Julia sets by Buff and Chéritat. As a by-product, we prove the existence of rational maps having a nowhere dense Julia set of positive area for which these maps have no irrationally indifferent periodic points, no Herman rings, and are not renormalizable.

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.

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.

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.

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.

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.

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.

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.

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.