Speakers:

Nov. 13 2020 - **Weixiao
Shen (****沈维孝****)** (Fudan
University)

Time: 3:00-4:00pm (Beijing Time) Location: Room 1114, Science Building 1,
Peking University

Title: Primitive tuning via quasiconformal
surgery****

Abstract: In 1980s, Douady-Hubbard developed a
complex counterpart of the Feigenbaum renormalization theory for quadratic-like
maps and used this theory to prove existence of small copies in the Mandelbrot
set. Inou and Kiwi have extended most of Douady-Hubbard's theory to higher
degree polynomial-like maps, but a key surjectivity property was left as a
conjecture. We will show how to use quasiconformal surgery to prove this
surjectivity conjecture by Inou-Kiwi, under a primitive assumption. This is
a joint work with Wang Yimin.

Nov. 20 2020 - **Guizhen
Cui (****崔贵珍****)** (Chinese
Academy of Sciences)

Time: 3:00-4:00pm (Beijing Time) Location: Room 1114, Science Building 1,
Peking University

Title: Rational maps with constant Thurston maps*

Abstract: A rational map with marked points induces
a holomorphic map between Teichmüller spaces of punctured spheres, which is
called the Thurston pullback map. One natural problem is to classify the marked
rational maps with constant Thurston maps. In this talk, I will introduce the
background about this problem at first. Then present a new class of rational
maps with constant Thurston pullback maps. This result can be stated in
elementary geometry. In the last I will present some relations about this
problem with monodromy groups of rational maps.

Nov. 27 2020 - **Robert
Tang** (Xi’an Jiaotong-Liverpool University)

Time: 10:00-11:00am (Beijing Time) Location: Zoom (ID: 647 1604 0205,
Password: 112487)

Title: Affine diffeomorphism groups are undistorted

Abstract: Studying the distortion properties of
naturally occurring subgroups of the mapping class group plays an important
part in understanding its large-scale geometry. In this talk, I will focus on
the affine diffeomorphism group Aff(S,q) of a half-translation surface (S,q).
This is the group of self-diffeomorphisms with constant differential away from
the singularities. This group also coincides with the stabilizer of the
associated Teichmüller disc under the action of the mapping class group on Teichmüller
space. Our main result is that any finitely generated subgroup of Aff(S,q) is
undistorted in the mapping class group.

Dec. 4** **2020
-** Mario Bonk** (UCLA)

Time: 11:00am-12:00pm (Beijing Time) Location: Zoom (ID: 649 4519 4579,
Password: 143386)

Title: Sullivan's dictionary and beyond*****

Abstract: The classical theory of holomorphic
dynamics includes the theory of Kleinian groups and the iteration theory of
rational maps. In the 1980s Sullivan pointed out several analogies between
these two subjects. For example, the concept of a Julia set of a rational map
corresponds to the concept of a limit set of a Kleinian group. Even if one goes
beyond holomorphic dynamics, an “extended” Sullivan dictionary can be very
helpful as a guide for investigations. In my talk I will give an introduction to
this subject and will discuss some recent developments.

Dec. 10** **2020
- **Huiping Pan (****潘会平****)** (Jinan University)

Time:
4:00-5:00 pm (Beijing Time) Location:
Zoom (ID: 653 2910 3639, Password: 175707)

Title: On the geometry of the saddle connection
graph

Abstract: Every half-translation surface induces a
singular flat metric. A saddle connection on a half-translation surface is an
open geodesic segment which connects singular points and which contains no
singular points in the interior. The saddle connection graph is a graph whose
vertices are saddle connections and edges are pairs of disjoint saddle
connections. In this talk, we will discuss the geometry of saddle connection
graphs. We will show that on the one hand saddle connection grpahs are isometrically
rigid, namely two saddle connection graphs are isometric if and only if the
underlying half-translation surfaces are affine equivalent. While on the other
hand all saddle connections graphs are uniformly quasi-isometric to the regular
countably infinite-valent tree. This talk is partially based on a joint work
with Valentina Disarlo, Anja Rendecker and Robert Tang.

Dec. 25** **2020
- **Yan Gao (****高延****) **(Sichuan University)

Time: 10:00-11:00 am (Beijing Time) Location: Zoom (ID: 690 5359 1120,
Password: 294214)

Title: Boundedness and compactification of
hyperbolic components for Newton maps

Abstract: In complex dynamics, an interesting
question is to determine which kinds of hyperbolic components are bounded in
the moduli space of rational maps. In this talk, we study this problem in a
well-known slice called Newton family. We prove that, in the moduli space of
quartic Newton maps, a hyperbolic component is bounded if and only if it is not
type-IE(immediately escaping); furthermore, the GIT-compactification of each
type-IE hyperbolic component at infinity boundary is either an analytic closed disk
or one point. The proof is based on a convergence theorem of internal rays we
establish for degenerate Newton sequences. This is a joint work with Hongming
Nie.

Jan. 1** **2021
- Break for New Year

Jan. 8** **2021
- **Wenbo Li (****李文博****) **(University of Toronto)

Time: 11:00am-12:00pm (Beijing Time) Location: Zoom (ID: 649 1586 5872, Password: 416250)

Title: Conformal dimension and minimality of
stochastic objects

Abstract: In this talk, we discuss the
conformal dimension of some stochastic objects. The conformal dimension of
a metric space is the infimum of the Hausdorff dimension of all its
quasisymmetric images. We call a metric space minimal if its
conformal dimension equals its Hausdorff dimension. We begin with a
construction of a graph of a random function which is a.s. minimal. Inspired by
this, we apply the same techniques to the study of 1-dimensional
Brownian graphs. The main tool is the Fuglede modulus. This
is a joint work with Ilia Binder and Hrant Hakobyan.

Jan. 13** **2021
-** Tim Mesikepp **(University of
Washington)

Time: 11:00am-12:00pm (Beijing Time) Location: Zoom (ID: 649 1586 5872, Password: 416250)

Title:How to weld. Or, towards a unified approach
to weldings, driving functions, and energies.

Abstract: How do you conformally weld? That is,
given a welding homeomorphism $\varphi:\mathbb{S}^1 \rightarrow \mathbb{S}^1$,
how do you compute the associated conformal maps? The measurable Riemann
mapping theorem provides a non-constructive answer for quasisymmetric weldings,
but sometimes an explicit approximation is helpful. A beautiful algorithm
developed by Don Marshall and Lennart Carleson in the early 1980's, called the
zipper, provides a numerically robust approach by composing large numbers of
"slit" mappings. We give the first convergence results for the zipper
by shifting the problem from weldings to driving functions. In the process, we
also give some results on minimal-energy curves and, if time permits, draw
connections to universal Teichmuller space. There will be many pictures and the
talk should be widely accessible.

Mar. 24** **2021 –** Zhe Sun **(IHES)

Time: 16:00-17:00pm (Beijing Time) Location: Zoom (ID: 674 1443 0868, Password: 904596)

Title: Surface group representations, tropical points and
webs.

Abstract: Kuperberg introduced oriented 3-valent
graphs on the surface, called 3-webs, to study the SL_3-invariant tensor
products T of irreducible representations of SL_3. Then Kuntson-Tao found a
family of linear inequalities to characterize when T contains an invariant
vector. Let A be a variation of the SL_3 character variety which generalizes
the Penner's decorated Teichmuller space. Actually, Goncharov--Shen related
Kuntson-Tao inequalities to the positivity of A. On the surface, we identify
the space of 3-webs up to homotopy with certain lattice of A mapping class
group equivariantly. As a consequence, as predicted by Fock--Goncharov duality
conjecture, these tropical points parameterize a linear basis of the regular
function ring of the dual space explicitly. This is a joint work with Daniel
Douglas.

Mar. 30 2021 - **Suzhen
Han (****韩素珍****)** (Peking
University)

Time: 10:00am-12:00pm (Beijing Time) Location: Room 1114, Science Building 1,
Peking University

Title: The proper actions of relatively hyperbolic
groups on finite products of quasi-trees

Abstract: A finitely generated group has property
(QT) if it can act isometrically on a finite product of quasi-trees so that the
orbit map is a quasi-isometric embedding. This notion is introduced by M.
Bestvina, K. Bromberg and K. Fujiwara, who also established such property for
residually finite hyperbolic groups and mapping class groups. In a joint work
with H.T. Nguyen and Wenyuan Yang, we generalize their result for hyperbolic
groups, and showed that residually finite relatively hyperbolic groups have
property (QT) if their peripheral subgroups satisfy some conditions.

April 7** **2021
–** Hao Liang (****梁灏****) **(Sun Yat-sen University)

Time: 3:00-4:00pm (Beijing Time) Location: Zoom (ID: 611 5027 7004, Password: 329424)

Title: A factorization theorem for homomorphisms
into Fuchsian groups

Abstract: I will present a factorization
theorem for homomorphisms into Fuchsian groups (discrete subgroups of
the isometry group of the hyperbolic plane). It is a useful tool for
understanding the structure of the collection of all discrete representations
of any fixed finitely generated group G into PSL(2, R). It will also
have some applications in discrete limit groups of PSL(2, R). I will
discuss the background and application of the theorem and the main idea
of the proof.

April 12** **2021
–** Yusheng Luo (****骆宇盛****) **(University of Michigan – Ann Arbor / Stony Brook University)

Time: 9:00-10:00am (Beijing Time) Location: Zoom (ID: 677 2262 6461, Password: 304728)

Title: Circle packings, kissing reflection group
and critically fixed anti-rational maps

Abstract: Circle packings appear frequently in the
studies of dynamics, geometry. One can naturally associate a reflection group
to a finite circle packing, generated by reflections along the corresponding
circles. In this talk, we will establish an explicit correspondence between
such reflection groups with anti-holomorphic maps of the Riemann sphere where
all the critical points are fixed. We will explore the correspondence both in
the dynamical plane and the parameter spaces. In particular, we will explain
how the analogue of Thurston’s compactness theorem for acylindrical hyperbolic
3-manifold holds for critically fixed anti-rational maps.

We will also brief discuss some open questions
motivated by the correspondence.

April 21** **2021
–** Ma Biao (****马彪****) **(Université de Nice)

Time: 4:00-5:00pm (Beijing Time) Location: Zoom (ID: 821 5134 9008, Password: 370777)

Title: Boundary representations of mapping class
groups

Abstract: Let S = S_g be a closed orientable
surface of genus g > 1 and Mod(S) be the mapping class group of $S$. In this
talk, we show that the boundary representation of Mod(S) is ergodic, which
generalizes the classical result of Masur on ergodicity of the action of Mod(S)
on the projective measured foliation space PMF(S). As a corollary, we show that
the boundary representation of Mod(S) is irreducible.

May 12** **2021
–** Lizhi Chen (****陈立志****) **(Lanzhou University)

Time: 4:00-5:00pm (Beijing Time) Location: Zoom (ID: 895 8085 2996, Password: 339132)

Title:
Homology and homotopy complexity of manifolds via systolic geometry

Abstract: We discuss homology and homotopy
complexity of manifolds in terms of Gromov’s systolic inequality. The optimal
constant in systolic inequality is usually called systolic volume. A central
theorem in systolic geometry relates systolic volume to simplicial volume.
Since for hyperbolic manifolds there exist proportionality principle, this
theorem builds a bridge between systolic geometry and hyperbolic geometry. In
the talk, we will present some applications of this theorem to the problem of
homology and homotopy complexity of manifolds.

May 17** **2021
–** Roland Roeder **(Indiana University
– Purdue University Indianapolis)

Time: 9:00-10:00am (Beijing Time) Location: Zoom (ID: 813 3007 9112, Password: 828437)

Title:
Dynamics of groups of birational automorphisms of cubic surfaces and
Fatou/Julia decomposition for Painlevé 6

Abstract: We study the dynamics of the group of
holomorphic automorphisms of the affine cubic surfaces

S_{A,B,C,D} = {(x,y,z) \in C^3
: x^2 + y^2 + z^2 +xyz = Ax +
By+Cz+D},

where A,B,C, and D are complex parameters. It arises naturally in the dynamics on
character varieties and it also describes the monodromy of the famous Painlevé 6 differential equation. We explore the Fatou/Julia dichotomy for this
group action (defined analogously to the usual definition for iteration of a
single rational map) and also the Locally Discrete / Non-Discrete dichotomy (a
non-linear version from the classical discrete/non-discrete dichotomy for Lie
groups). The interplay between these two dichotomies allow us to prove several
results about the topological dynamics of this group. Moreover, we show the coexistence of
non-empty Fatou sets and Julia sets with non-trivial interior for a large set
of parameters.

This is joint work with Julio Rebelo.

May 26** **2021
–** Fei Yang (****杨飞****) **(Nanjing University)

Time: 4:00-5:00pm (Beijing Time) Location: Zoom (ID: 884 5933 7010, Password: 684103)

Title: Local
connectivity of the Julia sets with bounded type Siegel disks

Abstract: Let f be a holomorphic map containing an
irrationally indifferent fixed point z0. If f is locally linearizable at z0,
then the maximal linearizable domain containing z0 is called the Siegel disk of
f centered at z0. The topology of the boundaries of Siegel disks has been
studied extensively in past 3 decades. This was motivated by the prediction of
Douady and Sullivan that the Siegel disk of every non-linear rational map is a
Jordan domain.

For the topology of whole Julia sets of holomorphic
maps with Siegel disks, the results appear less. Petersen proved that the
quadratic Julia sets with bounded type Siegel disks are locally connected.
Later Yampolsky proved the same result by an alternative method based on the
existence of complex a prior bound of unicritical circle maps. A big progress
was made by Petersen and Zakeri in 2004. They proved that for almost all
rotation numbers, the quadratic Julia sets with Siegel disks are locally connected.
Recently J. Yang proved a striking result that the Julia set of any polynomial
(assumed to be connected) is locally connected at the boundary points of their
bounded type Siegel disks.

As a generalization of Petersen's result, we prove
that the Julia sets of a number of rational maps and transcendental entire
functions with bounded type Siegel disks are locally connected. This is based
on establishing an expanding property of a long iteration of a class of
quasi-Blaschke models near the unit circle.

June 9** **2021
–** Yulan Qing (****卿于兰****) **(Fudan University)

Time: 4:00-5:00pm (Beijing Time) Location: Zoom (ID: 857 3683 8743, Password: 566197)

Title: The
Large scale geometry of big mapping class groups

Abstract: In this talk, we introduce the framework
of the coarse geometry of non-locally compact groups in the setting of big
mapping class groups, as studied by Rosendal. We will discuss the
characterization results of Mann-Rafi and Horbez-Qing-Rafi that illustrate big
mapping groups' rich geometric and algebraic structures. We will outline the
proofs in these results and their implications. If time permits, we will
discuss some open problems in this area.

June 18** **2021
–** Xiaoguang Wang (****王晓光****) **(Zhejiang University)

Time: 10:00-11:00am (Beijing Time) Location: Room 1560, Science Building 1,
Peking University

Title: Boundaries of capture hyperbolic components

Abstract: In the field of complex dynamics, the
boundaries of higher dimensional hyperbolic components in meaningful families
of polynomals or rational maps are mysterious topological objects. In this talk, we discuss some typical
families of polynomials, and show that the boundary of a (high-dimensional)
capture hyperbolic component is homeomorphic to the sphere. More strikingly,
the Hausdorff dimension of this boundary can be given explicitly. This is a
joint work with Jie Cao and Yongcheng Yin.

July 2 2021 –**
Gaofei Zhang (****张高飞****) **(Nanjing
University)

Time: 3:00-4:00pm (Beijing Time) Location: Zoom (ID:
822 0642 3584, Password: 822747)

Title: On
the local connectivity of the Julia sets of rational maps with bounded type
Siegel disks***

Abstract: In the talk, I will sketch the idea of
the proof of the following result.
Suppose f is a rational map with bounded type Siegel disks such that any
infinite critical orbit intersects either the closure of some bounded type
Siegel disk or the basin of some attracting periodic point. Suppose additionally the Julia set *J(f)* is connected. Then *J(f)* is locally connected. This was previously proved under the
assumption that the boundaries of attracting bastions do not intersect the
boundary of any Siegel disk.

September 24 2021 –** Jinsong Liu (****刘劲松****) **(Chinese
Academy of Sciences)

Time: 2:00-3:00pm (Beijing Time) Location: Room 77201, Jingchunyuan 78,
BICMR, Peking University

Title: Extensions
of quasi-isometries between complex domains***

Abstract: In this talk, by using the
Gehring-Hayman-type theorem on some complex domains. We will give some results
on bi-Holder extensions not only for biholomorphisms, but also for more general
Kobayashi metric quasiisometries between these domains.

October 28 2021 –** Jianyu Chen (****陈剑宇****) **(Soochow
University)

Time: 3:00-4:00pm (Beijing Time) Location: Zoom (ID: 886 6229 2968,
Password: 733229)

Title:
Inducing schemes with finite weighted complexity

Abstract: We consider a measurable map of a compact
metric space which admits an inducing scheme. Under the finite weighted
complexity condition, we establish a thermodynamic formalism for a parameter
family of potentials $\varphi+t\psi$ in an interval containing $t=0$.
Furthermore, if there is a generating partition compatible to the inducing
scheme, we show that all ergodic invariant measures with sufficiently large
pressure are liftable. Our results are applicable to the Sinai dispersing
billiards with finite horizon, that is, we establish the equilibrium measures
for the family of geometric potentials in a slightly restricted class. This is
a joint work with Fang Wang and Hong-Kun Zhang.

November 9 2021 –** Renxing Wan (****万仁星****) **(Peking
University)

Time: 3:10-5:00pm (Beijing Time) Location: Room 1114, Science Building 1,
Peking University

Title:
Uniform growth in groups of exponential growth

Abstract: This talk is based on two surveys about
the growth of groups by Grigorchuk and de la Harpe. At first, I will briefly
introduce some notions and list known results and open problems. Later, we will
focus on some basic examples including free groups, non-elementary hyperbolic
groups, some solvable groups , mapping class group and so on.

November 26 2021 –** Bin Yu (****余斌****) **(Tongji
University)

Time: 3:00-5:00pm (Beijing Time) Location: Zoom (ID:
879 8424 0017,
Password: 928052)

Title:
R-covered Anosov flows and freely homotopic periodic orbits

Abstract: In the last 30 years, by using foliation
theory and group action, Barbot and Fenley developed a powerful tool to
qualitatively understand 3-dimensioanl Anosov flows. In this seminar talk, we will introduce some basic ideas due to
Fenley in this direction. In particular,
we will explain a significant property discovered by Fenley: every free
homotopy class of an R-covered Anosov
flow on a hyperbolic 3-manifold contains infinitely many periodic orbits.

December 3 2021 –** Inhyeok Choi **(Seoul National University)

Time: 3:00-4:00pm (Beijing Time) Location: Zoom (ID:
871 9276 2553,
Password: 897719)

Title: Limit
laws for random walks on mapping class groups

Abstract: Random walks on groups acting on
non-positively-curved spaces have been studied in depth. Various limiting
behaviors of these random walks, including the positive escape rate, laws of
large numbers and central limit theorems, were known under some geometric
assumptions on the space and the group action. However, the actions of mapping
class groups on Teichmüller spaces and curve complexes do not satisfy these
assumptions, the former one being not Gromov hyperbolic and the latter one
being not locally compact. Nonetheless, recent developments imply that many
limiting behaviors of random walks on mapping class groups follow directly from
the (partial) hyperbolicity of Teichmüller spaces and curve complexes. In this
talk, I will focus on the principle behind these results and two possible
applications: the regularity of the harmonic measure and the counting problem
in mapping class groups. Partially joint with Hyungryul Baik and Dongryul M.
Kim.

December 24 2021 –** Jinsong Zeng (****曾劲松****) **(Guangzhou University)

Time: 3:00-4:00pm (Beijing Time) Location: Zoom (ID:
875 7470 1894,
Password: 101624)

Title:
Decomposition of rational maps

Abstract: This talk is mainly about the dynamics of
rational maps on the Riemann sphere. We will show that every postcritically
finite rational map with non-empty Fatou set can be decomposed into bubble
rational maps and Sierpinski rational maps. Based on this theory, an invariant
and finite connected graph can be constructed in the Julia set. This is a joint
work with Guizhen Cui and Yan Gao.

March 12 2022 –**
Zhuchao Ji (****冀诸超****) **(Shanghai
Center for Mathematical Sciences)

Time: 2:00-3:00pm (Beijing Time) Location: Quanzhai 9, Jingchunyuan 78,
BICMR, Peking University

Title: On
wandering domains in higher dimensions

Abstract: See
here.

May 26 2022 –**
Tim Mesikepp **(University of Washington)

Time: 10:00-11:00am (Beijing Time) Location: Zoom (ID:
826 4953 9587,
Password: 083683)

Title: A
deterministic approach to Loewner-energy minimizers

Abstract: The Loewner energy is a
conformally-invariant functional of curves in the upper half plane H that has
connections to hyperbolic geometry, SLE theory, Teichmuller theory and
geometric function theory. In this talk we study "one-point"
minimizers of Loewner energy, asking what curves minimize the energy among all
which pass through a given point in H, and what curves minimize the energy
among all which weld given points x<0<y. The former question was
partially studied by Yilin Wang, who used SLE techniques to calculate the
minimal energy and show it is uniquely attained. We revisit the question
using a purely deterministic methodology, however, and re-derive the energy
formula and also obtain further results, such as an explicit computation of the
driving function. Our approach also yields existence and uniqueness of
minimizers for the welding question, as well as an explicit energy formula and
explicit driving function. We discuss parallel properties of both
families, such as "universality" (all curves in either family can be
generated by a single driving function), energy usage, and connections to SLE
with forcing.

May 26 2022 –**
Jingyin Huang **(Ohio State University)

Time: 8:00-9:00pm (Beijing Time) Location: Zoom (ID:
860 9860 6310,
Password: 399654)

Title: The non-positive curvature geometry of some
fundamental groups of complex hyperplane arrangement complements

Abstract: A complex hyperplane complement is a
topological space obtained by removing a collection of complex codimension one
affine hyperplanes from C^n (or a convex cone of C^n). Despite the simple
definition, these spaces have highly non-trivial topology. They naturally
emerge from the study of real and complex reflection groups, braid groups and
configuration spaces, and Artin groups. More recently, the fundamental groups
of some of these spaces start to play important roles in geometric group theory,
though most of these groups remain rather mysterious. We introduce a geometric
way to understand classes of fundamental groups of some of these spaces, by
equivariantly “thickening” these groups to metric spaces which satisfy a
specific geometric property that is closely related to convexity and
non-positive curvature. We also discuss several algorithmic, geometric and
topological consequences of such a non-positive curvature condition. This is
joint work with D. Osajda.

June 21 2022 –**
Fabrizio Bianchi **(Université de
Lille)

Time: 4:00-5:00pm (Beijing Time)
Location: Zoom (ID: 823 4284 1727, Password: 366599)

Title: A spectral gap for the transfer operator on complex projective
spaces

Abstract: We study the transfer (Perron-Frobenius) operator on Pk(C)
induced by a generic holomorphic endomorphism and a suitable continuous weight.
We prove the existence of a unique equilibrium state and we introduce various
new invariant functional spaces, including a dynamical Sobolev space, on which
the action of f admits a spectral gap. This is one of the most desired
properties in dynamics. It allows us to obtain a list of statistical properties
for the equilibrium states such as the equidistribution of points, speed of
convergence, K-mixing, mixing of all orders, exponential mixing, central limit
theorem, Berry-Esseen theorem, local central limit theorem, almost sure
invariant principle, law of iterated logarithms, almost sure central limit
theorem and the large deviation principle. Most of the results are new even in
dimension 1 and in the case of constant weight function, i.e., for the operator
f_*. Our construction of the invariant functional spaces uses ideas from
pluripotential theory and interpolation between Banach spaces. This is a joint
work with Tien-Cuong Dinh.

September 20 2022 – **Carlangelo Liverani **(University of Rome Tor Vergata)

Time: 4:00-5:00pm (Beijing Time) Location: Room 77201, Jingchunyuan 78,
BICMR, Peking University; Zoom (ID: 812 0977 4828, Password:
813004)

Abstract: I will discuss some recent and less recent results concerning the
statistical properties of hyperbolic billiards, with particular emphasis on
properties motivated
by fundamental questions in non-equilibrium statistical mechanics.

October 7 2022 – **Wenbo Li (****李文博****)** (Peking
University)

Time: 10:30-11:30am (Beijing Time) Location: Room 77201, Jingchunyuan 78,
BICMR, Peking University

Abstract: A Hyperbolic space is a metric space whose geodesic triangles are
"thin" and a hyperbolic group is a group whose Cayley graph is
hyperbolic. We focus on the boundary of hyperbolic groups in this talk and go
through topics around two rigidity conjectures: The Cannon Conjecture and the
Kapovich-Kleiner Conjecture. Roughly speaking, these two conjectures ask about
whether special topological restrictions on the boundary of hyperbolic groups
will uniquely determine them up to quasisymmetries. In an effort to answer
these questions, many people have studied them from different approaches. We
will go through the work by Bonk and Kleiner. In the end, we provide our trying
on these conjectures. In particular, we have constructed a special case of
metric Sierpinski carpet, dyadic slit carpets, and completely characterize its
planar quasisymmetric embeddability.

October 14 2022 – **Xiaolei Wu (****伍晓磊****) **(Fudan
University)

Time: 3:00-4:00pm (Beijing Time) Location: Tencent (ID: 522-848-241,
Password: 221014)

Abstract: Asymptotic mapping class groups was introduced by Funar and
Kapoudjian. They can be treated as a generalization of the Thompson groups in
the mapping class group setting. We will discuss how these groups are
constructed and show that they are in fact of type F_infinity. The proof boils
down to prove certain subsurface complexes are highly connected. This is based
on joint work with Javier Aramayona, Kai-Uwe Bux, Jonas Flechsig and Nansen
Petrosyan.

October 21 2022 – **Ilia Binder **(University of Toronto)

Time: 8:00-9:00pm (Beijing Time) Location: Zoom (ID: 870 3163 5732, Password: 955080)

Abstract: In this talk, I will discuss recent advances in the computability
of various objects arising in Complex Dynamics and the Theory of Univalent
functions. After a brief introduction to the general Computability Theory, I
will talk about the computation properties of polynomial Julia sets, conformal
maps, and Harmonic measure. Based on joint work with M. Braverman (Princeton),
A. Glucksam (Northwestern), C. Rojas ( Pontificia Universidad Catolica de
Chile), and M. Yampolsky (University of Toronto).

October 27 2022 – **Bobo Hua (****华波波****) **(Fudan
University)

Time: 3:00-4:00pm (Beijing Time) Location: Tencent (ID: 464-146-395,
Password: 221028)

Abstract: Ollivier introduced a curvature notion on graphs via the optimal
transport, which is a discrete analog of the Ricci curvature on manifolds.
Analytic properties of discrete harmonic functions are closely related to the
Ollivier curvature. We prove that the number of ends is at most two for an
infinite graph with nonnegative Olliver curvature. This is joint work with
Florentin Muench.

November 4 2022 –MariuszUrbański(University of North Texas)

Time: 9:00-10:00am (Beijing Time) Location: Zoom (ID: 827 6082 3142, Password: 057934)

Abstract: We consider the subclass of class $\mathcal B$ consisting of
meromorphic functions $f\C\to\C$ for which infinity is not an asymptotic value
and whose all poles have orders uniformly bounded from above. This class was
introduced in [BwKo2012] and the Hausdorff dimension HD(I(f)) of the set I(f)
of all points escaping to infinity under forward iteration of $f$ was estimated
therein. In this lecture, based on joint paper with Volker Mayer, we provide a
closed formula for the exact value of HD(I(f)) identifying it with the critical
exponent of the natural series introduced in [BwKo2012]. This exponent is very
easy to calculate for many concrete functions. In particular, we construct a
function from this class which is of infinite order and for which HD(I(f))=0.

November 25 2022 –Daniel Meyer(University of Liverpool)

Time: 5:00-6:00pm (Beijing Time) Location: Zoom (ID: 838 0851 2867, Password: 686499)

Abstract: Quasisymmetric maps map ratios of distances in a controlled way.
They generalize conformal maps. The quasisymmetric uniformization theorem asks
if a certain metric space is quasisymmetrically equivalent to some model space.
Of particular interest in this context is the question to characterize
quasispheres, i.e., metric spaces that are quasisymmetrically equivalent to the
standard $2$-sphere. A simple class of fractal spheres are ``snowballs'', which
are topologically $2$-dimensional analogs of the van Koch snowflake curve.

A Thurston map is a topological analog of a rational map (i.e., a
holomorphic self-map of the Riemann sphere). Thurston gave a criterion when
such a map ``is'' rational. Given such a map $f$ that is expanding, we can
equip the sphere with a ``visual metric''. With respect to this metric, the
sphere is a quasisphere if and only if $f$ ``is'' rational.

This is joint work with Mario Bonk (UCLA).

December 2 2022 –Sang-hyun Kim(KIAS)

Time: 3:00-5:00pm (Beijing Time) Location: Zoom (ID: 565 147 0713, Password: 944743)

Abstract: We prove that two compact connected manifolds are homeomorphic
only if their homeomorphism groups are elementarily equivalent, i.e. have
exactly the same set of the true first-order group-theoretic sentences . This
is a work in progress. (Joint with Thomas Koberda and Javier de la
Nuez--Gonzalez).

March 17 2023 –Sylvester Eriksson-Bique(University of Oulu)

Time: 4:30-5:30pm (Beijing Time) Location: Zoom (ID: 892 5622
7347,
Password: 094591)

Abstract: What is in common between random or dynamically generated limit
superior sets and boundaries of sets of finite perimeter. In both settings
problems can be reduced to estimating the Hausdorff content of some set which
has a type of self-similarity. We introduce a new lemma that yields a Hausdorff
content lower bound in both of these settings, and which seems useful in
further applications. We illustrate its usefulness by deriving three well known
conclusions from it: the Federer characterization of sets of finite perimeter,
the Hausdorff dimension of random limit superior sets and the
Beresnevitch-Velani Mass Transference Principle.

April 14 2023 –Insung Park(Brown University)

Time: 10:00-11:00am (Beijing Time) Location: Zoom (ID: 856 2139
9573,
Password: 111111)

Abstract: The Ahlfors-regular conformal dimension, ARCdim(X), of a compact
metric space X is the infimal Hausdorff dimension in the Ahlfors-regularly
quasi-symmetric class of X. As a fractal embedded in the Riemann sphere, the
Julia set J_f of a hyperbolic rational map f has Ahlfors-regular conformal
dimension between 1 and 2. We have ARCdim(J_f)=2 iff J_f is the entire Riemann
sphere. The other extreme case ARCdim(J_f)=1, however, contains a variety of
Julia sets, including Julia sets of critically finite polynomials and Newton
maps. In this talk, we show that for a critically finite hyperbolic rational
map f, ARCdim(J_f)=1 if and only if there exists an f-invariant graph G
containing all the critical points such that the topological entropy of the
induced dynamics on G is zero. We also show that for a (possibly
non-hyperbolic) critically finite rational map f, ARCdim(X)=1 is attained as
the minimal Hausdorff dimension if and only if f is conjugate to the monomial
map z^{\pm} or the Chebyshev polynomial. This talk is partially based on joint
work with Angela Wu.

April 21 2023 –John Mackay(University of Bristol)

Time: 5:00-6:00pm (Beijing Time) Location: Zoom (ID: 859 6470
2961,
Password: 111111)

Abstract: Isometries of real hyperbolic spaces correspond to Mobius
transformations of their spheres at infinity. Thanks to Paulin and
Bonk-Schramm, quasi-isometries of Gromov hyperbolic spaces correspond to
quasisymmetries between their boundaries at infinity. After discussing
this background, I'll describe how certain maps between relative hyperbolic
groups correspond to maps between their Bowditch boundaries, and an
application. Joint work with Alessandro Sisto.

May 19 2023 –Kevin M. Pilgrim(Indiana University Bloomington)

Time: 8:00-9:00pm (Beijing Time) Location: Zoom (ID: 847 5703
2480,
Password: 111111)

Abstract: We show that a contracting recurrent self-similar group of
polynomial activity growth is amenable. The proof proceeds by first
showing that the associated limit space has conformal dimension equal to 1. We
then show this implies the orbital graphs of its standard action are recurrent.
Finally, we deduce amenability by applying results of Juschenko, Nekrashevych,
and de la Salle. Examples of such groups include the iterated monodromy
groups of critically finite hyperbolic “crochet” rational functions;
these are maps for which any two points in the Fatou set are joined by a curve
that meets the Julia set in an at most countable set.

May 24 2023 –Yulan Qing(卿于兰) (Fudan University)

Time: 1:00-3:00pm (Beijing Time) Location: Room
1114, Sciences Building No. 1

Abstract: Gromov boundary provides a useful compactification for all
infinite-diameter Gromov hyperbolic spaces. It consists of all geodesic rays
starting at a given base-point and it is an essential tool in the study of the
coarse geometry of hyperbolic groups. In this study we introduce a
generalization of the Gromov boundary for all finitely generated groups. We
construct the sublinearly Morse boundaries and show that it is a QI-invariant
topological space and it is metrizable. We show the geometric genericity of
points in this boundary using Patterson Sullivan measure on the visual boundary
of CAT(0) spaces. As an application we discuss the connection between the
sublinearly Morse boundary and random walk on groups. We answer open problems
regarding QI-invariant models of random walk on CAT(0) groups and on mapping
class groups. If time permits, we also will also look at how this boundary
behaves under sublinear bilipschitz equivalences.

May 26 2023 –Araceli M. Bonifant(University of Rhode Island)

Time: 8:00-9:00pm (Beijing Time) Location: Zoom (ID: 889 4828
5549,
Password: 111111)

Abstract: For each p>0 there is a family S_p of complex cubic maps with
a marked critical orbit of period p. For each q>0, I will describe a
dynamically defined tessellation of S_p. Each face of these tessellations is
associated with one particular behavior for periodic orbits of period q. (Joint
work with John Milnor.)

Jun 5 2023 –Tim Mesikepp(Peking University)

Time: 11:00am-12:00pm (Beijing Time) Location: Room 1114, Sciences Building No.
1

Abstract: The Loewner energy is a functional on Jordan curves that is the
large-deviations rate function for Schramm-Loewner Evolutions (SLE's). Recent work has shown it also has surprising
connections to other areas of probability, complex analysis, geometric measure
theory, hyperbolic geometry, and even Teichmuller theory. In the first part of this talk we survey some
of these connections, providing an "appetizer" for Yilin Wang's
upcoming summer school.

In the second part of the talk we show that Loewner-energy minimizers give
insight into the regularity of the inverse Loewner transform (the map taking a
driving function to its associated curve).
In particular, we use energy minimizers to exhibit curves of differing
regularity that have driving functions of the same regularity, thus showing a
theorem of Carto Wong is sharp. This
portion of the talk will be deterministic in nature.

Jun 5 2023 to Jun 12 2023 –Mario Bonk(UCLA)

Time & Location:

Jun 5 2023 9:00-11:00am (Beijing Time) Room 1114, Sciences Building No. 1

Jun 10 2023 10:00am-12:00pm (Beijing Time) Room 1556, Sciences Building No. 1

Jun 12 2023 3:00-5:00pm (Beijing Time) Room 1556, Sciences Building No. 1

Abstract: A postcritically-finite branched covering map on a topological
2-sphere is known as a Thurston map. It induces a holomorphic map on an
associated Teichmüller space. This map is known as the Thurston pull-back map. In this mini
course, I will give a quick introduction to necessary background (Teichmüller spaces and Thurston maps), outline a proof of the holomorphicity of the
pull-back map, and discuss a landmark theorem by Thurston on the
characterization of postcritically-finite rational maps.

I’ll discuss concrete examples for the pull-back maps for Thurston maps
with four postcritical points. As will be seen in the course, this leads to a
certain type of polymorphic functions related to modular functions and forms.
If time permits, I will also discuss ongoing research about the existence of a
global finite curve attractor in this framework.

June 7 2023 –Mario Bonk(UCLA)

Time: 4:00-5:00pm (Beijing Time)
Location: Room 1114, Sciences Building No. 1

Abstract: A Thurston map is a branched covering map on a topological 2-sphere for
which the forward orbit of each critical point under iteration is finite. Each
such map gives rise to a fractal geometry on its underlying 2-sphere. The study
of these maps and their associated fractal structures links diverse areas of
mathematics such as dynamical systems, ergodic theory, classical conformal
analysis, hyperbolic geometry, Teichmüller theory, and analysis on metric
spaces. In my talk I will give an introduction to this subject and report on
ongoing research.

June 16 2023 –Sabyasachi Mukherjee(Tata Institute of Fundamental Research)

Time: 4:00-5:00pm (Beijing Time) Location: Zoom (ID: 852 0183
6977,
Password: 111111)

Abstract: There are two frameworks for mating Kleinian groups with rational
maps on the Riemann sphere: an algebraic correspondence framework due to
Bullett-Penrose-Lomonaco and an orbit equivalence mating framework using
Bowen-Series maps. The latter is analogous to the Douady-Hubbard theory for
polynomial mating. We will discuss how these two frameworks can be unified and
generalized. As a consequence, we will construct holomorphic correspondences
that are matings of hyperbolic orbifold groups (including Hecke groups) with
Blaschke products. Time permitting, we will introduce an analog of a Bers slice
of the above orbifolds in the algebraic parameter space of correspondences.
Based on joint work with Mahan Mj.

June 30 2023 – **Scott Sutherland **(Stony Brook University)

Time: 9:00-10:00pm (Beijing Time) Location: Zoom (ID: 871 7088
4305,
Password: 111111)

Abstract: We consider the family or degree two rational maps of the Riemann
sphere to itself that preserve the real line, with a goal of understanding the
parameter space of all such mappings. As
in the case of polynomial maps, the
postcritically finite case (which are Thurston maps) play an important role; we
also discuss the associated combinatorics of such maps. [Joint work with Araceli Bonifant and John
Milnor.]

July 14 2023 – **Dylan Thurston **(Indiana University, Bloomington)

Time: 2:30-3:30pm (Beijing Time) Location: BICMR 77201

Abstract: How does a (post-critically finite) rational map act on the
Riemann sphere, topologically and as a dynamical system? We give a concrete way
to understand this using spines, appropriate graphs embedded in the sphere. The
entire dynamics of the map can be faithfully captured combinatorially by a pair
of graphs and a pair of maps between them.

July 14 2023 – **Dylan Thurston **(Indiana University, Bloomington)

Time: 4:00-5:00pm (Beijing Time) Location: BICMR JiaYiBing Lecture Hall

Abstract: We start with a detour to recall part of the Nielsen-Thurston
classification of surface self-maps, and in particular how curves on the
surface are stretched under iteration. The analogous notion for a general
dynamical system, which we call the expansivity, appears to be captured for
rational maps by a stretch factor naturally arising from the graphical
description, and the minimal Lipschitz stretch of maps between graphs.

July 15 2023 – **Dylan Thurston **(Indiana University, Bloomington)

Time: 10:00-11:00am (Beijing Time) Location: BICMR 77201

Abstract: When does a topological self-map of the sphere come from an
actual rational map? Can we give a positive combinatorial certificate that this
happens? Expansivity from the previous lecture gives one constraint, but to
give the full answer we need another "energy" associated to graph
maps, the stretch factor related to rubber bands and, with some caveats, to
electrical networks.

September 8 2023 – **Oliver Jenkinson **(Queen Mary, University of London)

Time: 4:00-5:00pm (Beijing Time) Location: Room 1114, Sciences Building No.
1

Abstract: For a given dynamical system, and a given real-valued function,
the field of Ergodic Optimization seeks to understand those orbits (or
invariant measures) that realise the largest possible ergodic average. Rather
often, these orbits turn out to be in some sense `simple', for example
periodic, or non-periodic but of low complexity (e.g. Sturmian). Ergodic
dominance is one strategy, appealing to ideas from stochastic dominance, for
understanding the optimizing orbits and measures for certain classes of
functions. One application is to constrained optimization problems for digit
expansions: e.g. if we fix the mean value of the decimal digits of a number, or
equivalently fix the arithmetic mean of an orbit under the map x-> 10x (mod
1), how can we minimize the variance around the mean, and how can we maximize the
geometric mean?

October 10 2023 – **Ilia Binder **(University
of Toronto)

Time: 4:00-5:00pm (Beijing Time) Location: BICMR 77201

Abstract: In this talk, I will discuss computability questions arising in
Complex Dynamics and Complex Analysis. More specifically, I will discuss the
existence of non-computable polynomial Julia sets and of polynomial Julia sets
with arbitrarily high computational complexity. I will also discuss the
conditions necessary and sufficient for the computability of conformal maps and
their Caratheodory extensions. The computability of another central object of
Complex Analysis, the Harmonic measure, will also be discussed. A crash course
in general Computability Theory will be given. The talk is based on joint
projects with M. Braverman (Princeton), A. Glucksam (Northwestern), C. Rojas
(Universidad Catolica de Chile), and M. Yampolsky (University of Toronto).

October 27 2023 – **Guy C. David **(Ball State University)

Time: 9:00-10:00am (Beijing Time) Location: Zoom (ID: 861 9382 6449, Password: 647898)

Abstract: We discuss two results on bi-Lipschitz embeddings of metric
spaces. The first is a (negative) answer to a 1997 question of Heinonen and
Semmes asking whether spaces that can be “folded” into Euclidean space can be
embedded bi-Lipschitzly. The second is a more general theorem showing that
spaces with “thick” curve families cannot have bi-Lipschitz embeddings in
Euclidean space, unless they admit some “infinitesimal splitting”. This latter
result has some applications to the study of conformal dimension. This is joint
work with Sylvester Eriksson-Bique (University of Jyvaskyla).

November 27 2023 – **Zachary Smith **(UCLA)

Time: 10:00-11:00am (Beijing Time) Location: Zoom (ID: TBC, Password:
TBC)

Abstract: A Thurston map f: (S^2, A) → (S^2, A) with marking set A induces
a map σ_f on Teichmüller space by pulling back complex structures. One way to
study this pullback map is via a correspondence on moduli space. In this talk I
will discuss both the general correspondence and some special cases.
Specifically, I will give a careful description of the correspondence for the
case where A has four points and the postcritical set (which is a subset of A)
has three points. The diagram I will present has been previously used by Kelsey
and Lodge to study quadratic Thurston maps with few postcritical points, and I
also use it in a forthcoming paper proving special cases of the curve attractor
problem for rational maps with four marked points. Time permitting I will
further discuss the results of this paper.

Time: TBC (Beijing Time) Location: Zoom (ID: TBC, Password:
TBC)

Abstract: TBC

Last
Updated: 11/21/2023