Quasiworld

• Wednesday, February 5th:

8-9 am (PST):  Nicolas Matte Bon (CNRS, Université Claude Bernard Lyon 1)

Conformal dimension and Liouville property for iterated monodromy groups

The dynamics of every expanding (branched) self cover of a compact space (e.g. a hyperbolic complex rational function in restriction to its Julia set) can be encoded by a group, called its iterated monodromy group, which is an example of contracting self-similar group. The original dynamical system can be reconstructed from the group through a construction called limit dynamical system (this theory was extensively developed by V. Nekrashevych and others). The amenability of all iterated monodromy groups is a long-standing open question. In joint work with V. Nekrashevych and T. Zheng, we show that the iterated monodromy group of every hyperbolic rational function is amenable. This follows from a criterion based on quasi-conformal geometry. The limit dynamical system of a contracting self-similar group admits a canonical quasi-conformal structure, so that its (Ahlfors-Regular) conformal dimension is well defined (for the iterated monodromy group of a hyperbolic rational function it is equal to the conformal dimension of the Julia set). If the conformal dimension of this space is less than 2, then every symmetric random walk with finite second moment on the group has trivial poisson boundary (this property, also called the Liouville property, implies amenability).

9-10 am (PST): Dzmitry Dudko (Stony Brook University)

Non-uniform hyperbolicity of PCF correspondences on Riemann surfaces and applications to the problem of finite global curve/graph attractors for PCF rational maps

One of the challenges in the theory of postcritically finite (PCF) rational maps is that the associated modular correspondence is not (in general) uniformly hyperbolic. We will introduce a notion of weak hyperbolicity, called ``contraction along X-rays'', and justify it for (non-exceptional) PCF correspondences between Riemann surfaces. Roughly, non-uniform contraction eventually dominates any additive correction. Consequently, a (non-Lattes) PCF rational map with 4 postcritical points has finite global curve/graph attractors. We will also discuss motivations and related questions.

Joint work with Laurent Bartholdi and Kevin M. Pilgrim, arXiv:2407.15548.

• Wednesday, March 5th:

NOTE THE SPECIAL TIME:

7-8 am (PST): Juliana Xavier (IMERL-FING-UdelaR)

Indecomposability of Julia sets from a topological viewpoint

It is a longstanding question in one-dimensional complex dynamics if it exists a rational function whose Julia set is an indecomposable continuum.  We explore this problem from a topological viewpoint: is it even possible for a branched covering of the sphere of degree d>1?  This problem relates to another old conjecture about C^1 maps of the sphere S^2 of degree d>1, stating that the number of fixed points of the iterates $f^n$ should have an exponential growth rate.

The aim of this talk is to introduce the two problems, show how they interact and tell the little bits of truth that I know of. 

8-9 am (PST):  Nikolai Prochorov (Aix-Marseille University)

Towards Transcendental Thurston Theory

In the 1980’s, William Thurston obtained his celebrated characterization of post-critically finite rational maps. This result laid the foundation of such a field as Thurston's theory in holomorphic dynamics, which has been actively developing in the last few decades. One of the most important problems in this area is the characterization question, which asks whether a given topological map is equivalent to a holomorphic one. The result of W. Thurston and further developments allow us to answer this question quite effectively in the setting of (postcritically finite) maps of finite degree, and it has numerous applications for the dynamics of rational maps.

A similar question can be formulated for the maps of infinite degrees (i.e., in the transcendental setting), for instance, for entire or meromorphic postsingularly finite maps. However, the characterization problem becomes significantly more complicated, and the complete answer in the transcendental case is still not known.

In my talk, I am going to motivate the questions above and introduce the key notions of Thurston's theory in the transcendental setting.  I will present a result showing that a version of Thurston's theorem applies to a large class of transcendental maps. If time permits, I will also briefly discuss a "relative" version of Thurston's theorem, which works in complete generality in both finite and infinite degree cases.

• Wednesday, April 2nd:

8 - 9 am (PST): Ville Salo (University of Turku)

Groups of cellular automata on groups

The full shift is the Cantor space A^G for finite alphabet A and countably infinite group G, endowed with the shift action of G. The automorphism group Aut(A^G) of the full shift is the group of G-commuting self-homeomorphisms (equivalently, reversible cellular automata). In recent work, we have shown that Aut(A^{F_2}) does not embed in Aut(A^\Z) (which answers a question of Barbieri, Carrasco-Vargas and Rivera-Burgos), and Aut(A^{\Z^D}) does not embed in Aut(A^{\Z^d}) if D is sufficiently larger than d (which answers a question of Hochman).

There are two rather orthogonal new ideas involved in the proofs:

1) To prove noninclusions into Aut(A^{\Z^d}), we develop arguably the first non-trivial method for showing that a group does not embed in Aut(A^{\Z^d}). Specifically, we show that it takes a lot of time to build elements of large finite simple subgroups of this group.

2) On the other hand, in the left-hand side groups Aut(A^{\Z^D}), Aut(A^{F_2}), we have to show that an element of a large finite alternating group can be built very quickly. This is based on ideas from computer science: we write a "short program" (in a suitable computational model) that uses the inherent parallelism of cellular automata to describe a single element of a very large simple group.

By programming more elaborate things, we can show that word problems of finitely-generated subgroups of Aut(A^G) jump from being always in co-NP on groups of polynomial growth, to being PSPACE-hard when the growth function of the group is at least stretched exponential. Under Grigorchuk's Gap Conjecture and NP \neq PSPACE, this is a strong dichotomy result.

In the talk, I explain and motivate these results, and, if time allows, explain some key ideas from the proof of Aut(A^(F_2)) \not\leq Aut(A^\Z).

9-10 am (PST): Rachel Skipper (University of Utah)

Maximal Subgroups of Thompson's group V

Maximal subgroups of a group provide a range of information about the group. First, maximal subgroups correspond to primitive actions of a group. Secondly, in a finitely generated group every proper subgroup is contained in a maximal one. In this talk, we will discuss some ongoing work with Jim Belk, Collin Bleak, and Martyn Quick to understand and classify maximal subgroups of Thompson's group V. 

• Wednesday, May 7th:

8 - 9 am (PST): Xianghui Shi (Beijing International Center for Mathematical Research and Peking University)

  TBA

9-10 am (PST): Alex Rodriguez (Stony Brook University)

Every circle homeomorphism is the composition of two conformal weldings.

Conformal welding homeomorphisms are circle homeomorphisms that arise naturally in Teichmuller theory, Mathematical physics and dynamics. It is well known that not every circle homeomorphism is a conformal welding. However, in this talk we will see that every orientation-preserving circle homeomorphisms is the composition of two conformal weldings, which implies that conformal weldings are not closed under composition. Our approach uses the log-singular maps introduced by Bishop. The main tool that we introduce are log-singular sets, which are zero capacity sets that admit a log-singular map that maps their complement to a zero capacity set.

• Wednesday, October 2nd:

8-10 am (PST):  Sylvester Eriksson-Bique (University of Jyväskylä) and Riku Anttila (University of Jyväskylä)

Combinatorial Loewner spaces which are not Quasisymmetric to Loewner

We will give a joint talk on our recent work resolving Kleiner's conjecture. For Loewner spaces there is a rich and somewhat rigid theory of quasisymetric mappings, which was already developed by Heinonen and Koskela. This theory is also useful whenever X is a metric space which is quasisymmetric to a Loewner space. One would hope to characterise such spaces, and there is a very natural necessary condition: the Q-combinatorial Loewner property. The combinatorial Loewner property (CLP) is a direct analogue of the continuous Loewner property, and is obtained by replacing continuous modulus with discrete modulus in the relevant estimtes. It was conjectured by Kleiner, and others, that any self-similar space with CLP would be quasisymmetric to a Loewner space. This would lead to many new examples of Loewner spaces, since there are many spaces, such as the Sierpinski carpet, for which CLP is known, but no known Loewner structure has been found. We found counter examples to the conjecture. In this talk we will explain these counter examples and how they were discovered. We will also describe the construction that leads to our counter-examples: Iterated Graph Systems. Such constructions seem interesting for a variety of other problems. While these give a negative answer, the original conjecture remains interesting for many other fractals. Pursuing this, we discuss also recent joint work with Guy C. David, where we established that positive answer for Kleiner for a certain pillow space would imply the existence of a ``analytically one dimensional sphere''. This either suggests another counter-example, or an entirely new type of analytic structure on the plane. 

• Wednesday, November 6th:

8-9 am (PST): Xiangdong Xie (Bowling Green University)

Rigidity  of pattern-preserving quasi-isometries  of the Heisenberg  groups

Quasiisometries of the Heisenberg groups are very flexible. However, if the quasiisometries preserve additional structures then they could be very rigid.  Let H be the first Heisenberg group and identify it with the three dimensional Euclidean space. With this identification the x-axis and y-axis are 1-dimensional subgroups.  If f is a self quasiisometry of H and there is a constant D such that the image of each coset of the x-axis or the y-axis  under f is at Hausdorff distance at most D from a coset of the x-axis or the y-axis, then f  is a finite distance from an affine map of the Heisenberg group. Similar statements also hold for quasiisometries of the discrete Heisenberg group  and the higher Heisenberg groups. Quasiisometries that preserve  such structures arise naturally when one studies the quasiisometries of  a graph of groups where the vertex groups are discrete Heisenberg groups and the edge groups are infinite cyclic.  As a consequence we obtain quasiisometric rigidity result of the fundamental groups of   such graphs of groups.  This is ongoing joint work with   Mitra Alizadeh.

9-10 am (PST):  Mathav Murugan (University of British Columbia)

Sobolev spaces and energy measures on the Sierpinski carpet

We describe the construction of (1,p)-Sobolev space, the corresponding p-energies and p-energy measures on the Sierpinski carpet.  An important motivation for the construction of energy measures is to determine whether or not the Ahlfors regular conformal dimension is attained on the Sierpinski carpet. If the Ahlfors regular conformal dimension is attained, we show that any optimal Ahlfors regular measure attaining the Ahlfors regular conformal dimension must necessarily be a bounded perturbation of the p-energy measure of some function in our Sobolev space, where p is the Ahlfors regular conformal dimension. This is joint work with Ryosuke Shimizu. 

• Wednesday, Dec 4th:

8 - 9 am (PST): Pascale Roesch (Université Paul Sabatier, Toulouse)

COR curves and connectedness loci 

This is joint work with Hiroyuki Inou  and Jan Kiwi. The main result is the presence of homeomorphic copies of the connectivity loci of one-parameter families when restricted to well-chosen curves of dimension 1, given by critical orbit relations. 

9-10 am (PST): Vasiliki Evdoridou (The Open University)

Approximation theory and wandering domains

The first use of Approximation Theory in Complex Dynamics was in 1987 when Eremenko and Lyubich in 1987 produced transcendental entire functions with different types of Fatou components. A variety of results in Approximation theory have been used since then mostly to construct examples of wandering domains, which are Fatou components that are not eventually periodic. We will discuss some of these results and present a new and more general way of using approximation theory to obtain transcendental functions with wandering continua. We will see how the fact that the resulting function is conjugate on the closure of a wandering domain to the model map can be used to produce examples with interesting dynamics. This is joint work with D. Marti-Pete and L. Rempe.

• Wednesday, February 7th:

8-9 am (PST):  Adi Glücksam (Northwestern University) 

Multi-fractal spectrum of planar harmonic measure

In this talk, I will define various notions of the multi-fractal spectrum of harmonic measures and discuss finer features of the relationship between them and properties of the corresponding conformal maps. Furthermore, I will describe the role of multifractal formalism and dynamics in the universal counterparts. This talk is based on a joint work with I. Binder.

9-10 am (PST):  Tom Kojar (Univeristy of Toronto) 

Gluing independent disks with the Inverse of Gaussian Multiplicative chaos

We will do an overview of the properties of the Inverse of one-dimensional Gaussian Multiplicative chaos and how they were put together to obtain the existence of a random Jordan loop from gluing two disks with lengths given by two independent 1d-GMCs. This is en route of providing a parallel perspective to the 2016-work of S.Sheffield "Conformal weldings of random surfaces" using the technology of degenerate Beltrami equation and Lehto welding.

• Wednesday, March 6th:

8-9 am (PST): Mario Bonk (University of California, Los Angeles) 

Elliptic integrals, modular forms, and the Weierstrass zeta-function

A standard topic in a somewhat more advanced graduate course in complex analysis are elliptic functions. These are doubly-periodic meromorphic functions in the complex plane. According to Liouville's basic theorems, each elliptic function has to have poles and if there are only poles at the points of the period lattice, then they cannot be of first order. Now in his systematic theory of elliptic functions, Weierstrass introduced his zeta function as a meromorphic function with only first order poles at the points of a given rank-2 lattice. So this zeta function cannot be doubly-periodic according to Liouville. But can it be periodic?  I will answer this question and show how this relates to many classical themes such as elliptic integrals, the hypergeometric ODE,  Schwarz triangle functions, modular forms, etc. The talk will provide entertainment for a broad mathematical audience.

9-10 am (PST): Zhiqiang Li (Peking University)

Ergodic optimization and visual metrics

In this talk, we discuss some recent progresses in an area in ergodic theory called ergodic optimization. The focus of ergodic op- timization is on the properties of invariant measures that maximizes the integral of potential functions, which has close connections to zero temperature behaviors of dynamical systems, weak KAM theory, and maximal mean-cycle problems on random graphs. We will focus on the study of ergodic optimization in a non-uniformly expanding setting in complex dynamics and show how visual metrics introduced by Bonk– Meyer and Ha ̈ıssinsky–Pilgrim from geometric group theory are crucially used here.

• Wednesday, April 3rd:

8 - 10 am (PST): Moon Duchin (Tufts, Boston)
                            Jamie Tucker-Foltz (Harvard University)

Random Graph Partitions Via Random Spanning Trees   (combined talk of both speakers) 

Finding balanced graph partitions -- dividing a graph into a fixed number of parts under an equal-weight constraint -- is a problem with many applications, including in political redistricting. A popular family of algorithms for this task uses spanning trees: If a tree has an edge whose complementary components have equal weight, its removal gives a balanced bipartition. In Part 1 of this talk we will consider UST, the uniform distribution on the spanning trees of a graph, and present new lower bounds on the likelihood of splittable trees on grids and grid-like graphs. This establishes the first provably polynomial-time algorithms for sampling balanced tree-weighted partitions on a nontrivial class of graphs. Part 2 of this talk will shift attention to MST, the distribution on trees induced by minimizing random edge weights, as in Kruskal's algorithm. MST is ubiquitous in applications because it is fast and flexible, but its mathematical properties are far less understood than UST. Taken together, the parts of this talk will give us insight into properties of tree-based partitions.

• Wednesday, May 8th:

8 - 9 am (PST): Jani Onninen (Syracuse Universitry)

Quasiregular values

Quasiregular maps form a higher-dimensional class of maps with many similar properties to holomorphic maps, such as continuity, openness, discreteness, and versions of the Liouville and Picard theorems. In this talk, we give a pointwise definition of quasiregularity. We show that this condition yields counterparts to many fundamental properties of quasiregular maps at a single point. The studied maps have already shown to play a key part in various important 2D results. Joint work with Ilmari Kangasniemi.

9 - 10 am (PST):  Daniel Faraco (Universidad Autónoma de Madrid)

Geometric Function Theory, Burkholder functionals, and lower semicontinuity

The central question in the vectorial Calculus of Variations is to characterize integral functionals that are lower semicontinuous with respect to the weak topology of the appropriate Sobolev Space. In 1952, Morrey answered the question by introducing the notion of quasiconvexity. However, Morrey's result had two drawbacks. Firstly, quasiconvexity is very difficult to verify from its definition but easily implies a local notion of convexity along rank-one lines, i.e., rank-one convexity. It is still open whether rank-one convexity is equivalent to quasiconvexity for planar deformations. The second drawback is that his theorem required growth conditions typically incompatible with real models of strain energies in hyperelasticity. Hyperelasticity requires that the energy blows up as the determinant of the deformation tends to 0.

Researchers in Geometric Function Theory became interested in this theory partially because if a particular functional, the Burkholder functional, which is rank-one convex, were quasiconvex, it would yield, as a corollary, a proof of T. Iwaniec's conjecture on the exact norm of the Ahlfors-Beurling transform between Lebesgue spaces.

In this talk, I will describe how ideas from geometric function theory prove the quasiconvexity of the Burkholder functionals for the corresponding quasiconformal deformations. Moreover, when the exponent tends to 2, this yields the quasiconvexity of certain rank-one convex functionals consistent with the requirements of hyperelasticity. In this setting, quasiconvexity does not imply lower semicontinuity, but we introduce a new notion, "principal quasiconvexity," emanating from Stoilow factorization, which yields lower semicontinuity for functionals in the context of hyperelasticity. If time permits, I will also show how principal quasiconvexity mixes well with blow-up techniques and yields lower semicontinuity in borderline scenarios.

This is a program developed with K. Astala (U. Helsinki), A. Guerra (Eth), A. Koski (U. Aalto), and J. Kristensen (Oxford).

• Monday, June 3rd:

8 - 9 am (PST): Han Peters (University of Amsterdam) 

Title: Equilibrium measures for transcendental dynamics.

In ongoing work with Leandro Arosio, Anna Miriam Benini and John Erik Fornaess, we study the entropy of transcendental maps, both in one and two variables. Following a suggestion of Nessim Sibony, we aim to prove that the entropy of transcendental maps is infinite. In previous work we treated topological entropy. In current work we aim to construct analogues of the unique measure of maximal entropy for rational maps.

For rational maps, the unique measure of maximal entropy can be constructed in a number of different ways: via equidistribution of preimages or periodic cycles, by taking the Laplacian of the Green's function for polynomials, and for particularly nice maps, by using symbolic dynamical systems. None of these methods easily generalize to arbitrary transcendental maps. In this talk I will discuss different one-dimensional transcendental functions for which either symbolic dynamics or equidistribution methods lead to ergodic measures of infinite entropy. For these examples the support of the measure equals the entire Julia set. 

9 - 10 am (PST):  Núria Fagella (Universitat de Barcelona)

Title: Quasiconformal surgery on wandering domains.

In this talk, we present a surgery construction that replaces the interior dynamics in an orbit of wandering domains with the non-autonomous dynamics of a sequence of Blaschke products, as long as both are uniformly hyperbolic. The surgery is performed in infinitely many domains at once, despite mapping to each other with a degree larger than one. As an application, we construct an entire function with a wandering domain for which discrete and indiscrete grand orbit relations coexist, in a way that is not possible for periodic Fatou components. Understanding grand orbit relations in the different types of Fatou components is a key step in the study of quasiconformal deformations of holomorphic maps.

This is joint work with Vasiliki Evdoridou, Lukas Geyer and Leticia Pardo-Simon

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