# Quasiworld

Welcome to the Quasiworld!

This is a virtual international seminar organized by Mario Bonk, Sylvester Eriksson-Bique, Mikhail Hlushchanka, and Annina Iseli. The seminar focuses on quasiconformal geometry, complex dynamics, and analysis and geometry on metric spaces. We construe these broadly and invite anyone interested to participate, and to volunteer talks.

The seminar started in April 2020 as a weekly seminar with the purpose of connecting the community during the first lockdown and was continued since. We took a break from these weekly meetings during Spring semester 2022 when the organizers and many of the attendees spent a semester at MSRI Berkeley. The Quasiworld seminar has continued on a monthly basis since Fall 2023 generally on the first Wednesday of each month. (exceptions below in red).

Dates for the Spring Semester 2024:

- Wednesday, February 7th, 8-10am Pacific time (5-7pm Central European time)

- Wednesday, March 6, 8-10am Pacific time (5-7pm Central European time)

- Wednesday, April 3, 8-10am Pacific time (5-7pm Central European time)

- Wednesday, May 8th, 8-10am Pacific time (5-7pm Central European time)

- Monday, June 3, 8-10am Pacific time (5-7pm Central European time)

Follow us on Youtube: Quasiworld Channel, @Quasiworld5564

## Titles and abstracts for Spring 2024

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)

## Titles and abstracts for Fall 2023

Wednesday, October 4th:

8-9 am: Piotr Hajłasz (University of Pittsburgh)

Title: Hölder continuous mappings, differential forms and the Heisenberg groups

I will show a new proof of Gromov's theorem about non-existence of Holder embeddings into the Heisenberg groups. The talk will be based on my joint work with A. Schikorra.

9-10 am: Chris Gartland (UCSD)

Title: Stochastic Embeddings of Hyperbolic Metric Spaces

Abstract: This talk is based on ongoing work of the speaker. We will discuss the stochastic embeddability of snowflakes of finite Nagata dimensional spaces into ultrametric spaces and the induced stochastic embeddings of their hyperbolic fillings into trees. Several results follow as applications, for example: (1) For any uniformly concave gauge $\omega$, the Wasserstein 1-metric over $([0,1]^n,\omega(\|\cdot\|))$ biLipschitzly embeds into $\ell^1$. (2) The Wasserstein 1-metric over any finitely generated Gromov hyperbolic group biLipschitzly embeds into $\ell^1$.

Wednesday, November 1st:

8-9 am: Dan Margalit (Vanderbilt University)

Title: A Tale of Two Theorems of Thurston

In the 20th century, Thurston proved two classification theorems, one for surface homeomorphisms and one for branched covers of surfaces. While the theorems have long been understood to be analogous, we will present new work with Belk and Winarski showing that the two theorems are in fact special cases of one Ubertheorem. We will also discuss joint work with Belk, Lanier, Strenner, Taylor, Winarski, and Yurttas on further algorithmic and theoretical aspects of Thurston’s theorems.

9-10am: Willie Rush Lim (Stony Brook University)

Title: From Herman Rings to Herman Curves

A maximal invariant domain of a rational map is called a Herman ring if it is an annulus on which the map is conjugate to irrational rotation. By adapting the near-degenerate machinery designed by Kahn, Lyubich, and D. Dudko, we show that Herman rings of bounded type rotation number of the simplest configuration satisfy a priori bounds, that is, the boundaries are quasicircles with dilatation independent of their conformal moduli. As a major application, we study the limits of degenerating Herman rings and construct the first general examples of rational maps admitting a "Herman curve" (a rotation curve that is not contained in the closure of any rotation domain) of arbitrary degree and combinatorics.

Wednesday, December 6th:

8-9 am: Malavika Mukundan (University of Michigan)

Title: Constructing dynamical approximations for entire functions

Postsingularly finite holomorphic functions are entire functions for which the forward orbit of the set of critical and asymptotic values is finite. Motivated by previous work on approximating entire functions dynamically by polynomials, we ask the following question:

Given a postsingularly finite entire function f, can f be realised as the locally uniform limit of a sequence of postcritically finite polynomials?

In joint work with Nikolai Prochorov and Bernhard Reinke, we answer this question in the affirmative.

9-10 am: Pjotr Buys (University of Amsterdam)

Title: Using complex dynamics to study graph partition functions

In this talk we consider the independence polynomial of graphs, in physics also known as the hard-core model. In the 1950s it was shown that if there is a zero-free region in the complex plane surrounding a real parameter for graphs in a certain graph class that the graph class does not undergo a phase transition at that parameter. In the last decade it was shown that such a zero-free region also implies that approximating the partition function is computationally "easy" at that parameter. This talk will highlight an inverse result, namely that the presence of zeros implies that approximating the partition function is computationally hard (#P-hard). The key of this connection is relating both the presence of zeros and hardness of approximation to the dynamic behavior of a related set of rational functions.

## Quasiworld Workshop 2023 in Helsinki

There was an in-person conference called the Quasiworld Workshop in August 2023 at the University of Helsinki.

This conference was an occasion to celebrate Mario Bonk's 60th birthdays and his many impactful contributions to the mathematics and the math community.

Please see the official website for more information: Quasiworld Workshop 2023

## Seminar Logistics and Email List

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

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

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

The easiest way you can join our email list is by sending a blank email to

quasiworld-seminar+subscribe@googlegroups.com.

If you wish to view past emails, and have a google account, you can access these at https://groups.google.com/d/forum/quasiworld-seminar (requires joining first with a google account).

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

Picture by M.C. Escher