Quasiworld

• Thursday, February 5th:

8-9 am (PST):  Sergiy Merenkov  (The City College of New York)

 Quasiregular maps of carpet Julia sets

I plan to discuss an extension of a result by Bonk-Lyubich-Merenkov on quasisymmetries of carpet Julia sets to the quasiregular setting. Namely, if f and g are postcritically finite rational maps with Sierpinski carpet Julia sets J(f) and J(g), respectively, and q is a quasiregular map of the sphere such that q^{-1}(J(g))=J(f), then q|_{J(f)} has a unique extension to a rational map Q. Moreover, if f=g above and the degree of q is at least 2, then Q satisfies f^k Q^l=f^{2k} for some k and l, where the powers are iterative. Finally, we give examples, such as rational maps whose Julia sets are the basilica or a gasket, for which the above conclusions fail. This is joint work with Letian Shen. 

9-10 am (PST):  Janne Junnila (KTH)

On singularity lines of complex projective structures with real holonomy

I will discuss certain curve systems on the punctured Riemann sphere that are induced by complex projective structures with real holonomy. Such curves appear e.g. in semiclassical limits of the Schramm-Loewner evolution. A central question is whether every isotopy class of a curve system can be realized as the singularity lines of some projective structure, and whether it is unique. With Bonk, Rohde and Wang we answered this affirmatively in the special case when the projective structure is induced by a differentiable Jordan curve passing through the punctures and satisfying the following geodesic property: Every arc on the curve is a hyperbolic geodesic in the domain bounded by all the other arcs. These curves turn out to be Loewner energy minimizing in their isotopy classes, and we also proved that the accessory parameters of the associated Schwarzian derivative satisfy an identity analogous to the famous formula conjectured by Polyakov and proven by Takhtajan and Zograf in the Fuchsian case. In addition to explaining these results, I will also showcase a few examples of other such curve systems whose projective structures can be explicitly described thanks to certain symmetries and holonomical constraints.

• Thursday, March 5th:

8-9 am (PST):  Jamie Juul  (Colorado State University)

Arboreal Galois groups of postcritically finite quadratic polynomials

We provide an explicit construction of the arboreal Galois group for the postcritically finite polynomial f(z)=z^2+c, where c belongs to some arbitrary field of characteristic not equal to 2. This is joint work with Benedetto, Ghioca, and Tucker.

9-10 am (PST):  Dean Wardell  (Leiden University)

Profinite iterated monodromy groups of postcritically finite cubic polynomials 

Interest in profinite iterated monodromy groups is motivated by their connections to the arboreal representations of Galois groups of number fields. For such groups, we generally know the images of elements under the representation only up to conjugation, which naturally leads to the notion of invariable generation. In this talk, we will discuss the profinite geometric iterated monodromy groups associated with postcritically finite cubic polynomials defined over number fields. Under a mild dynamical condition, these groups turn out to be finitely invariably generated, and they are completely determined (up to conjugation) by the polynomial's ramification portrait. In contrast to other cases studied in the literature, our methods explicitly make use of an odometer. We will also highlight certain group-theoretic properties of such groups. This talk is based on joint work with Mikhail Hlushchanka and Olga Lukina.

• Thursday, April 2nd:

8-9 am (PST):  Pekka Pankka (University of Helsinki)

Uniformization of Thurston-type maps

In two dimensions it is a theorem of Bonk-Meyer and Haïssinky-Pilgrim that the visual sphere associated to an expanding Thurston map of the 2-sphere is quasisymmetric to the standard sphere if and only if the map is conjugate to a rational map.  Thurston maps are post-critically finite branched covers and the post-critical finiteness restricts the theory of (interesting) Thurston maps to two dimensions. Also, when considering higher dimensional analogs of the uniformization theorem, the rational maps need a replacement. In this talk, I will discuss replacements for Thurston maps and rational maps — and such a uniformization theorem — on higher dimensional Riemannian manifolds. This is joint work with Zhiqiang Li and Hanyun Zheng (Peking U.)

9-10 am (PST):  Julia Münch (University of Liverpool)

Extending rational expanding Thurston maps

In this talk I will present an extension result. It states that one can extend a certain class of holomorphic maps on the Riemann sphere f:S^2→ S^2 to a uniformly quasi-regular map  F: R^3→R^3. There are two main motivations, one comes from the theory of quasi-conformal mappings and one comes from generalising complex dynamics.

Quasi-conformality is not trivially preserved under products, but there is a classical result that one can extend a given quasi-conformal map f:R^n → R^n to a quasi-conformal map F: R^(n+1) → R^(n+1). Our setting fits in this context, but we drop the assumption that the map we start with is a homeomorphism. The second motivation is to generalise holomorphic dynamics to higher dimensions. Quasi-regular mappings on R^n are a natural generalisation of holomorphic maps in C. The dynamics is particularly nice if the same eccentricity bound on ellipses holds for all iterates of the map, i.e., if we restrict to uniformly quasi-regular mappings, but it is difficult to find interesting examples of such maps.

In a second part of the talk I will present an application of the extension towards an entry of Sullivan’s dictionary, i.e., illustrate similarities between the theory of Kleinian groups and rational dynamics.

This is joint work with Daniel Meyer, partially about work in progress.

• Thursday, May 7th:

8-9 am (PST):  Volodymyr Nekrashevych (Texas A&M University)

Conformal dimension and group theory

We will discuss different characterizations and properties of the Ahlfors regular conformal dimension of the limit space of a contracting self-similar group. In the case of iterated monodromy groups of rational functions, it is the classical conformal dimension of the Julia set. In particular, we will discuss its relation with the algebraic properties of the group and applications to amenability and random walks.

9-10 am (PST):  John Mackay (University of Bristol)

Connecting conformal dimension and Poincaré profiles

We'll discuss some connections between two different ways of measuring how well-connected an infinite group is.  On the one hand, Benjamini, Schramm and Timár quantified how well-connected an infinite graph is in terms of its "separation profile", where one considers the cut size of finite subgraphs.  There is an "L^p" version of this that uses Poincaré inequalities to measure the connectivity of finite subgraphs, which was studied in previous work with Hume and Tessera.  On the other hand, for Gromov hyperbolic groups, there is Pansu's conformal dimension of the boundary at infinity.  I'll discuss recent work with Hume where we further study the connection between these two notions.

• Thursday, June 4th:

8-9 am (PST):  James Belk (University of Glasgow)

Quasisymmetries of finitely ramified Julia sets

I will outline a theory of quasisymmetries for finitely ramified fractals, including finitely ramified Julia sets. Such fractals admit a natural class of “undistorted metrics” that are all quasi-equivalent, and it follows that piecewise-defined homeomorphisms that locally preserve the cell structure are quasisymmetries. Applying this to Julia sets of hyperbolic quadratics, we prove that every such Julia set has infinitely many quasisymmetries, generalizing a result of Lyubich and Merenkov.  We will also discuss extensions to cubic polynomials as well as certain rational maps.  This is joint work with Bradley Forrest.

9-10 am (PST):  Gunther Cornelissen (Utrecht University)

Limit configurations of zeros of modular forms

We study the limit configuration of the zero sets of modular forms, more specifically, Eisenstein series for arbitrary congruence groups, as the (even) weight tends to infinity. Adapting a method from statistical physics, we show that the limit is always a configuration of geodesic segments in the complex upper half plane. Using more precise complex analytic tools, we show that for principal congruence groups of odd level, the counting measure of the zeros converges weakly to the restriction of the Haar measure to a segment of the complex unit circle. Although the setup is different, the results show a striking resemblance with theorems about limits of zeros of graph polynomials, random real polynomials, etc. No familiarity with modular forms will be assumed. 

(joint work with Sebastian Carrillo and Berend Ringeling) 

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