### Next seminar

Speaker: **TBA**

Title: **TBA**

Time: TBA

Place: Room MATH-210, hybrid format (via Zoom)

Abstract: TBA

### Past seminars

Speaker: **Nikolai Prochorov** (Aix-Marseille Université)

Title: **Towards Transcendental Thurston Theory - 2**

Time: Thursday 2nd May 2024, 13:00-14:00

Place: Room MATH-210, hybrid format (via Zoom)

Abstract: This is a continuation of my talk given two weeks ago. The main goal of this part is to explain how one can prove that for certain families of transcendental Thurston maps, we have a ''Levy cycle criteria": Thurston maps f is realized if and only if it has no Levy cycles. Such criteria would provide a satisfactory answer to the characterization problem, i.e., the problem of determining whether a given Thurston map is realized by an entire postsingularly finite map.

To begin, I will revisit essential concepts introduced earlier, including Thurston maps, combinatorial equivalence, Levy cycles, and the characterization problem in the setting of transcendental Thurston maps.

Further, I will explain how to every Thurston map, one can associate an operator σ_f (called Thurston sigma map) acting on a suitable chosen Teichmuller space. I will show how the dynamic properties of σ_f are linked with those of the Thurston map f. In particular, I will establish that σ_f possesses a fixed point if and only if f can be realized by an entire postsingularly finite map.

Further, I will show that a Levy cycle is an obstruction for a Thurston map to be realized. I will outline techniques for identifying Levy cycles for f through the iteration of the Thurston sigma map σ_f. Finally, I will showcase the application of these methods in the context of exponential-like Thurston maps (treated in the work of Hubbard-Schleicher-Shishikura), and if time permits, I will extend the discussion to the uncountable family of Thurston maps introduced in the previous talk.

Speaker: **Matthieu Astorg** (University of Orléans)

Title: **Horn maps of semi-parabolic Hénon maps**

Time: Thursday 25th April 2024, 13:00-14:00

Place: Room MATH-210, hybrid format (via Zoom)

Abstract: Bedford, Smillie and Ueda have introduced a notion of horn maps for polynomial diffeomorphisms of C2 with a semi parabolic fixed point, generalizing classical results from parabolic implosion in one complex variable. We prove that these horn maps satisfy a weak version of the Ahlfors island property. As a consequence, we obtain the density of repelling cycles in their Julia set, and we prove the existence of perturbations of the initial Hénon map for which the forward Julia set J^+ has Hausdorff dimension arbitrarily close to 4. Joint work with Fabrizio Bianchi.

Speaker: **Nikolai Prochorov** (Aix-Marseille Université)

Title: **Towards Transcendental Thurston Theory**

Time: Thursday 18th April 2024, 13:00-14:00

Place: Room MATH-210, hybrid format (via Zoom)

Abstract: 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 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. Further, I am going to explain the main techniques to attack the characterization problem. Finally, I report about new classes of transcendental maps, where we managed to obtain an answer to this question.

Speaker: **Leticia Pardo-Simón **(University of Manchester)

Title: **Grand orbit relations in wandering domains**

Time: Thursday 23rd November 2023, 13:00-14:00

Place: Room MATH-210, hybrid format (via Zoom)

Abstract: The Teichmüller space of quasiconformal deformations of an entire map f can be decomposed into smaller subspaces. Such partition depends, in particular, on whether for each point z in its Fatou set, its grand orbit- that is, the set of w such that f^n(z) = f^m(w) for some naturals n, m- is a discrete or indiscrete subset of the plane. We provide criteria to conclude when orbits of points in wandering domains are discrete, and show that, unlike for periodic Fatou components, points with discrete and indiscrete orbits can coexist in a wandering domain. This is based on joint work with V. Evdoridou, N. Fagella and L. Geyer.

Speaker: **Rod Halburd **(University College London)

Title: **Singularity and integrability: beyond the Painlevé property**

Time: Thursday 16th November 2023, 13:00-14:00

Place: Room MATH-210, hybrid format (via Zoom)

Abstract: I will discuss various ways in which the singularity structure of solutions of differential equations in the complex domain can be used to detect integrable (in some sense, exactly solvable) equations. The Painlevé property and associated tests have led to the identification of integrable cases of many families of equations. In this talk I will describe some generalisations of the Painlevé property that allow for the identification of a wider set of integrable equations. Various necessary conditions will be discussed and implemented. Examples from Newtonian and relativistic stellar models will be analysed. I will also show how to find special solutions of some generally non-integrable equations, using methods from Nevanlinna theory.

Speaker: **Elefterios Soultanis **(University of Warwick)

Title: **Curve fragments, plan-modulus duality, and differentiability of Lipschitz functions**

Time: Thursday 2nd November 2023, 13:00-14:00

Place: Room MATH-210, hybrid format (via Zoom)

Abstract: Rademacher's Theorem states that a Lipschitz function on Euclidean space is differentiable a.e. with respect to the Lebesgue measure. This statement is no longer true if we replace the Lebesgue measure with a singular measure. I will discuss a weaker form of differentiability, which remains true for singular measures, phrased in terms of curve fragments, i.e. (bi)-Lipschitz images of compact subsets of R. key ingredients in such weak differentiability results are decompositions of measures into curve fragments, and their duality with modulus.

Speaker: **Mikhail Hlushchanka** (University of Amsterdam)

Title: **Rational maps, Julia sets, and iterated monodromy groups: complexity and decomposition**

Time: Thursday 14th September 2023, 13:00-14:00

Place: Room MATH-210, hybrid format (via Zoom)

Abstract: There are various classical and more recent decomposition results in mapping class group theory, geometric group theory, and complex dynamics (which include celebrated results by Bill Thurston). The goal of the talk is to introduce a novel decomposition of rational maps based on the topological structure of their Julia sets (obtained jointly with Dima Dudko and Dierk Schleicher). Namely, we will discuss the following result: every postcritically-finite rational map with non-empty Fatou set can be canonically decomposed into crochet maps (these have very "thinly connected" Julia sets") and Sierpiński carpet maps (these have very "heavily connected" Julia sets). At the end of the talk, we will briefly consider connections of this decomposition to geometric group theory and self-similar groups.

### Seminars from previous years

- Year 2022-23
- Year 2021-22
- Year 2020-21
- Year 2019-20
- Year 2018-19
- Year 2017-18
- Year 2016-17
- Year 2015-16

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