Date/Time : 03 May 2023 15:00
Venue : S17 #04-04 (SR3)
Speaker : Marco Abate, University of Pisa

In this talk we shall describe the asymptotic behaviour of sequences of functions obtained by iteratively left-composing or right-composing holomorphic self-maps of a hyperbolic Riemann surface $X$. We shall consider in detail two cases: when the holomorphic self-maps to be composed have values in a Bloch subdomain of~$X$; and when the holomorphic self-maps to be composed are sufficiently close to a given self-map. This is a joint work with Argyrios Christodoulou and Ian Short.

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Date/Time : 19 Apr 2023 15:00
Venue : S17 #04-04 (SR3)
Speaker : Lucas Kaufmann, University of Orleans

Consider a (singular) holomorphic foliation F on a complex manifold. In the 1970s, Baum-Bott constructed residue classes associated with each singular component of F. These classes satisfy an index theorem, which computes characteristic classes of F.

A natural question since their work is how to compute or find natural representatives of such cohomology classes. Apart from the case where the foliation is of rank one and has isolated singularities, no general result is known. In this talk I will show that each Baum-Bott class can be naturally represented by a residue current supported by the singular component of F. This is a joint work with Richard Lärkäng and Elizabeth Wulcan.

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Date/Time : 08 Feb 2023 15:30
Venue : S17 #04-04 (SR3)
Speaker : Fabrizio Bianchi, University of Lille

We show that every Hölder observable satisfies the Central Limit Theorem (CLT) with respect to the measure of maximal entropy of every complex Hénon map. The proof is based on a general criterion to ensure the validity of the CLT, valid for abstract measurable dynamical systems. This is a joint work with Tien-Cuong Dinh.

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Date/Time : 25 Jan 2023 15:30
Venue : S17 #04-05 (SR2)
Speaker : Wang Tianqi, National University of Singapore

Minsky shows that there is an open set of PSL(2,C) characters of a free group F_n, namely the primitive-stable character, which is strictly larger than the convex-cocompact character and on which the action of Out(F_n) is properly discontinuous. In this talk, we introduce the notion of simple Anosov representations of closed surface groups and give an analogue of Minsky’s theorem, that is, the set of simple Anosov representations of the fundamental group of a closed hyperbolic surface into a higher rank Lie group is open in the character variety, which is strictly larger than the set of Anosov representations, with the mapping class group action properly discontinuous. This is joint work with Nicholas Tholozan.

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Date/Time : 18 Jan 2023 15:30
Venue : S17 #04-05 (SR2)
Speaker : Yap Jit Wu, Harvard University

Let f,g be two rational maps on P^1. Then Baker-Demarco, and independently Yuan-Zhang, showed that either they have exactly the same set of preperiodic points, or only finitely many in common. More recently, Demarco-Krieger-Ye showed the existence of an uniform bound on number of common preperiodic points for the quadratic family z^2 + c and also the Lattes family. In this talk, we will discuss the ideas behind these two results along with some work in progress by the author on related questions.

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Date/Time : 11 Jan 2023 15:30
Venue : S17 #04-04 (SR3)
Speaker : Omar Kidwai, University of Birmingham

Starting from the data of a quadratic (or higher) differential, Gaiotto-Moore-Neitzke’s famous study of four-dimensional N=2 supersymmetric QFTs outputs data known as a BPS structure (the same structure describes the output of Donaldson-Thomas theory). To solve a totally different problem in the theory of matrix models, Chekhov and Eynard-Orantin introduced the topological recursion, which takes initial data called a spectral curve and recursively produces (through residue calculus and combinatorics) an infinite tower of geometric invariants, often with an enumerative interpretation.

We will describe recent work connecting these two different theories. In the simplest cases, when the spectral curve is of degree two and genus zero, we describe a simple explicit formula for the “free energies” of the topological recursion purely in terms of a corresponding BPS structure constructed from the same initial data. Time permitting, we will sketch the relation to the WKB analysis of quantum curves, and suggest a wide range of generalizations. Based on joint work with K. Iwaki and K. Osuga.

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Date/Time : 04 Jan 2023 15:30
Venue : S17 #04-04 (SR3)
Speaker : Asbjorn Nordentoft, Université Sorbonne Paris Nord

Duke, Imamoḡlu, and Tóth have recently constructed a canonical hyperbolic orbifold whose boundary is equal to a given closed geodesic on the modular surface. Furthermore, they have shown that the projection of these hyperbolic orbifolds onto the modular surface equidistributes (as the discriminant of the closed geodesic tends to infinity). We will describe an extension to the case of modular surfaces of non-trivial genus relying on Morse coding with respect to special fundamental polygons as introduced by Kulkarni. We also obtain an equidistribution result which requires a detailed study of the homology of the modular curves. This is joint work with Peter Humphries.

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Date/Time : 17 Nov 2022 15:30
Venue : S17 #04-04 (SR3)
Speaker : Renaud Leplaideur, University of New Caledonia

This is a joined work with A. Baraviera from UFRGS (Brazil). We introduce a notion of substitution on the colored binary trees.

In the first step I will present the motivtions for this work. Strangely (perhaps), they come from questions in ergodic theory and thermodynamic formalism.

In the second step I will present how we define these substreetutions, and then present one example.

The main result is that the associated “attractor” is minimal and non-periodic. I will also present many open questions about these objects.

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Date/Time : 10 Nov 2022 15:30
Venue : S17 #05-11 (SR5)
Speaker : Fabrizio Bianchi, University of Lille

Abstract:  We discuss the stability of holomorphic dynamical systems under perturbation. In dimension 1, the theory is now classical and is based on works by Lyubich, Mané-Sad-Sullivan and DeMarco. I will review this theory and present a recent generalisation valid in any dimension. Since classical 1-dimensional techniques no longer apply in higher dimensions, our approach is based on ergodic and pluripotential methods.

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