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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Date/Time : 27 Oct 2022 15:30
Venue : S17-05-11 (SR 5)
Speaker : Takashi Kishimoto, Saitama University

Abstract: We address the following questions: ”(1) How to construct a Mori fiber space (Mfs) X containing the affine space Ank as an open subset ?” and ”(2) In which Mfs as in (1), the translations on Ank can be extended to automorphisms on X, namely X can be an equivariant completion of the vector group Gna,k ?” As for (1), we will present a certain machinery to construct Mfs over P1k, which contains the affine space, with birationally rigid Fano varieties being general fibers. Note however that such completions X are never equivariant ones, more precisely, the unipotent radical of Aut0(X) is at most of dimension one. Actually, it turns out that there are few equivariant completions. Then we will focus on forms defined over any field of characteristic zero of quintic del Pezzo varieties. The main result lies in clarifying which ones among them can be equivariant completions and determining their automorphism groups. The talk is based on two joint works, one of which is with Adrien Dubouloz and Karol Palka, and the other of which is with Adrien Dubouloz and Pedro Montero.

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Date/Time : 18 Oct 2022 15:30
Venue : S17 #06-11 (SR6)
Speaker : Richard Canary, University of Michigan

Abstract: The Hitchin component of representations of a closed surface group into SL(d,R) is one of the primary examples of a Higher Teichmuller space (a component of the space of representations of a surface group into a Lie group which consists entirely of discrete, faithful representations). We discuss a theory of cusped Hitchin representations of geometrically finite Fuchsian groups into SL(d,R). These cusped Hitchin representations arise naturally by “pinching” Hitchin representations. We show that cusped Hitchin representations are cusped Borel Anosov and establish counting and equidistribution results.

The long term goal of this project is to develop a metric theory of the augmented Hitchin component which generalizes the fact that augmented Teichmuller space is the metric completion of Teichmuller space with the Weil-Petersson metric. (This is joint work with Tengren Zhang and Andy Zimmer, and with Harry Bray, Nyima Kao and Giuseppe Martone).

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