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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Date/Time : 13 Oct 2022 15:30
Venue : S17 #05-11 (SR5)
Speaker : Chen Junkai Alfred, National Taiwan University

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https://nus-sg.zoom.us/j/87511476637?pwd=VURaUDVzaHpMYmduN2Fod0h0Y29sQT09

Meeting ID: 875 1147 6637
Passcode: 468399

Abstract: In minimal model program, divisorial contractions, flip and flops are considered to be the elementary maps. Divisorial contractions to points are well-understood and can be realized as weighted under certain embeddings. In this talk, we will present the recent developments toward the classification of divisorial contractions to curves. We will provide an outline of our proposed program and demonstrate some new examples. If time permits, we will talk about their factorizations as well. The talk is based on the joint work in progress with Hsin-Ku Chen and Jheng-Jie Chen.

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Date/Time : 01 Sep 2022 15:00
Venue : S17 #04-04 (SR3)
Speaker : Nhat Minh Doan, Vietnam Academy of Science and Technology

Abstract:  We investigate the terms arising in an identity for hyperbolic surfaces proved by Luo and Tan, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity properties. Using these properties, we deduce two results. As a first application, we show how to deduce a theorem of Thurston which states, in particular for closed hyperbolic surfaces, that if a simple length spectrum “dominates” another, then in fact the two surfaces are isometric. As a second application, we show how to find upper bounds on the number of pairs of pants of bounded length that only depend on the boundary length and the topology of the surface. This is joint work with Hugo Parlier and Ser Peow Tan.

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