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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Date/Time : 25 Aug 2022 15:00
Venue : S17 #04-04 (SR3)
Speaker : Timothée Bénard, University of Cambridge

Abstract:  A famous theorem due to Hopf-Tsuji-Sullivan-Kaimanovich asserts that the geodesic flow and the Brownian motion on a rank-one symmetric space of the non-compact type $\Lambda \backslash G/K$ are either both recurrent ergodic or both transient. In this talk, I will explain how this dichotomy can be extended to a much larger class of random walks, namely any right random walk on $\Lambda \backslash G$ induced by a Zariski-dense probability measure on $G$ with finite first moment.

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Date/Time : 18 Aug 2022 15:00
Venue : S17 #04-04 (SR3)
Speaker : Farhad Babaee, University of Bristol

Abstract: Tropical geometry is an effort to simplify algebraic subvarieties of the complex torus through the tropicalisation process to obtain tropical varieties. The upshot is that tropical varieties still retain a lot of information about the initial algebraic varieties. In this talk, I will explain basic notions in tropical geometry, and discuss how natural and useful it is to look at tropical geometry through the lens of complex geometry and holomorphic dynamics.

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Date/Time : 16 Aug 2022 15:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Tianqi, National University of Singapore

Abstract: The notion of Anosov representations of a finitely generated group into a semisimple Lie group, introduced by Labourie, generalizes the notion of convex-cocompact representations to higher rank Lie groups. On the other hand, there are many non-discrete, and hence non-Anosov, representations that have good geometric properties, such as the primitive-stable representations introduced by Minsky. In this talk, we introduce a weakening of the notion of Anosov representations by requiring the Anosov condition to hold only on a chosen invariant closed geodesic subflows. This weakening allows for such non-discrete representations to arise as examples. We discuss several properties of this type of representations that are analogous to the properties of classical Anosov representations, such as stability and the limit maps being Hölder continuous in a weaker sense

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Date/Time : 11 Aug 2022 15:00
Venue : via Zoom
Speaker : Alexander Stoimenow, KAIST

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

Meeting ID: 875 1147 6637
Passcode: 468399

Abstract: The braid groups $B_n$ were introduced in the 1930s in the work of Artin. An element in the braid group $B_n$ is called an $n$-braid. Alexander related braids to knots and links in 3-dimensional space, by means of a closure operation. In that realm, it became important to understand the braid representatives of a given link. Markov’s theorem relates these representatives by two moves, conjugacy in the braid group, and (de)stabilization, which passes between braid groups. Markov’s moves and braid group algebra have become fundamental in Jones’ pioneering work and its later continuation towards quantum invariants. Conjugacy is, starting with Garside’s, and later many others’ work, now relatively well group-theoretically understood. In contrast, the effect of (de)stabilization on conjugacy classes of braid representatives of a given link is in general difficult to control. Only in very special situations can these conjugacy classes be well described.

We are concerned with the question when infinitely many conjugacy classes of $n$-braid representatives of a given link occur. Birman and Menasco introduced a move called exchange move, and proved that it necessarily underlies the switch between many conjugacy classes of $n$-braid representatives of a given link.

After some very brief general introduction to knots and braids and their applications, we will discuss several results when the exchange move is also sufficient for generating infinitely many conjugacy classes of braid representatives.

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