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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Date/Time : 27 Jul 2022 14:00
Venue : S17 #04-04 (SR3)
Speaker : Paolo Cascini, Imperial college

Abstract:  I will survey some recent results on the study of the birational geometry of foliations over complex projective varieties, focusing, in particular, on the case of algebraically integrable foliations.

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Date/Time : 04 Jul 2022 16:00
Venue : S17 #06-11 (SR6)
Speaker : Sang-hyun Kim, Korea Institute of Advanced Study

Abstract:  We prove that for each finite index subgroup H of the mapping class group of a closed hyperbolic surface, and for each real number r>1 there does not exist a faithful C^r-action (in Hölder’s sense) of H on a circle. For this, we partially determine the optimal regularity of faithful actions by right-angled Artin groups on a circle. (Joint with Thomas Koberda and Cristobal Rivas).

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Date/Time : 19 Apr 2022 16:00
Venue : via Zoom
Speaker : Nicholas Tholozan, École Normale Supérieure

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

Meeting ID: 875 1147 6637
Passcode: 468399

Abstract: Let G/H be a reductive symmetric space. In 2016, Kobayashi and Kassel conjectured that the discrete subgroups of G acting properly discontinuously and cocompactly on G/H are « sharply » embedded in G. Here, I will present a proof of this conjecture, obtained jointly with Fanny Kassel, and discuss its many consequences (spectral theory of these compact quotients, stability under deformation and, in some cases, non-existence of such quotients).

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Date/Time : 12 Apr 2022 10:00
Venue : via Zoom
Speaker : Minju Lee, Institute of Advanced Study

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

Meeting ID: 875 1147 6637
Passcode: 468399

Abstract : This is joint work with Samuel Edwards and Hee Oh. Let $G$ be a connected semisimple real algebraic group, and $\Gamma<G$ be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We describe the asymptotic behavior of matrix coefficients $\langle (\exp tv). f_1, f_2\rangle$ in $L^2(\Gamma\backslash G)$ as $t\to \infty$ for any $f_1, f_2\in C_c(\Gamma\backslash G)$ and any vector $v$ in the interior of the limit cone of $\Gamma$. These asymptotics involve higher rank analogues of Burger-Roblin measures which will be introduced in this talk. As an application, for an affine symmetric subgroup $H$ of $G$, we obtain a bisector counting result for $\Gamma$-orbits with respect to the corresponding generalized Cartan decomposition of $G$. Moreover, we obtain analogues of the results of Duke-Rudnick-Sarnak and Eskin-McMullen for counting discrete $\Gamma$-orbits in affine symmetric spaces $H\backslash G$.

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Date/Time : 05 Apr 2022 16:00
Venue : S17 #04-04 (SR3)
Speaker : John Loftin, Rutgers University

Abstract: Consider a Riemann surface S of genus g at least 2 equipped with a holomorphic cubic differential U. For a given g, such pairs (S,U) can be used to parametrize the Hitchin component of the space of representations of the fundamental group into PSL(3,R), and also the space of convex projective structures on S. This generalizes and extends the space Fuchsian representations into PSL(2,R) (the Teichmuller space of hyperbolic structures), which corresponds to the locus U=0. This parametrization of the Hitchin component is not explicit but involves the Hitchin system of PDEs. For nonzero U, we study the case sU as s approaches infinity. In particular, we show the geometry in this limit can be read off explicitly from U, in terms of an embedding of the universal cover of S into the real building given by the asymptotic cone of the symmetric space SL(3,R)/SO(3). We are able to provide explicit pictures for most triangle groups.

This is joint work with Andrea Tamburelli and Mike Wolf.

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