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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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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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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Date/Time : 22 Mar 2022 16:00
Venue : via Zoom
Speaker : Takahiro Shibata, National University of Singapore

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Meeting ID: 875 1147 6637
Passcode: 468399

Abstract: In this talk, we deal with finite quotients of abelian varieties (fqav for short), that is, the quotients of abelian varieties by finite group actions. We mainly focus on the positivity of the canonical divisor of such a variety. Those varieties with Q-linearly trivial (resp. anti-ample) canonical divisors are called Q-abelian varieties (resp. Q-Fano fqav). We gives characterizations of those varieties and see that they are building blocks of general fqav’s.

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Date/Time : 15 Mar 2022 16:00
Venue : via Zoom
Speaker : Hugo Parlier, University of Luxembourg

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Meeting ID: 875 1147 6637
Passcode: 468399

Abstract: On closed surfaces of positive genus, through classical work of Dehn, simple closed curves can be described using intersection numbers. Now what if you want to describe curves with self-intersections in a similar way? This talk will be on joint work with Binbin Xu about this question, and where we end up constructing and studying so-called k-equivalent curves. These are distinct curves that intersect all curves with k self-intersections the same number of times.

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Date/Time : 08 Mar 2022 16:00
Venue : via Zoom
Speaker : Yang Pengyu, ETH Zurich

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Meeting ID: 875 1147 6637
Passcode: 468399

Abstract: We study translates of natural measures on curves/submanifolds under diagonal flows in homogeneous spaces, and describe geometric/arithmetic obstructions to equidistribution. Our basic strategy is to use Ratner’s unipotent rigidity theorem, and I will discuss the new ingredients from geometric invariant theory.

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Date/Time : 15 Feb 2022 10:00
Venue : via Zoom
Speaker : Min Ru, University of Houston

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Meeting ID: 875 1147 6637
Passcode: 468399

Abstract: In this talk, I’ll first discuss an Nevanlinna-Diophantine inequality obtained by Ru-Vojta. Then I’ll describe its applications in the study of the GCD problems (the work of Guo-Wang, and Wang-Yasufuku), as well as in the study of the divisibility, hyperbolicity and integral points problems (the work of Rousseau, Turchet and Wang).

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Date/Time : 08 Feb 2022 10:00
Venue : via Zoom
Speaker : Theodore Weisman, University of Texas, Austin

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Meeting ID: 875 1147 6637
Passcode: 468399

Abstract: Anosov representations are a higher-rank generalization of convex cocompact subgroups of rank-one Lie groups. They are only defined for word-hyperbolic groups, but recently Kapovich-Leeb and Zhu have suggested possible definitions for an Anosov representation of a relatively hyperbolic group – aiming to give a higher-rank generalization of geometrical finiteness.

In this talk, we will introduce a more general version of relative Anosov representation which also interacts well with the theory of convex projective structures. In particular, the definition includes projectively convex cocompact representations of relatively hyperbolic groups, and allows for deformations of cusped convex projective manifolds (including hyperbolic manifolds) in which the cusp groups change in nontrivial ways.

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