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
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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
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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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Date/Time : 25 Jan 2022 16:00
Venue : via Zoom
Speaker : Feng Zhu, Technion - Israel Institute of Technology

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Meeting ID: 875 1147 6637
Passcode: 468399

Abstract: Putting a hyperbolic metric on a closed surface gives us a linear representation (the holonomy representation) with many nice geometric and dynamical properties. Anosov representations generalise this picture, or more generally the theory of convex cocompact subgroups of rank-one Lie groups such as PSL(2,R) or PSL(2,C), to higher-rank Lie groups such as PSL(d,K) for d>2. They give us an open subset of the space of representations consisting of quasi-isometric embeddings of word-hyperbolic groups.

There are many equivalent characterisations of Anosov representations; in particular, Labourie’s original definition uses a flow coming from the hyperbolic group to obtain additional control over the image subgroup. In this talk, I will introduce a relative notion of Anosov representations, similarly formulated in terms of flows, which allows for (certain) parabolic elements / controlled breakdowns of hyperbolicity.

This generalises work of Canary–Zhang–Zimmer and is joint work in progress with Andrew Zimmer.

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Date/Time : 18 Jan 2022 10:00
Venue : via Zoom
Speaker : Ara Basmajian, City University of New York

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Meeting ID: 875 1147 6637
Passcode: 468399

Abstract:  There are various contexts for which it is natural to study spaces that look the same from any point of the space (that is, “homogeneous spaces”). For example, the simply connected Riemann surfaces, namely the Riemann sphere, complex plane, and unit disc are conformally homogeneous Riemann surfaces. In fact they are the only ones. On the other hand, given any surface it is not difficult to cook up a diffeomorphism between any two points of the surface. Hence one needs a notion that is not as strong as conformality and not as weak as differentiability. The key observation is that while smooth maps can distort infinitesimal circles to ellipses with unbounded eccentricity (the ratio of the major to the minor axis can be arbitrarily large), conformal maps do not distort infinitesimal circles at all. This leads to the notion of a homeomorphism being K-quasiconformal (has eccentricity bounded by K). Conformal homeomorphisms are 1-quasiconformal.

A Riemann Surface X is said to be K-quasiconformally homogeneous if for any two points x and y on it, there exists a K-quasiconformal self-mapping taking x to y. If such a K exists we say that X is a QCH Riemann surface. After introducing the basics, the focus of this talk will be on Riemann surface structures that are QCH, and their connections to the topology and hyperbolic geometry of the surface.

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