Date/Time : 08 Apr 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Chris Leininger, Rice University

A billiard trajectory in a polygon in the Euclidean or hyperbolic plane is the path of a particle in the polygon, following a geodesic line until it hits the side, reflects off, and continues. This trajectory gives rise to a symbolic coding, its “bounce sequence”, which is (generically) a biinfinite sequence recording the sides encountered (which we assume are labeled). In joint work with Erlandsson, Duchin, and Sadanand, we prove that generically the “bounce spectrum” (i.e. the set of all bounce sequences of biinfinite trajectories), determines the shape of the polygon. The “non-generic” case is also understood, and we can completely determine when two polygons have the same bounce spectrum. In the talk, I’ll explain this theorem and describe how it is related to non-positively curved cone metrics on surfaces.

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Date/Time : 03 Apr 2024 15:30
Venue : S17 #05-12 (SR4)
Speaker : DO Thai Duong, National University of Singapore

We will present some properties of the higher complex Sobolev spaces. These spaces share several similar properties with the complex Sobolev space; in particular, we can build their corresponding Vigny’s functional capacity and prove the Moser-Trudinger inequality for these spaces. We also discuss the relation of these spaces with the complex Monge-Ampere equation. This talk is based on my joint work with Nguyen Duc-Bao.

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Date/Time : 27 Mar 2024 14:30
Venue : S17 #05-12 (SR4)
Speaker : Wu Hao, National University of Singapore

Let X be a compact Riemann surface and L be a positive line bundle on it. We study the conditional zero expectation of all the holomorphic sections of L^n which do not vanish on D for some fixed open subset D of X. We prove that as n tends to infinity, the zeros of these sections are equidistributed outside D with respect to some probability measure. This gives rise to a surprising forbidden set.

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Date/Time : 20 Mar 2024 15:30
Venue : S17 #05-12 (SR4)
Speaker : Fabrizio Bianchi, Università di Pisa

The theory of parabolic implosion – i.e., the study of perturbations of parabolic fixed points – has been at the centre of the research in one-dimensional holomorphic dynamics in the last several decades. I will discuss recent progress on the generalization of this theory in two complex variables, as well as some applications.

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Date/Time : 13 Mar 2024 15:30
Venue : S17 #05-12 (SR4)
Speaker : Yasushi Yamashita, Chuo University

The Riley slice of Schottky space is a family of Kleinian groups generated by two parabolic elements. It was studied in detail by Keen and Series. In this talk, we consider families of Kleinian groups generated by two elliptic elements, and show some computer generated pictures related to the groups.

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Date/Time : 06 Mar 2024 15:30
Venue : S17 #04-05 (SR2)
Speaker : Viet-Anh Nguyen, Universite de Lille

Let $X$ be a compact normal complex space of dimension $n$ and $L$ be a holomorphic line bundle on $X$. Suppose that $\Sigma=(\Sigma_1,\ldots,\Sigma_\ell)$ is an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$, $\tau=(\tau_1,\ldots,\tau_\ell)$ is an $\ell$-tuple of positive real numbers, and let $H^0_0(X,L^p)$ be the space of holomorphic sections of $L^p:=L^{\otimes p}$ that vanish to order at least $\tau_jp$ along $\Sigma_j$, $1\leq j\leq\ell$. If $Y\subset X$ is an irreducible analytic subset of dimension $m$, we consider the space $H^0_0 (X|Y, L^p)$ of holomorphic sections of $L^p|_Y$ that extend to global holomorphic sections in $H^0_0(X,L^p)$. Assuming that the triplet $(L,\Sigma,\tau)$ is big in the sense that $\dim H^0_0(X,L^p)\sim p^n$, we give a general condition on $Y$ to ensure that $\dim H^0_0(X|Y,L^p)\sim p^m$. When $L$ is endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces $H^0_0(X|Y,L^p)$ converge to a certain equilibrium current on $Y$. We apply this to the study of the equidistribution of zeros in $Y$ of random holomorphic sections in $H^0_0(X|Y,L^p)$ as $p\to\infty$.

This is a joint-work with Dan Coman and George Marinescu.

Keywords:
(Partial) Bergman kernel function, big line bundle, singular Hermitian metric, holomorphic section, big cohomology class.

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Date/Time : 07 Feb 2024 16:00
Venue : S17 #05-11 (SR5)
Speaker : Didac Martinez-Granado, University of Luxembourg and National University of Singapore

We define a notion of hyperplane for a hyperbolic group G and study G-invariant measures on the space of hyperplanes of its Gromov boundary, which we call “co-geodesic currents´´. Co-geodesic currents are induced by many classical objects such as geodesic currents for surface groups, certain cocompact actions of hyperbolic groups on CAT(0) cube complexes, quasi-fuchsian representations on a surface group, etc. Moreover, there is a natural intersection pairing between co-geodesic currents and the more classical geodesic currents, introduced by Bonahon, i.e. G-invariant measures on the square of the boundary of G. This pairing generalizes Bonahon’s intersection number when G is a surface group. Furthermore, every co-geodesic current \mu induces natural dual pseudo-metric space, in the sense that the intersection of \mu with the current determined by the conjugacy class of an element g in G recovers the stable length of g in the pseudo-metric space. This is joint work in progress with Eduardo Reyes.

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Date/Time : 26 Jan 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Tong Zhang, East China Normal University

It is known by Chen-Chen-Jiang that every complex smooth threefold X of general type with p_g(X) >= 11 satisfies the Noether inequality Vol(X) >= 4/3 p_g(X) – 10/3. If X attains the equality, it is said to be on the Noether line. In this talk, I will discuss the explicit structure of threefolds on the Noether line, and talk about what we know so far about their moduli space. This is a joint work in progress with S. Coughlan, Y. Hu, and R. Pignatelli.

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Date/Time : 25 Jan 2024 16:00
Venue : S17 #04-06 (SR1)
Speaker : Sungwoon Kim, Jeju National University

Sullivan sketched a proof of his structural stability theorem for differentiable group actions satisfying certain expansion-hyperbolicity axioms. In this paper, we relax Sullivan’s axioms and introduce a notion of meandering hyperbolicity for group actions on geodesic metric spaces. This generalization is substantial enough to encompass actions of certain nonhyperbolic groups, such as actions of uniform lattices in semisimple Lie groups on flag manifolds. At the same time, our notion is sufficiently robust, and we prove that meandering-hyperbolic actions are still structurally stable. We also prove some basic results on meandering-hyperbolic actions and give other examples of such actions. This is a joint work with Jaejeong Lee and Misha Kapovich.

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