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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Date/Time : 24 Jan 2024 15:30
Venue : S17 #04-06 (SR1)
Speaker : Bong Lian, Brandeis University

We will consider a class of (typically) singular Calabi-Yau varieties given by cyclic branched covers of a fixed semi Fano manifold. The first prototype example goes back to Euler, Gauss and Legendre, who considered 2-fold covers of $\P^1$ branched over 4 points. Two-fold covers of $\P^2$ branched over 6 lines have been studied more recently by many authors, including Matsumoto, Sasaki, Yoshida and others, mainly from the viewpoint of their moduli spaces and their comparisons. I will outline a higher dimensional generalization from the viewpoint of mirror symmetry, and discuss the Riemann-Hilbert problem for periods of these singular varieties. The new insight here is the idea of `abelian gauge fixing’ and `fractional complete intersections’ that leads to a new interpretation of those classical results. The idea further points to a construction of large class of Calabi-Yau mirror pairs.

The lecture is based on joint work with S. Hosono, T.-J. Lee, M. Romo, L. Santilli, H. Takagi, S.-T. Yau.

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Date/Time : 18 Jan 2024 15:30
Venue : S17 #04-06 (SR1)
Speaker : Biao Ma, Technion - Israel Institute of Technology

Let Mod(S) be the mapping class group of a connected, closed, oriented surface S of genus g ≥ 2. Based on ideas of Margulis, Athreya-Bufetov-Eskin-Mirzakhani gave an asymptotic formula for orbit growth of Mod(S) on the Teichmuller space with respect to the Teichmuller metric. In this talk, I will explain how we can get asymptotic formulas for orbit growth of certain subgroups of Mod(S) based on Roblin’s ideas. More precisely, let G be a non-elementary subgroup of Mod(S) so that the associated Bowen-Margulis measure is finite. We will discuss an asymptotic growth formula for G with respect to the Teichmuller metric. This talk is based on a joint work with Ilya Gekhtman.

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Date/Time : 17 Jan 2024 16:00
Venue : S17 #06-11 (SR6)
Speaker : Tan Sheng Li, East China Normal University

Foliations are generalizations of fibrations on an algebraic surface. Some invariants of fibrations are generalized to foliations, e.g., Kodaira dimensions, geometric genus, Chern numbers, volume and slope. Fibrations with minimal slope must be hyperelliptic, which are double covers of ruled surfaces. We have no slope inequality for foliations, so it is interesting to construct some examples with small slopes. A nature way is to consider double covers of some transcendental foliations, we need to generalize the theory of double covers of fibrations to foliations. We will present an example whose slope is 12/7. This is a joint work with Jun LU and Shi XU.

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Date/Time : 05 Dec 2023 15:30
Venue : S17 #04-04 (SR3)
Speaker : Lu Qi, Princeton University

In this talk, I will discuss some convexity properties of multiplicities of filtrations on a local ring. In particular, the multiplicity function is convex along geodesics. As a major application, this gives a new proof of a theorem due to Xu and Zhuang on the uniqueness of normalized volume minimizers. In order to characterize strict convexity, we introduce the notion of saturation of a filtration, which turns out to be useful in other settings. For example, it allows us to generalizes a theorem by Rees on characterization of when two filtrations have equal multiplicities. It also allows us to introduce a metric on the space of filtrations. This talk is based on joint work with Harold Blum and Yuchen Liu and some ongoing work.

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