Date/Time : 13 May 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Sheng Meng, East China Normal University

A critical point x of a differentiable self-map f of X=R^n or manifolds means the determinant of the Jacobian |J_f(x)|=0. We collect these points and call it the critical set C_f. This set C_f also carries algebraic data, denoted as R_f which is nothing but the zeros (counting multiplicities) of the equation |J_f(x)|=0, called the ramification.

In the algebraic geometric setting, R_f turns out to be an effective divisor, but not necessarily smooth anymore. In this talk, I shall show you the behaviour of the coefficients of R_f (multiplicity in codimension 1), and also the much more complicated, the singularities of R_f (multiplicity in higher codimension).
Some applications will be discussed if time permits.

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Date/Time : 23 Apr 2024 16:00
Venue : S8 #03-14
Speaker : Philipp Kunde, Jagiellonian University

One of the oldest and most influential problems in ergodic theory is the classification of invertible measure-preserving transformations (MPT’s) up to isomorphism or Kakutani equivalence. Starting from the late 1990s, so-called anti-classification results use concepts from descriptive set theory to show precisely that classification for ergodic MPT’s is impossible in terms of computable invariants, even with a very inclusive understanding of “computability”. In this talk, we present more details about our techniques to prove such anti-classification results.

In the search for classification or anti-classification results, we also place increasingly more stringent conditions on the family of transformations being considered for classification. Bernoulli shifts, which can be viewed as the most random type of transformations, have the strongest type of classification, a complete Borel invariant (namely measure-theoretic entropy). But what about classes of transformations that are more random than ergodic transformations, but less so than Bernoulli shifts, such as weakly mixing transformations, mixing transformations, and K-automorphisms?

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Date/Time : 17 Apr 2024 15:30
Venue : S17 #05-12 (SR4)
Speaker : Shilpak Banerjee, Indian Institute of Technology Tirupati

A fundamental theme in dynamics is the classification of systems up to appropriate equivalence relations. For instance, the equivalence relation of topological conjugacy preserves the qualitative behavior of topological dynamical systems. Smale’s celebrated program proposes to classify topological or smooth dynamical systems up to topological conjugacy. In Ergodic Theory the isomorphism problem dates back to von Neumann’s foundational paper and asks to classify measure-preserving transformations up to measure isomorphism.

These classification problems not only turn out to be hard but sometimes even to be impossible. In this talk, we give an overview of classification as well as anti-classification results and present some related projects.

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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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