Date/Time : 19 Mar 2025 14:00
Venue : S17 #04-04 (SR3)
Speaker : Viet-Anh Nguyen, University de Lille

In 1992 Demailly raised the problem of developing a theory of intersection for positive closed currents. During the period 2009-2012, Dinh and Sibony gave two remarkable answers to this problem via their two new theories, namely, the super-potential theory and the theory of tangent currents. We focus more on the latter theory. More concretely, given m (≥2) positive closed currents T_i of bidegree (p_i,p_i) for 1≤i≤m on a compact Kahler manifold X, under theoretically reasonable assumptions, Dinh-Sibony can define the intersection T1 ⋏ . . . ⋏ Tm in the sense of the theory of tangent currents. In continuation of Dinh-Sibony’s pioneering works, we address/study, in this talk, the following basic questions:

Problem 1 (Existence of the intersection). Find an effective sufficient condition so that m(≥ 2) positive closed currents T_i of bidegree (p_i,p_i) for 1≤ i≤ m on a compact Kahler manifold X are wedgeable in the sense of the theory of tangent currents, that is, the current T1 ⋏ . . . ⋏ Tm exists.

Problem 2 (Continuity of the intersection). For m≥ 2 and for 1≤i≤m let (T_{i,n})_{n\in\N} be a sequence of positive closed currents of bidegree (p_i,p_i) on a compact Kahler manifold X so that \lim_{n\to\infty} T_{i,n} exists in the sense of currents. Find an effective sufficient condition imposed on these $m$ sequences so that the following equality holds:

\lim_{n\to\infty} ( T_{1,n}⋏ …⋏ T_{m,n}) =(\lim_{n\to\infty} T_{1,n})⋏…… ⋏\lim_{n\to\infty} T_{m,n})

Our approach relies on a new tool: the generalized Lelong numbers.

Key Words: positive closed/pluriharmonic/plurisubharmonic current, horizontal dimension, tangent currents, Lelong-Jensen formula, generalized Lelong numbers.

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Date/Time : 13 Mar 2025 14:00
Venue : S17 #06-11 (SR6)
Speaker : Didac Martinez Granado, University of Luxembourg

In geometric group theory, one typically studies whether a given group G admits particular geometric structures—well-behaved actions on real trees, measured wall structures, or admissible random walks—all of which give rise to (invariant) metrics on G. This mini course adopts a broader perspective: rather than focusing on each structure in isolation, we consider *all* possible metrics on G at once and investigate the global geometry of this “space of metrics.” Introduced by Furman in the early 2000s, it has recently undergone an intense study by Reyes, Cantrell-Tanaka, Cantrell-Reyes, and myself with collaborators, among others.

When G is a surface group, a motivating analogy comes from the Teichmüller space of a surface, which parametrizes all marked hyperbolic metrics and is a finite-dimensional ball. However, the space of metrics of a surface group is infinitely dimensional, incorporating diverse constructions such as hyperbolic metrics, word-metrics, geodesic currents and Green metrics. We will describe some of these metrics, focusing on geodesic currents as a motivating example. Another avenue to obtain metrics on G from measures is from certain “admissible” random walks with respect to probability measures on G. These induce “Green metrics” which naturally sit in this space. We will introduce and analyze a natural distance on the space of metrics, discussing in particular how both word-metrics and those arising from admissible random walks are dense (Cantrell-MG-Reyes).

We believe this course could be of interest to people interested in hyperbolic geometry, geometric group theory, dynamics, or probability.

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Date/Time : 12 Mar 2025 14:00
Venue : S17 #04-04 (SR3)
Speaker : Luo Muhan, National University of Singapore

We prove that under the natural assumption over the dynamical degrees, the saddle periodic points of a Henon-like map in any dimension equidistribute with respect to the equilibrium measure. Our work generalizes results of Bedford-Lyubich-Smillie, Dujardin and Dinh-Sibony along with improvements of their techniques. We also investigate some fine properties of Green currents associated with the map. This is a joint work with Qi Zhou.

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Date/Time : 12 Mar 2025 15:00
Venue : S17 #05-12 (SR4)
Speaker : Svetlana Katok, Pennsylvania State Unviersity

Given a closed, orientable surface of constant negative curvature and genus g ≥ 2, we study a family of generalized Bowen–Series boundary maps and their two dynamical invariants: the topological entropy and the measure-theoretic entropy with respect to their smooth invariant measure. Each such map is defined for a particular fundamental polygon and a particular multi-parameter. We prove two strikingly different results: rigidity of topological entropy and flexibility of measure-theoretic entropy. The topological entropy is constant in this family and depends only on the genus of the surface. We give an explicit formula for this entropy and show that it stays constant both within our parameter space and within the Teichmüller space of the surface. We obtain an explicit formula for the measure-theoretic entropy for the smooth invariant probability measure on the circle that only depends on the genus of the surface and the perimeter of the (8g−4)-sided fundamental polygon, and prove that it varies in the Teichmüller space and takes all values between 0 and maximum that is achieved on the surface which admits a regular fundamental (8g-4)-gon, and stays constant on a subset of the parameter space. The rigidity proof uses conjugation to maps of constant slope, while the flexibility proof – valid only for certain multi-parameters – uses the realization of the geodesic flow on the surface as a special flow over the natural extension of the boundary map, and the Reduction Theory for maps associated to Fuchsian groups.

In particular, we show that the smooth invariant probability measure is not the measure of maximal entropy (the Parry measure) for the considered boundary maps.

This is joint work with Adam Abrams and Ilie Ugarcovici.

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Date/Time : 07 Mar 2025 16:00
Venue : S17 #05-11 (SR5)
Speaker : Sheng MENG, East China Normal University

Let f be a surjective endomorphism of a projective variety X defined over a number field. The Kawaguchi-Silverman conjecture asserts that the arithmetic degree of a point with a Zariski dense forward orbit under f equals the first dynamical degree of f. In this talk, I will first survey previous results (primarily those based on Junyi Xie’s recursive approach) and then focus on our recent progress concerning the case where X is a smooth projective variety with Picard number 2.

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Date/Time : 21 Feb 2025 13:00
Venue : S17 #04-04 (SR3)
Speaker : Weikun He, AMSS, Chinese Academy of Sciences

On homogeneous spaces such as SL_2(R)/SL_2(Z), there are interesting random walks which are driven by group action. We focus on asymptotic behaviour of such random walk, especially on the property of equidistribution in law. In this talk, I will present a method developed in a joint work with Timothée Bénard to estimate the rate of convergence of such equidistribution. It involves ideas from fractal geometry and it constitutes a crucial ingredient in the study of Diophantine approximation on fractal sets, presented in the YMLS talk.

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Date/Time : 19 Feb 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : Meng Chen, Fudan University

In this lecture, I will provide an outline to prove the following theorem: the inequality K^3 ≥ 4/3 p_g – 10/3 holds for any complex 3-folds of general type. Based on this inequality, I will explain an effective method to study moduli spaces of canonical 3-folds with boundary volumes.

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Date/Time : 12 Feb 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : Ni Lei, University of California, San Diego

In this talk, I will review some applications of Lie theory in geometry such as Cartan’s work on symmetric spaces, Hamilton’s observation and Boehm-Wilking’s constructions utilizing Lie brackets and some highly useful considerations from Lie theory. Then I will talk about some more recent applications in my joint/solo works.

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Date/Time : 12 Feb 2025 16:00
Venue : S17 #04-04 (SR3)
Speaker : Fukushi Kenji, Kyoto University

The extension of Morse theory to S^1 valued Morse functions, more generally, closed 1-form is called Morse-Novikov theory. Morse-Novikov theory gives a new invariants of manifolds, the Novikov homology. As in Morse homology, the essence is the trajectory of the gradient flow connecting the critical points.

Based on this, Michael Farber extended Lusternik-Schnirelmann category to closed 1-form.

Unlike the ordinary result of Lusternik-Schnirelmann category about the number of critical points of functions, this extended category no longer gives the lower bound of the number of zeros of closed 1-forms. However, if the number of zeros of closed 1-form is less than this category, the dynamics of the gradient flow of the closed 1-form must have a Homoclinic cycle.

In this talk, I will explain the Lusternik-Schnirelmann category of closed 1-form, and I will introduce the extension of the Farber’s result to non compact manifolds.

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