Date/Time : 18 Jun 2025 14:00
Venue : S17 #04-04 (SR3)
Speaker : Khanh Le, Rice University

For a hyperbolic surface X with non-empty boundary, the ortho-length spectrum of X is the multi-set of lengths of (ortho)-geodesic arcs meeting the boundary of X perpendicularly at both endpoints. The spectrum was introduced by Basmajian for hyperbolic n-manifolds in the context of studying co-dimensional one embedded geodesic submanifolds. The ortho-length spectrum has been studied extensively in relation to geometric identities, hyperbolic geometry of surfaces and inverse geometry. In this work, we initiate a new investigation into the arithmetic properties of the ortho-length spectrum. In particular, we consider hyperbolic surfaces whose cosh-of-length of ortho-geodesics are all integers, ortho-integral surfaces. We prove that there are only finitely many ortho-integral surfaces for a fixed topological type. We also constructed infinitely many pairwise non-commensurable ortho-integral surfaces. In the talk, I will motivate the consideration of such surfaces and discuss some ideas and tools going into the proof. If time permits, I will also discuss some further questions that we would like to investigate. This is joint work with Minh Doan.

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Date/Time : 08 May 2025 11:00
Venue : S17 #04-04 (SR3)
Speaker : Tianqi Wang, Yale University

We construct an example of discrete, faithful and simple-stable representation of a closed hyperbolic surface groups into $SO^+(4,1)$ that is not convex-cocompact but deforms to a Fuchsian one continuously. Therefore the simple-stable characters forma domain of discontinuity for the mapping class group action on the character variety, who has a connected component strictly containing the convex-cocompact characters. We further talk about some general notions of ending currents in the study of discrete faithful representations into $SO^+(4,1)$.

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Date/Time : 30 Apr 2025 15:15
Venue : S17 #04-04 (SR3)
Speaker : Fabrizio Bianchi, University of Pisa

For a rational map, the classical Mané-Manning formula relates the entropy, the Lyapunov exponent, and the Hausdorff dimension of any invariant measure. I will describe a generalisation of this formula valid for invariant measures for endomorphisms of P^k(C) in any dimension with strictly positive Lyapunov exponents. This is joint work with Y. M. He.

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Date/Time : 30 Apr 2025 14:00
Venue : S17 #04-04 (SR3)
Speaker : Lin Shu, Peking University

There are a variety of interesting chaotic dynamical systems in negatively curved spaces. Entropies related to these systems are important quantities for understanding not only the complexities of their dynamics but also the complexities of the underlying spaces. This partially explains why the study of entropies in negatively curved spaces has always been a vibrant topic in dynamical systems. In this talk, we will give a biased introduction to this heavy topic by focusing on some natural diffusions (for instance, foliated Brownian motion) associated with the geodesic flow in a negatively curved space. We will mention some of our results (joint with F. Ledrappier) on stochastic entropies of these diffusions, in an attempt to use these quantities to characterize the geometry of the underlying space, to find connections between classical dynamical measures, and to understand the linear response theory for heat conduction, etc.

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Date/Time : 16 Apr 2025 14:00
Venue : S17 #04-05 (SR2)
Speaker : Paolo Cascini, Imperial College

I will review recent progress on extending the Minimal Model Program to algebraically integrable foliations. My focus will be on some recent results toward the boundedness of Fano foliations.

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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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