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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Date/Time : 06 Feb 2025 13:30
Venue : S17 #04-06 (SR1)
Speaker : Asbjørn Christian Nordentoft, Université Paris-Saclay

In the case of the modular surface Duke-Imamouglu-Toth defined a thin subgroup associated to each closed geodesic and studied the equidistribution of the projection of the associated hyperbolic orbifold with boundary (i.e. the partial coverings). In this talk, I will give a purely geometric generalization to any hyperbolic surface. Furthermore, I will explain that the equidistribution of closed geodesics in the unit tangent bundle implies equidistribution of the associated partial coverings. This yields a new proof of the result of Duke-Imammouglu-Toth as well as many other new equidistribution results.

This is joint work with Ser Peow Tan.

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Date/Time : 22 Jan 2025 14:00
Venue : S17 #04-04 (SR3)
Speaker : Gaofeng Huang, University of Bern

In this talk we survey a few results in the study of large holomorphic automorphism groups. First we give a short historical development of the so-called Andersen–Lempert theory, namely the study of large holomorphic automorphism groups. This field nowadays focuses on the so-called density property and a criterion for discovering more examples, many of which are smooth affine algebraic varieties.

In the second part we will consider the symplectic setting, namely complex symplectic manifolds with large group of holomorphic symplectic automorphisms. An example is the even dimensional Euclidean space endowed with the standard complex symplectic form. As a new example, the Calogero–Moser space, also admits the symplectic density property, which in turn leads to a strong approximation result for symplectic automorphisms in the Carleman sense.

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Date/Time : 21 Jan 2025 14:00
Venue : S17 #04-05 (SR2)
Speaker : Ara Basmajian, City University of New York

There are many topological invariants one can associate with homotopy classes of closed curves. These include algebraic and geometric self-intersection number, intersection with curves in a class of curves (for example, simple ones), the Goldman bracket, complementary component types of a curve, mapping class group stabilizers of a curve, and many others. How these invariants interact and determine the curve type (mapping class group orbit) is an active area of research today. In this talk, we focus on the so called inf invariant (shortest length metric) associated to a filling curve, its relationship with the geometric self-intersection number, and its relation to the optimal metric that is tailored to produce the minimum length. While clearly the geometric self-intersection number is a type invariant, we address whether the inf invariant can distinguish between curves that have the same self-intersection. This is joint work with Sayantika Mondal.

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