Date/Time : 27 Nov 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Marco Vergamini, Scuola Normale Superiore

Let f be a complex Hénon map and µ its unique measure of maximal entropy. Recently, Bianchi-Dinh proved that µ is exponentially mixing of all orders for all Hölder observables, and that all such observables satisfy the central limit theorem with respect to µ. De Thélin-Vigny generalized these results for a certain class of bounded plurisubharmonic observables. We prove that these properties hold for all, not necessarily bounded, plurisubharmonic observables. This is a joint work with Hao Wu.

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Date/Time : 06 Nov 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Lei Yang, National University of Singapore

In this talk, I will give a new (and shorter) proof of Strombergsson’s Theorem on effective equidistribution of unipotent orbits in ASL(2, R)/ASL(2,Z). The proof is based on a general framework developed in recent works by Lindenstrauss-Mohammadi-Wang and myself, which relates the problem to incidence geometry. It relies on the study of a Kakeya-type model. This is joint with Elon Lindenstrauss.

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Date/Time : 09 Oct 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Enrique ARTAL BARTOLO, Universidad de Zaragoza

In this talk we discuss how algebraic (or arithmetic) and geometric properties of plane curves affect the topology. There are some classical results starting from Zariski where the position of some singular points with respect to curves of some fixed degree has surprising consequences on the topology of these curves. We discuss the state of art of the subject, leading to cases where for curves defined over a number field, the action of the Galois group has also influence on the topology. We will present recent discoveries on the subject, specially with my collaborators and from the Hiro Tokunaga’s team, and also open problems.

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Date/Time : 02 Oct 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Hideo Kojima, Niigata University

A cuspidal curve means a projective curve whose singular points are locally irreducible. The study of rational cuspidal plane curves has had a long and interesting history. In the first half of the talk, I will introduce some of recent progress on the study. Then, I will give some results on rational cuspidal curves on Q-homology projective planes. Here, a Q-homology plane is a normal projective complex surface with quotient singularities having the same rational homology as the projective plane. Some results on rational cuspidal plane curves will be generalized.

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Date/Time : 09 Sep 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : Ichiro Shimada, Hiroshima University

The Ap\’ery-Fermi pencil is a certain one-dimensional family of complex affine surfaces related to a Fermi surface in solid state physics. This pencil is also connected to Ap\’ery’s proof of the irrationality of the zeta value at s=3.

A general member of the Ap\’ery-Fermi pencil is birational to a K3 surface with Picard number 19. We calculate the automorphism group of this K3 surface by means of Borcherds’ method.

I also present some general algorithms that may be useful in computational algebraic geometry.

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Date/Time : 04 Sep 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Jia Jia, Tsinghua University

We build bridges between moduli theory of sheaf stable pairs on one hand and birational geometry on the other hand. We will in particular treat moduli of sheaf stable pairs on smooth projective curves in detail and present some calculations in low degrees.

This is based on a joint work with Caucher Birkar and Artan Sheshmani.

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Date/Time : 04 Sep 2024 14:00
Venue : S17 #05-12 (SR4)
Speaker : Fabrizio Bianchi, University of Pisa

Let f be a holomorphic endomorphism of P^k (C) of algebraic degree d. The Julia set J of f is the support of the unique measure mu of maximal entropy k log d. De Thélin and Dinh proved that every measure whose entropy is larger than (k-1)(log d) is also supported on J. Their proof crucially relies on the existence of the Green current. In particular, it follows from a delicate induction which makes use of the successive self-intersections T^j of the Green current T. It is then unclear how to generalize this result to more general non-algebraic settings, where a dynamical Green current does not exist.

I will present a new proof of this theorem which does not rely on the existence of a Green current, but instead on a quantified speed of convergence of preimages of points towards mu, which was established by Dinh and Sibony. In particular, the result can be extended to the more general setting of polynomial-like maps of large topological degree.

This is a joint work with Sardor Bazarbaev and Karim Rakhimov.

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Date/Time : 04 Sep 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Wilderich Tuschmann, Karlsruhe Institute of Technology

Consider a smooth manifold with a Riemannian metric satisfying some sort of geometric constraint like, for example, positive scalar curvature, non-negative Ricci or negative sectional curvature, being Einstein, Kähler, Sasakian, of special holonomy, etc. A natural question to ponder is then what the space of all such metrics does look like. Moreover, one can also study this question for the corresponding moduli spaces of metrics,i.e., quotients of the former by the diffeomorphism group of the manifold, acting by pulling back metrics. These spaces are customarily equipped with the topology of smooth convergence on compact subsets and the quotient topology, respectively, and their topological properties then provide the right means to measure ‘how many’ different metrics and geometries the given manifold actually does exhibit.

The history of the subject as a whole indeed goes back more than a century, and since H. Weyl’s early result on the connectedness of the space of positive Gaussian curvature metrics on the two-sphere and the foundings of Teichmüller, infinite- dimensional manifold and Lie group theory, uniformization and geometrization, the study of spaces of metrics and their corresponding moduli has been a topic of interest for differential geometers, global and geometric analysts and topologists alike.

In my talk, I will provide a gentle introduction to the subject with a focus on lower curvature bounds and present recent results and open questions about the global topological properties of moduli spaces of nonnegatively curved Riemannian metrics on manifolds, and as well, if time permits, also corresponding facts and questions for the moduli of nonnegative curvature metrics on RCD spaces.

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Date/Time : 29 Aug 2024 16:00
Venue : S17 #04-06 (SR1)
Speaker : Minsung Kim, Centro di Ricerca Matematica Ennio De Giorgi

The deviation of the Birkhoff integral for area-preserving flow on compact surfaces was first studied by Forni. He proved that the deviation spectrum was determined by the Lyapunov exponents of a renormalization cocycle so-called Kontsevitch-Zorich cocycle. This deviation result was later proved again by Bufetov and Frączek-Ulcigrai for translation flows and locally Hamiltonian flows for non-degenerate types. In this talk, we study the spectrum for deviations of Birkhoff integrals for locally Hamiltonian flows beyond the case of Forni where the observable vanishes at the singularities. Our new developments include a better understanding of the asymptotics at singularities (degenerate type) and the appearance of a new deviation spectrum. This is a joint work with Krzysztof Frączek.

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