Seminar in Arithmetic and Algebraic Dynamics
About the seminar
This research seminar on algebraic and arithmetic dynamics is jointly organized by R. Dujardin (LPSM) and C. Favre (CMLS). It will be focused on algebraic and arithmetic dynamics in a broad sense. Talks will be 1h30 long and can be either in french or in english. We will enable online participations.
Reach out C. Favre if you want to be included in the newslist concerning this seminar.
A companion seminar in complex dynamics held by M. Astorg, N.B. Dang, and L. Kaufmann is held at IHP once a month.
We also advertise the Séminaire de dynamique de l’IMJ which runs every Friday afternoon in Jussieu.
Location
Salle Jacques Neveu n.113, Jussieu couloir 16.26, 1er étage, Sorbonne Université
Program
- September 29, 2023 (10 am): Nguyen-Bac Dang (Université Paris-Saclay).
Asymptotic behavior of the Hausdorff dimension for degenerating families of Schottky groups and Poincare series. (joint work with V. Mehmeti)
Abstract
We study how the Hausdorff dimension of the limits sets vary for degenerating families of Schottky groups. Inspired by the work of Boucksom-Jonsson, Favre-Dujardin and Favre, I will explain how the degenerating family extends continuously to a family of groups acting on the Berkovich line defined over a Banach ring (which contains both usual Riemann sphere as well as some trees when the field is non-archimedean) Under some constraints on the family, the Hausdorff dimension is then continuous. This is a special case of general statement where we show that this dimension varies continuously over the moduli space of Schottky groups defined over Z constructed by Poineau-Turchetti. Naturally, the variation of the Hausdorff dimension is deeply linked to the pole of a certain Poincare series. In the course of our proof, we found that over non-archimedean fields, the Poincare series associated to Schottky groups extend meromorphically to the complex plane as quotient of exponential polynomials and analytically at zero. This first part is a generalization of Herzonsky-Hubbard’s result and the second is directly connected to the work of Anantharaman and analogous to a recent result of N-V. Dang-G. Rivière.
- November 10, 2023 Zhuchao Ji (Westlake univ., Hangzhou) and Junyi Xie (BICMR, Beijing)
The dynamical André-Oort conjecture for rational maps
Abstract
We prove the Dynamical André-Oort (DAO) conjecture proposed by Baker and DeMarco for families of rational maps parameterized by an algebraic curve. In fact, we prove a stronger result, which is a Bogomolov type generalization of DAO for curves.
- December 8, 2023 (10 am): Susanna Zimmermann (Université Paris-Saclay)
Sarkisov program in dimension 2 over imperfect fields (work in progress with F. Bernasconi, A. Fanelli and J. Schneider)
Abstract
Any birational map of the plane decomposes into so-called Sarkisov links, which are basically: blow-up of a point on a minimal surface, an elementary transformation of conic fibrations and the exchange of two fibrations. This result was shown by Corti and Iskovskih over any perfect field after an idea of Sarkisov. In this talk I want to present the generalisation of the Sarkisov program to imperfect fields and also what makes up the difficulties of working over imperfect fields instead of perfect fields.
- January 19, 2024 (10 am) Julien Marché (IMJ -Sorbonne Université)
Valuations sur les variétés de caractères des surfaces et laminations mesurées (Travail commun avec Christopher-Lloyd Simon)
Abstract
depuis les travaux de Morgan et Shalen, on sait compactifier la variété des caractères d’une surface dans SL_2 à l’aide de valuations, mais pas de manière très explicite. Je vais expliquer que la compactification de l’espace de Teichmüller à l’aide de laminations mesurées se réalise à l’aide de valuations simples et en donnerai des applications.
- February 9, 2024 (10 am) Luc Pirio (USVQ)
Généralisations de l’équation du logarithme via les surfaces de del Pezzo
Abstract
Nous commencerons par présenter un travail en collaboration avec Ana-Maria Castravet dans lequel nous construisons une famille d’identités fonctionnelles hyperlogarithmiques qui généralisent l’équation fonctionnelle du logarithme Log(x)+Log(y)=Log(xy) et du dilogarithme (relation d’Abel à 5 termes). L’approche est géométrique et repose sur des faits classiques relatifs à la géométrie des surfaces de del Pezzo et en particulier à l’action du groupe de Weyl d’une surface de del Pezzo dP_d de degré d sur l’ensemble des droites qu’elle contient.
Ensuite, nous discuterons des très nombreuses similitudes entre les cas associés à dP_5 et dP_4 puis nous esquisserons les grandes lignes d’une approche géométrique à la Gelfand-MacPherson des ces identités fonctionnelles, qui passe par la considération de certains espaces homogènes dans lesquels se plongent naturellement les “variétés de Cox” des surfaces de del Pezzo considérées.
- March 15 2024 (10 am) Shengyuan Zhao (Institut de Mathématiques de Toulouse)
Applications rationnelles twistées
Abstract
Dans un travail en commun avec Hongming Nie, on considère la dynamique de la composition d’une fonction rationnelle avec une action de Galois sur la droite projective sur un corps non archimédien. Il se trouve que ce type de dynamique ressemble beaucoup à celle de fonctions rationnelles usuelles bien qu’il y aie des différences fondamentales. On éspère des application éventuelles en dynamique fibrée en dimension complexe deux.
- April 26 2024 (10 am) Gaëtan Leclerc (Institut de Mathématiques de Jussieu)
Fourier dimension and Julia sets
Abstract
Consider the triadic Cantor set equipped with the Cantor law. It happens that its cumulative distribution function, the devil’s staircase, is Hölder regular, and its best exponent of regularity is ln(2)/ ln(3), which is exactly the Hausdorff dimension of the Cantor set. Moreover, one can show that the Fourier transform of the Cantor law decay like |ξ|^{− ln 2/ ln 3} “on average”. This is no coincidence, and hint for a deeper link between Fractal Geometry and Fourier Analysis. In this talk we will detail and explore this link through the notion of Fourier Dimension. We will introduce the Fourier dimension, compute it on some easy examples, quote some natural questions that arise, and then discuss a (partial) state of the art on the topic. We will take the time to discuss a recent result in particular: the Fourier dimension of hyperbolic julia sets (in the riemann sphere) is positive (as long as the Julia set isn’t included in a circle).
- June 7, 2024 (10 am) Ekaterina Amerik (HSE à Moscou et LMO à Orsay)
Isotropic boundary of the ample cone
Abstract
Joint work with M. Verbitsky and A. Soldatenkov. It follows from an old result of Kovacs that the ample cone of a projective K3 surface of Picard number at least three is either “round” (equal to the positive cone), or has “no round part”, that is the boundary of the ample cone is nowhere dense in the isotropic cone. With a help of hyperbolic geometry, this easily generalizes to the hyperkähler case. We pursue this a bit further and prove that any real analytic curve in the projectivization of the isotropic boundary of the ample cone lies in a sphere contained in this isotropic boundary. We discuss how the union of maximal such spheres, the ``Apollonian carpet’‘, equal to the union of all analytic curves on the projectivized isotropic boundary, can look like.