Institut de recherche mathématique avancée

Résumé : En dynamique holomorphe, on peut naturellement associer à chaque point périodique d'une fraction rationnelle un nombre complexe appelé le multiplicateur. Dans cet exposé, je présenterai plusieurs résultats montrant qu'une classe de conjugaison de fractions rationnelles est caractérisée par la collection des multiplicateurs en ses points périodiques.

Résumé : Soit X un schéma sur un trait S. Sous des hypothèses convenables, Spencer Bloch a conjecturé une formule (dite "formule du conducteur de Bloch") qui identifie la dimension totale de la cohomologie évanescente à l'aide des formes différentielles algébriques. Le cas où la fibre spéciale est lisse en dehors d'une singularité isolée avait apparu quelques ans avant dans une conjecture due à Pierre Deligne et qui est connue comme "formule de Deligne--Milnor". Dans cet exposé, je vais parler de la preuve de quelques nouveau cas (d'une généralisation) de la formule du conducteur de Bloch, y compris le cas d'une singularité isolée. Cet exposé se base sur un travail en collaboration avec Dario Beraldo.

Résumé : In this talk, we will discuss Engel structures with a split through three examples. These geometric structures appear on differential manifolds of dimension four as rank-two distributions together with a decomposition into two preferred line fields. We will first consider the configuration space of a car moving in a plane, then we will look at third-order differential equations and finally we will present the geometry of pointed lightlike geodesics in three-dimensional Lorentzian manifolds. This talk is based on the article https://arxiv.org/pdf/1908.01169 by C. D. Hill and P. Nurowski.

Résumé : Zhen Gao (Augsburg)

Title: Morse-Bott Floer homology and rectangular pegs

Abstract: The rectangular peg problem, an extension of the square peg problem, is easy to outline but challenging to prove through elementary methods. In this talk, I discuss how to show the existence and a generic multiplicity result assuming the Jordan curve is smooth, utilizing Morse-Bott Floer homology. In particular, we obtain a convenient formula for computing the algebraic intersection number of cleanly intersecting Lagrangian submanifolds, which is well consistent with the Euler characteristic of Morse-Bott Floer homology in the spirit of ``categorification''.

Zihong Chen (MIT)

Title: The exponential type conjecture for quantum connection

Abstract: The (small) quantum connection is one of the simplest objects built out of Gromov-Witten theory, yet it gives rise to a repertoire of rich and important questions such as the Gamma conjectures and the Dubrovin conjectures. There is a very basic question one can ask about this connection: what is its formal singularity type? People's expectation for this is packaged into the so-called exponential type conjecture, and I will discuss a proof in the case of closed monotone symplectic manifolds. My approach follows a reduction mod p argument, by combining Katz's classical result on differential equations and the more recent quantum Steenrod operations.

Jonghyeon Ahn (UIUC)

Title: S^1-equivariant relative symplectic cohomology and relative symplectic capacities

Abstract: In this talk, I will construct an S^1-equivariant version of the relative symplectic cohomology developed by Varolgunes. As an application, I will construct a relative version of Gutt-Hutchings capacities and a relative version of symplectic (co)homology capacity. We will see that these relative symplectic capacities can detect the diplaceability and the heaviness of a compact subset of a symplectic manifold. We compare the first relative Gutt-Hutchings capacity and the relative symplectic (co)homology capacity and prove that they are equal to each other under a convexity assumption.

Résumé : Given a closed compact surface S and a reductive Lie group G, Goldman introduced a symplectic structure on the character variety Hom(pi_1(S),G)//G (the space of representations modulo conjugation). For Teichmüller space, the form coincides with the Weil-Petersson form. The symplectic structure gives rise to Hamiltonian flows associated to functions on character varieties, and therefore gives ways of deforming representations. I will talk about joint work with Anna Wienhard and James Farre, where we focus on a particular type of function on the character variety, whose input includes families of curves on S. The result I will present states that the Hamiltonian flows of such functions are what we call subsurface deformations, which roughly means that the flow is concentrated on a subsurface of S that depends on the curves defining the function. If time permits, I will discuss some applications to Hamiltonian flows of length functions associated to some self intersecting curves.