**Responsable d’équipe :** François Loeser**Responsables adjoints :** Penka GEORGIEVA, Maxime ZAVIDOVIQUE**Gestionnaire :** Julienne PASSAVE

Adresse postale :

IMJ-PRG – UMR7586 Université Pierre et Marie Curie Boite courrier 247 Couloir 15-25 5e étage 4 place Jussieu, 75252 Paris Cedex 05 |

L’après-midi de rentrée de l’équipe aura lieu le 12/10/2023 en salle **15.16.413**.

**Programme :**

15h30 – 18h00 : Exposés des post-doctorants de l’équipe (titres et résumés plus bas)

15h30 – 16h10 : Yuan Yao

16h20 – 17h00 : Panrui Ni

17h10 – 17h50 : Brain Hepler

18h00 – 18h15 : Réunion de l’équipe

**À partir de 18h15 : pot dans la salle 15.16.417**

**Résumés des exposés :**

Yuan Yao

Title : Fixed points of symplectic maps

Abstract : I’ll first give a general exposition of how symplectic topologists study symplectic maps using fixed-point Floer cohomology.

Then I will explain some of my joint work with Ziwen Zhao and Maxim Jeffs on computing algebraic operations in fixed point Floer cohomology on surfaces, and its connection to nodal elliptic curves in algebraic geometry. If time permits I will describe how direct limits in fixed point Floer cohomology may be used to describe a version of Gromov-Witten invariants for hypersurface singularities.

Panrui Ni

Title : Hamilton-Jacobi equations depending Lipschitz continuously on the unknown function

Abstract : In this talk, I will discuss the ergodic problem, the large time behavior and the vanishing discount problem for Hamilton-Jacobi equations depending Lipschitz continuously on the unknown function.**Brian Hepler**Title : What does it mean to be a solution to a differential equation?

Abstract : I will give a brief overview of what, to me, is one of the foundational problems in algebraic analysis: the Riemann-Hilbert correspondence. This correspondence has over 100 years of history, beginning with Hilbert’s 21st problem concerning the existence of ordinary differential equations with regular singularities on a Riemann surface with prescribed monodromy groups. Loosely, it is the “correct” generalization of the correspondence between vector bundles with flat connection and local systems to the singular setting, which is itself a massive generalization of the local existence and uniqueness of ordinary differential equations.

The problem of extending the Riemann-Hilbert correspondence to cover holonomic D-modules with irregular singularities has recently been settled by Kashiwara-D’Agnolo in general, and many others in special cases along the way (e.g., Deligne, Malgrange, Mochizuki, Sabbah, and Kedlaya to name just a few). These objects correspond topologically to « perverse enhanced ind-sheaves » (and several other equivalent Abelian categories, following Deligne‘s « Stokes–perverse sheaves », Kuwagaki’s « irregular perverse sheaves », etc.). I will discuss some of the advantages and disadvantages of working in these different frameworks, and time-permitting I will discuss my own research on nearby and vanishing cycles in the irregular setting, and possible future applications to Clausen-Scholze’s recent theory of condensed mathematics.

**Programme de l’après-midi de l’équipe AA, le 09/03/2023**

**En salle 15.16.413**

15h30 – 18h00 : Exposés des post-doctorants de l’équipe (titres et résumés plus bas)

15h30 – 16h10 Mura Yakerson

16h20 – 17h00 Xiaohan Yan

17h10 – 17h50 Dustin Connery-Grigg

18h00 – 18h15 : Réunion de l’équipe

**À partir de 18h15 : pot dans la salle 15.25.502**

**Résumés des exposés :**

**Mura Yakerson**

Title : Universality of cohomology theories in algebraic geometry

Abstract : Motivic homotopy theory provides a framework for studying various cohomology theories of algebraic varieties. In this talk, we will discuss how many interesting examples of these cohomology theories, such as algebraic K-theory or algebraic cobordism, acquire universality properties, which are based on certain covariance structures of these cohomology theories. This is a summary of joint projects with Tom Bachmann, Elden Elmanto, Marc Hoyois, Joachim Jelisiejew, Adeel Khan, Denis Nardin, Vladimir Sosnilo and Burt Totaro.

**Xiaohan Yan**

Title : Level structures in quantum K-theory

Abstract : Quantum K-theory studies a K-theoretic analogue of Gromov-Witten invariants defined as holomorphic Euler characteristics over the moduli spaces of stable maps. Inspired by Verlinde/Grassmannian correspondence, level structures are introduced into quantum K-theory as determinant-type twistings. Such structures admit various symmetries, reveal surprising connections of quantum K-theory to mock theta functions, and appear in the so-called quantum Serre duality as well.

**Dustin Connery-Grigg**

Title : Symplectic dynamics and Hamiltonian Floer theory

Abstract : Given a Hamiltonian dynamical system on a symplectic manifold, what is the relationship between the dynamical features which the system may (or must) exhibit, and the (symplectic) topology of the underlying manifold? In 1989, Andreas Floer introduced an approach to doing relative Morse theory for the Hamiltonian action functional which provided a lower bound for the number of 1-periodic orbits of the associated Hamiltonian system in terms of the (quantum) homology of the underlying symplectic manifold, answering a version of Arnold’s conjecture in the process. The tools introduced by Floer in this work have since become a cornerstone of modern symplectic geometric research, but the finer-grained relationship between the objects appearing in this theory and the underlying dynamics remain somewhat mysterious. In this talk, I will give an brief introduction to Floer’s theory, and discuss some results in low-dimensions which provide links between the qualitative dynamics of low-dimensional Hamiltonian systems and their Floer theory.