CNRS Paris Diderot Sorbonne Université

Equipe Analyse Algébrique

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

Adresse postale :

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

Après-midi de l’équipe 05/03/2024

 l’après-midi de l’équipe aura lieu le 05/03/2024 en salle 15.16.413.

Programme :

14h30 – 16h50 : Exposés des nouveaux arrivants dans l’équipe (titres et résumés plus bas)

14h30 – 15h10 Erman Cineli

15h20 – 16h00 Tudor Padurariu       

16h10 – 16h50 Amanda Hirschi

16h50 – 17h15 : Réunion de l’équipe

À partir de 17h30 : pot dans la salle 15.16.417

Résumés des exposés :

Erman Cineli

Title: Invariant Sets and Hyperbolic Periodic Orbits of Reeb Flows

Abstract: In this talk we will discuss the impact of hyperbolic (or, more generally, isolated as an invariant set) closed Reeb orbits on the global dynamics of Reeb flows on the standard contact sphere. We will discuss extensions of two results previously known for Hamiltonian diffeomorphisms to the Reeb setting. The first one asserts that, under a very mild dynamical convexity type assumption, the presence of one hyperbolic closed orbit implies the existence of infinitely many simple closed Reeb orbits. The second result is a higher-dimensional Reeb analogue of the Le Calvez-Yoccoz theorem, asserting that no closed orbit of a non-degenerate dynamically convex Reeb pseudo-rotation is isolated as an invariant set. The talk is based on a joint work with Viktor Ginzburg, Basak Gurel and Marco Mazzucchelli.

Tudor Padurariu

Title: Quasi-BPS categories for K3 surfaces

Abstract: BPS invariants and cohomology are central objects in the study of (Kontsevich-Soibelman) Hall algebras or in enumerative geometry of Calabi-Yau 3-folds. 

In joint work with Yukinobu Toda, we introduce a categorical version of BPS cohomology for local K3 surfaces, called quasi-BPS categories. When the weight and the Mukai vector are coprime, the quasi-BPS category is smooth, proper, and with trivial Serre functor etale locally on the good moduli space. Thus quasi-BPS categories provide (twisted) categorical (etale locally) crepant resolutions of the moduli space of semistable sheaves on a K3 surface for generic stability condition and a general Mukai vector. Time permitting, I will also discuss a categorical version of the \chi-independence phenomenon for BPS invariants.

Amanda Hirschi

Title : Going global

Abstract : Moduli spaces of pseudoholomorphic curves , while providing powerful symplectic invariants, are generally difficult to work with due to transversality problems. In 2021, Abouzaid-McLean-Smith achieved a breakthrough by constructing a nice representation, called a global Kuranishi chart, for the moduli space of stable maps to a symplectic manifold. While briefly sketching the construction, I will mainly focus on explaining some applications to pseudoholomorphic curve theory.

Rentrée de l’équipe 12/10/2023

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.