CNRS Paris Diderot Sorbonne Université

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. 

L’après-midi de l’équipe, le 09/03/2023

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.

L’après-midi de l’équipe, le 20/10/2022

Programme de l’après-midi de l’équipe AA, le 20/10/2022

En salle 15.16.413

15h00 – 17h30 : Exposés des post-doctorants de l’équipe (titres et résumés plus bas)

15h00 – 15h40 : Konstantinos Kartas

15h50 – 16h30 : Owen Rouille       

16h40 – 17h20 : Vukasin Stojisavljevic

17h30 – 18h : Réunion de l’équipe

À partir de 18h00 : pot dans la salle 15.25.502

Résumés des exposés :

Konstantinos Kartas

Title : Some model theory of the tilting correspondence

Abstract : The idea that p-adic fields are in many ways similar to Laurent series over finite fields is a powerful philosophy. This philosophy has had two formal justifications. On one hand, the classical model-theoretic work by Ax-Kochen/Ershov in the ’60s achieves a transfer principle when p goes to infinity.  On the other hand, perfectoid geometry suggests replacing local fields with certain highly ramified extensions; this has the effect of making the similarities between mixed and positive characteristic even stronger. I will first survey those two approaches and then mention some recent work with F. Jahnke in which we give a model-theoretic generalization of the Fontaine-Wintenberger theorem. As a new arithmetic application, we provide some examples in mixed characteristic verifying the Lang-Manin conjecture on the existence of rational points on nearly rational varieties over C1 fields. 

Owen Rouille

Title : Experimenting in mathematics: examples of data generation and visualisation.

Abstract : Mathematics include many complex objects which are difficult to understand. In particular, intuition is crucial in teaching and research, allowing to state and prove conjectures. Building this intuition requires looking at and analysing many examples, which constitutes an experimental approach to mathematics. Computers are very important tools at our disposal: they allow to generate and analyse large quantities of data, and visualise patterns that would be difficult to draw by hand. A notable recent example of the use of computers in mathematics consisted in using AI to look for correlations between quantities in knot theory, this resulted in a conjecture that was then proved by specialists of the domain [Davies 21]. However, using computers is not straightforward, and implementation raises many new questions, among which the encoding and the performances for instances.

The talk is dedicated to the use of computers to help with mathematics. In the first part, I will present part of the work I did during my PhD concerning the computation of two topological invariants for 3-manifolds (Turaev–Viro invariants and hyperbolic volume). These projects illustrate the use of computers in mathematics in a domain where the computations can be difficult and the dataset very large. In the second part, I will present the main project of my postdoc: the computation and the representation of limit sets in S^3, with a focus on triangular groups.

Vukasin Stojisavljevic

Title : Coarse nodal geometry and topological persistence

Abstract : Given a Laplace-Beltrami eigenfunction on a closed manifold,its nodal domains are connected components of the complement of its zero set. Courant’s nodal domain theorem is a classical result in spectral geometry which, together with Weyl’s law, provides an upper bound on the total number of nodal domains in terms of the corresponding eigenvalue. A well-known question asks whether this result generalizes to linear combinations of eigenfunctions. While direct generalizations fail to be true, we will show that by counting nodal domains in a coarse way, i.e. by ignoring small oscillations, one may prove a version of Courant’s theorem for linear combinations as well. In order to prove this version of Courant’s theorem, we use the theory of persistence modules and barcodes, combined with multiscale polynomial approximation in Sobolev spaces. The same method allows us to prove a coarse version of Bézout’s theorem for linear combinations of Laplace-Beltrami eigenfunctions. The talk is based on a joint work with L. Buhovsky, J. Payette, I. Polterovich, L. Polterovich and E. Shelukhin.

L’après-midi de l’équipe, le 10/03/2022

Programme de l’après-midi de l’équipe AA, le 10/03/2022

En salle 15.16.413

16h-17h30 : Exposés des post-doctorants de l’équipe (titres et résumés plus bas)
16h00 – 16h40 Erman Cineli16h50 – 17h30 Marvin Hahn
17h40 – 18h : Réunion de l’équipe

À partir de 18h15 : pot dans la salle de convivialité

Résumés des exposés :
Erman Cineli 
Title: Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

Abstract: In this talk I will introduce barcode entropy and discuss its connections to topological entropy. The barcode entropy is a Floer-theoretic invariant of a compactly supported Hamiltonian diffeomorphism, measuring, roughly speaking, the exponential growth under iterations of the number of not-too-short bars in the barcode of the Floer complex. The topological entropy bounds from above the barcode entropy and, conversely, the barcode entropy is bounded from below by the topological entropy of any hyperbolic locally maximal invariant set. As a consequence, the two quantities are equal for Hamiltonian diffeomorphisms of closed surfaces. The talk is based on a joint work with Viktor Ginzburg and Basak Gurel.

Marvin Hahn 
Title: Quasimodularity of weighted Hurwitz numbers
Abstract: Hurwitz numbers enumerate branched mophisms between Riemann surfaces with fixed numerical data. When the target surface is an elliptic curve, these enumerative invariants are intimately related to mirror symmetry, which e.g. predicts a quasimodular structure of the generating series of elliptic Hurwitz numbers. This prediction was confirmed in seminal work of Dijkgraaf in 1995. In the past years, several variants of Hurwitz numbers were introduced that arise in various different contexts, e.g. monotone Hurwitz numbers in random matrix theory or strictly monotone Hurwitz numbers in the theory of Grothendieck dessins d’enfants. Recently, in work of Guay-Paquet and Harnard a unified framework for these different variants was introduced under the name of weighted Hurwitz numbers. Here the idea is to consider a Hurwitz numbers-like enumeration that depends on a weight function. For different choices of this weight function, weighted Hurwitz numbers specialise to essentially all known variants of Hurwitz numbers. In this talk, we present results regarding the structure of elliptic Hurwitz numbers. In particular, we generalise Dijkgraaf’s work to this case and derive a quasimodular structure of generating series. Our methods heavily rely on tropical geometry. This talk is based on a joint work in progress with Danilo Lewanski and Jonas Wahl.

Journée de rentrée, le 21/10/2021 

Programme de la journée de rentrée de l’équipe AA, le 21/10/2021 

En salle 15.16.413

15h-17h20 : Exposés des post-doctorants de l’équipe (titres et résumés plus bas)

15h-15h40 : Dusan Joksimovic

15h50-16h30 : Bernhard Reinke

16h40-17h20 : Jinhe Ye

17h30-18h : Réunion de l’équipe

À partir de 18h15 : pot sous la barre 24-25 niveau Jussieu

Résumés des exposés :

Dusan Joksimovic
Title : No symplectic-Lipschitz structures on $S^{2n \geq 4}$

Abstract: One of the central questions in $C^0$-symplectic geometry is whether spheres (of dimension at least 4) admit symplectic topological atlas (i.e. atlas whose transition functions are symplectic homeomorphisms). In this talk, we will prove that the answer is « no » if we replace the word « topological » with « Lipschitz ». More precisely, we will prove that every closed symplectic-Lipschitz manifold has non-vanishing even degree cohomology groups with real coefficients. The proof is based on the fact that one can define analogs of differential forms and de Rham complex on Lipschitz manifolds which share similar properties as in the smooth setting.

Bernhard Reinke
Title : Connections between complex dynamics and algebra

Abstract: Complex dynamics and algebra are deeply connected. I will present two examples of their connection: the dynamics of root-finding methods, and iterated monodromy groups of transcendental maps.

Finding roots of a polynomial is a fundamental numerical problem. Many root-finding methods, such as the Newton’s method or the Weierstrass/Durand-Kerner method can be understood as complex dynamical systems. I will sketch how computer algebraic tools were used to show that the Weierstrass method is not generally convergent.

Iterated monodromy groups are self-similar groups associated to partial self-coverings. I will give an overview of iterated monodromy groups of post-singularly finite entire transcendental functions. These groups act self-similarly on a regular rooted tree, but in contrast to IMGs of rational functions, every vertex of the tree has countably infinite degree.

I will discuss the similarities and differences of IMGs of entire transcendental functions and of polynomials, in particular in the direction of amenability.

Jinhe Ye

Title : Curve-excluding fields 

Abstract : Consider the class of fields with Char(K)=0 and x^4+y^4=1 has only 4 solutions in K, we show that this class has a model companion, which we denote by CXF, curve-excluding fields. Curve-excluding fields provides examples to various questions. Model theoretically, they are model complete and algebraically bounded. Field theoretically, they are not large. This answers a question of Junker and Macintyre negatively. Joint work with Will Johnson and Erik Walsberg.

Journée de fin d’année, jeudi 1er juillet

Amphi 24

14h-16h : Exposés courts des doctorants de l’équipe (titres plus bas)

16h-17h : réunion de fin d’année de l’équipe

À partir de 17h : pot dans le patio de l’amphi 24

14h-14h15 Antoine Toussaint

Orientations complexes des surfaces algébriques réelles

14h15-14h30 Flavien Grycan-Gérard

Entropie polynomiale des systèmes intégrables hamiltoniens à singularités modérées

14h30-14h45 Perla Azzi

Distance aux strates d’isotropie appliquée à l’espace des tenseurs d’élasticité

14h45-15h Thibaut Mazuir

Algèbre supérieure de la théorie de Morse

15h-15h15 Mahya Mehrabdollahei

Les mesures de Mahler d’une famille de polynômes exacts

15h15-15h30 An Khuong Doan

Equivariant (derived) deformations of algebraic schemes and of complex compact manifolds

15h30-15h45 Benoît Joly

Codes barres d’homéomorphismes hamiltoniens de surfaces

Journée de rentrée, le 20/02/2020

Programme de la journée de rentrée de l’équipe AA, le 20/02/2020

En salle 15.25.502

15h30-16h10, Sébastien Biebler (IMJ-PRG)

Domaines errants en dynamique holomorphe (en commun avec Pierre Berger)

16h10-16h50, Yanqiao Ding (IMJ-PRG/Zhengzhou University)

Welschinger invariants and degeneration technique

16h50-17h30, Alban Quadrat (IMJ-PRG/INRIA)

Quelques résultats effectifs de la théorie des D-modules algébriques

17h30-18h, réunion d’équipe

En salle 15.16.417

18h15 – pot de l’équipe

Résumés des exposés

Sébastien Biebler

Titre : Domaines errants en dynamique holomorphe (en commun avec Pierre Berger)

Résumé : Pour une application holomorphe f : M→M d’une variété complexe M, on peut définir son ensemble de Fatou comme l’ensemble des points z∈M tels que la suite (f^n)_n des itérées de f est normale dans un voisinage de z. En particulier, c’est un ouvert, et c’est le lieu où la dynamique garde le même comportement en variant un peu les conditions initiales. A l’inverse, l’ensemble de Julia de f, défini comme le complémentaire de l’ensemble de Fatou, est le lieu où la dynamique peut changer drastiquement de comportement en variant les conditions initiales.

Je commencerai par rappeler un important résultat de Sullivan : en dynamique holomorphe en une variable, toute composante connexe de l’ensemble de Fatou d’une application rationnelle est envoyée en un temps fini sur une composante périodique. En particulier, comme ces dernières ont été classifiées, on comprend parfaitement la dynamique sur l’ensemble de Fatou.

Dans un second temps, je présenterai un travail récent en commun avec Pierre Berger où nous montrons que cette propriété n’est plus vraie en dimension supérieure pour des automorphismes polynomiaux de C^2.

Yanqiao Ding

Titre : Welschinger invariants and degeneration technique

Résumé : The Welschinger invariant provides a lower bound for the number of real irreducible rational curves in a given divisor class and passing through a set of real points in del Pezzo surfaces. In this talk, I will explain some computations of Welschinger invariants using a degeneration technique.

Alban Quadrat

Titre : Quelques résultats effectifs de la théorie des D-modules algébriques

Résumé : Le but de cet exposé est de montrer quelques résultats effectifs obtenus récemment dans l’étude des D-modules algébriques (modules sur des algèbres de Weyl, c-à-d sur des algèbres d’opérateurs différentiels à coefficients polynomiaux).

En particulier, en nous basant sur des méthodes de calcul formel (bases de Gröbner ou bases de Janet, théorie de l’élimination différentielle, etc.), nous étudierons le calcul effectif de la filtration d’un D-module par le grade, ainsi que la caractérisation effective de certaines propriétés des modules (avec torsion, sans torsion, réflexive, projective, stablement libre, libre). Nous illustrerons ces résultats en expliquant leurs intérêts en théorie mathématique des systèmes. Finalement, nous montrerons comment les implantations de ces résultats peuvent améliorer le solveur différentiel de Maple.

Finalement, en fonction du temps, nous évoquerons des versions effectives de théorèmes de Stafford obtenus pour les algèbres de Weyl (calcul de deux générateurs des idéaux, calcul d’éléments unimodulaires de D-modules, lemme de Swan, splitting-off de Serre, cancellation theorem de Bass).

Décès de Daniel Pecker

Notre collègue Daniel Pecker est décédé le samedi 14 septembre 2019, des suites d’une longue maladie, dans sa soixante-dixième année.

Daniel Pecker a été Maître de Conférences à l’Université Pierre et Marie Curie de 1986 à 2014, puis membre bénévole de l’IMJ-PRG. Il était membre de l’équipe Analyse Algébrique et membre associé de l’équipe-projet Ouragan.

Daniel Pecker était spécialiste de géométrie algébrique réelle. Il s’intéressait depuis quelques années aux représentations polynomiales des nœuds. Citons en particulier sa démonstration d’une conjecture de V. F. R. Jones : Il existe un corps convexe B tel que tout nœud soit isotope à une trajectoire de billard dans B [Pecker, 2012].

Journée de l’équipe 2018

Le 8 octobre 2018. La journée de l’équipe aura lieu en salle 15-25-502.

Programme :
16h30-17h10 : Exposé de Jean-Baptiste Teyssier.
17h15-17h55 : Exposé de Juho Leppänen.
18h00-18h10 : Point sur les finances de l’équipe.
18h30 : Pot en 15.16.417 !

Titres et résumés :

Jean-Baptiste Teyssier
Titre : Phénomène de Stokes et géométrie algébrique
Résumé : Dans cet exposé, on introduira le phénomène de Stokes pour les systèmes différentiels. Dans un second temps, on expliquera comment une conjecture de Deligne sur les représentations du groupe fondamental d’une variété sur un corps fini a suggéré la construction d’une variété algébrique paramétrant les données de Stokes des systèmes différentiels.

Juho Leppänen
Titre : Quasistatic dynamical systems
Résumé : Quasistatic dynamical systems (QDS), introduced by Dobbs and Stenlund in 2015, model dynamics that slowly transform over time due to external influences. They are generalizations of conventional dynamical systems and belong to the realm of deterministic non-equilibrium processes.

Conférences 2018

Une conférence en l’honneur de Pierre Schapira du 9 au 11 avril 2018 à l’IHP (Paris)

et une autre en l’honneur de François Loeser du 28 mai au 1er juin 2018 à Banyuls-sur-Mer

Journée de l’équipe 2017

Le 9 octobre 2017. La journée de l’équipe aura lieu en salle 15-16-413.

*Programme :*

16h00 : Présentation du budget par Viviane Baladi notre responsable d’équipe,
16h15-17h00 : Exposé de Lorenzo Fantini,
17h00-17h45 : Exposé de Shu Shen,
18h00 : Pot !

*Titres et résumés :*

/Lorenzo Fantini :/
Titre : Espaces de Berkovich et singularités.
Résumé : J’illustrerai à travers de quelques exemples le rôle joué par les valuations dans l’étude des singularités et de leurs résolutions. Dans ce cadre, la théorie des espaces analytiques non archimédiens à la Berkovich se montre souvent fructueuse.

/Shu Shen :/
Titre : Torsion analytique et la fonction zêta dynamique
Résumé : Dans mon exposé, j’explique la preuve de la conjecture de Fried sur l’espace localement symétrique qui affirme une égalité entre la torsion analytique et la valeur en zéro de la fonction zêta dynamique de Ruelle.

Jean-Jacques Risler 1940-2016

Jean-Jacques Risler nous a malheureusement quittés ce 17 février 2016.
Un texte a été écrit en sa mémoire :