## Prochainement

Mercredi 23 juin 14:00-16:00 Michal Wrochna (Cergy)
An index theorem on asymptotically static spacetimes with compact Cauchy surface

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Lieu : Demander le lien Zoom à jean-michel.bismut@universite-paris-saclay.fr

Résumé : A theorem due to Bär and Strohmaier (Amer. J. Math., 141 (5)) says that the Dirac operator on a Lorentzian manifold with compact Cauchy surface is Fredholm if Atiyah-Patodi-Singer boundary conditions are imposed at finite times. Furthermore, the index is given by a geometric formula that parallels as closely as possible the Atiyah-Patodi-Singer theorem in the Riemannian setting. In this talk I will report on joint work with Dawei Shen (Sorbonne Université) which extends this result to the infinite-time setting. Furthermore, we demonstrate that in the infinite-time situation, Fredholm inverses are Feynman parametrices in the sense of Duistermaat-Hörmander, a property which allows to show relationships with local aspects of the geometry.

Mercredi 30 juin 16:00-18:00 Simion Filip (University of Chicago )
Degenerations of Kahler forms on K3 surfaces, and some dynamics

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Lieu : Demander le lien Zoom à jean-michel.bismut@universite-paris-saclay.fr

Résumé : K3 surfaces have a rich geometry and admit interesting holomorphic automorphisms. As examples of Calabi-Yau manifolds, they admit Ricci-flat Kahler metrics, and a lot of attention has been devoted to how these metrics degenerate as the Kahler class approaches natural boundaries. I will discuss how to use the full automorphism group to analyze the degenerations and obtain certain canonical objects (closed positive currents) on the boundary. While most of the previous work was devoted to degenerating the metric along an elliptic fibration (motivated by the SYZ picture of mirror symmetry) I will discuss how to analyze all the other points. Time permitting, I will also describe the construction of canonical heights on K3 surfaces (in the sense of number theory), generalizing constructions due to Silverman and Tate.
Joint work with Valentino Tosatti.

## Passés

Jeudi 10 juin 14:00-15:00 Rafael Potrie (Montevideo, Uruguay)
Hyperbolicité partielle et dynamique pseudo-Anosov

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Lieu : https://webconf.imo.universite-paris-saclay.fr/b/jer-7cp-7mk

Résumé : Les difféomorphismes partiellement hyperboliques sont omniprésents quand on regarde des phénomènes de type robuste pour les systèmes dynamiques. En plus, il est possible de les détecter par un nombre fini d’itérés. En dimension 3, certaines techniques topologiques permettent d’aborder une classification. J’expliquerai comment la « dynamique grossière » (à grande échelle) permet d’avancer dans cette direction. En particulier, on obtient une classification de ces systèmes dans certaines 3-variétés et classes d’isotopie (notamment les 3-variétés hyperboliques).

Jeudi 3 juin 10:00-15:15 Yann Chaubet, Daniel Perez, Santiago Barbieri, Yassine Guerch, Santiago Martinchich (LMO)
Exposés doctorants

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Lieu : En salle 2L8 et sur https://webconf.imo.universite-paris-saclay.fr/b/jer-7cp-7mk

Résumé :

10h00-10h30 Yann Chaubet (visio)
Géodésiques fermées des surfaces et nombres d’intersection
10h40-11h10 Daniel Perez (présentiel)
Fonctions $\zeta$, arbres et et homologie persistante stochastique
11h30-12h00 Santiago Barbieri (présentiel)
Stabilité générique des systèmes hamiltoniens presque intégrables
14h00-14h30 Yassine Guerch (visio)
Phénomènes de rigidité dans le groupe des automorphismes d’un groupe de Coxeter universel
14h45-15h15 Santiago Martinchich (visio)
Unicité d’attracteurs pour des flots d’Anosov discrétisés

Notes de dernières minutes :

Jeudi 27 mai 14:00-15:00 François Béguin (U. Paris-Nord)
Sur l’existence de sections de Birkhoff

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Lieu : https://webconf.imo.universite-paris-saclay.fr/b/jer-7cp-7mk

Résumé : Dans cet exposé on s’intéressera à des flots non-singuliers sur des variétés fermées de dimension 3. Une situation idéale est celle où il existe une section globale, c’est-à-dire une surface fermée, transverse aux orbites, qui les coupe toutes. L’étude du flot se ramène alors à celle d’un difféomorphisme de surface : l’application de retour des orbites sur la section globale. Cette situation est hélas assez rare — elle implique notamment que la variété sous-jacente a une topologie très particulière. Ceci conduit à affaiblir la notion de section globale, en cherchant à des sections de Birkhoff. En très gros, une section de Birkhoff est une section globale « à un nombre fini d’orbites périodiques près ».
Birkhoff a découvert que les flots géodésiques sur les fibres unitaires tangents des surfaces hyperboliques fermées admettent des sections de Birkhoff. Puis D. Fried a prouvé que tout flot d’Anosov transitif admet une telle section. Ce résultat devient faux pour les flots d’Anosov non-transitifs ; j’expliquerai lesquels de ces flots admettent des sections de Birkhoff. J’essaierai également d’expliquer ce qu’on peut espérer pour des flots plus généraux.

Mercredi 16 juin 16:00-18:00 Omar Mohsen (Münster)
A pseudo-differential calculus for singular filtrations of the tangent bundle and index theorem

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Lieu : Demander le lien Zoom à jean-michel.bismut@universite-paris-saclay.fr

Résumé : Starting from Folland and Stein’s work, people have defined inhomogeneous principal symbols for differential operators in various generalisations. The goal of this symbol is to prove that operators like Hormander’s sum of squares are hypoelliptic. In this talk I will define an inhomogeneous principal symbol associated to a singular filtration of the tangent bundle (a filtration by locally finitely generated submodules of the module of vector fields). I will then define an associated deformation groupoid and a pseudo-differential calculus. In the end I will show how the inhomogeneous principal symbol computes the index of differential operators which are elliptic in such calculus (equivalently any maximally hypoelliptic differential operator)
This is joint work with Androulidakis, van-Erp, Yuncken.

Mercredi 9 juin 16:00-18:00 Shu SHEN (IMJ-PRG)
Coherent sheaves, superconnection, and the Riemann-Roch-Grothendieck formula

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Lieu : Demander le lien Zoom à jean-michel.bismut@universite-paris-saclay.fr

Résumé : In this talk, I will explain a construction of Chern character for coherent sheaves on a closed complex manifold with values in Bott-Chern cohomology. I will also show a corresponding Riemann-Roch-Grothendieck formula, which holds for general holomorphic maps between closed non-Kahler manifolds. Our proof is based on two fundamental objects : the superconnection and the hypoelliptic deformations. This is a joint work with J.-M. Bismut and Z. Wei arXiv:2102.08129.

Mercredi 2 juin 16:00-18:00 Lashi Bandara (Postdam)
Boundary value problems for general first-order elliptic differential operators

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Lieu : Demander le lien Zoom à jean-michel.bismut@universite-paris-saclay.fr

Résumé : The index theorem for compact manifolds with boundary, established by Atiyah-Patodi-Singer in 1975, is considered one of the most significant mathematical achievements of the 20th century.
An important and curious fact is that local boundary conditions are topologically obstructed for index formulae and non-local boundary conditions lie at the heart of this theorem.
Consequently, this has inspired the study of boundary value problems for first-order elliptic differential operators by many different schools, with a class of induced boundary operators taking centre stage in establishing non-local boundary conditions.
The work of Bär and Ballmann from 2012 is a modern and comprehensive framework that is useful to study elliptic boundary value problems for first-order elliptic operators on manifolds with compact and smooth boundary.
As in the work of Atiyah-Patodi-Singer, a fundamental assumption in Bär-Ballmann is that the induced operator on the boundary can be chosen self-adjoint.
All Dirac-type operators, which in particular includes the Hodge-Dirac operator as well as the Atiyah-Singer Dirac operator, are captured via this framework.
In contrast to the APS index theorem, which is essentially restricted to Dirac-type operators, the earlier index theorem of Atiyah-Singer from 1968 on closed manifolds is valid for general first-order elliptic differential operators.
There are important operators from both geometry and physics which are more general than those captured by the state-of-the-art for BVPs and index theory.
A quintessential example is the Rarita-Schwinger operator on 3/2-spinors, which arises in physics for the study of the so-called delta baryons.
A fundamental and seemingly fatal obstacle to study BVPs for such operators is that the induced operator on the boundary may no longer be chosen self-adjoint, even if the operator on the interior is symmetric.
In recent work with Bär, we extend the Bär-Ballmann framework to consider general first-order elliptic differential operators by dispensing with the self-adjointness requirement for induced boundary operators.
Modulo a zeroth order additive term, we show every induced boundary operator is a bi-sectorial operator via the ellipticity of the interior operator.
An essential tool at this level of generality is the bounded holomorphic functional calculus, coupled with pseudo-differential operator theory, semi-group theory as well as methods arising from the resolution of the Kato square root problem.
This perspective also paves way for studying non-compact boundary, Lipschitz boundary, as well as boundary value problems in the L^p setting.

Mercredi 19 mai 16:00-18:00 Gérard Freixas (IMJ-PRG)
Non-abelian Hodge theory and complex Chern-Simons

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Lieu : Demander le lien Zoom à jean-michel.bismut@universite-paris-saclay.fr

Résumé : In this talk I will propose a construction of complex Chern-Simons line bundles on moduli spaces of flat vector bundles on families of Riemann surfaces. The approach is based on Deligne’s functorial approach to characteristic classes in Arakelov geometry, where we replace hermitian metrics by relative flat connections and Bott-Chern secondary classes by Chern-Simons counterparts. Our construction requires an intermediate result on extensions of relative flat connections to global ones, which can be seen as a geometric avatar of Spinaci’s study of variations of twisted harmonic maps in non-abelian Hodge theory. I will discuss some applications to moduli spaces of curves, projective structures and Deligne pairings of line bundles. This is ongoing work with D. Eriksson (Chalmers University of Technology) and R. Wentworth (University of Maryland).

Mercredi 12 mai 16:00-18:00 Antoine Song (Berkeley)
Essential minimal volume and minimizing metrics

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Lieu : Demander le lien Zoom à jean-michel.bismut@universite-paris-saclay.fr

Résumé : One way to measure the complexity of a smooth manifold is to consider its minimal volume, denoted by MinVol, introduced by Gromov, which is simply defined as the infimum of the volume among metrics with sectional curvature between -1 and 1. I will introduce a close variant of MinVol, called the essential minimal volume, which has most of the good’’ properties of MinVol and has also some additional advantages : it is always achieved by some Riemannian metrics which in some sense generalize hyperbolic metrics, moreover it can be estimated for Einstein 4-manifolds and most complex surfaces in terms of topology.

 Département de Mathématiques Bâtiment 307 Faculté des Sciences d'Orsay Université Paris-Saclay F-91405 Orsay Cedex Tél. : +33 (0) 1-69-15-79-56 Département Fermeture du département Laboratoire Formation