The Zilber-Pink conjecture is a diophantine finiteness conjecture. It unifies and gives a far-reaching generalization of the classical Mordell-Lang and Andre-Oort conjectures, and is wide open in general.Point-counting results for definable sets in o-minimal structures provide a strategy for proving suitable cases which has had some success, in particular in its use in proving the Andre-Oort conjecture.The course will describe the Zilber-Pink conjecture and the point-counting approach to proving cases of it, eventually concentrating on the case of a curve in a power of the modular curve.We will describe the model-theoretic contexts of the conjectures and techniques, and the essential arithmetic ingredients.

Atttention : La première Leçon aura lieu à l’Ecole polytechnique, Amphithéâtre Becquerel, le 30 janvier à 15hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://www.fondation-hadamard.fr/en/events/advanced-courses/Abstract:Given a vector field in the euclidean space, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth. The theorem looses its validity as soon as the vector field is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields.The course presents a modern view, new results and open problems in the context of flows of irregular vector fields.We develop, in this framework, recent ideas and techniques such as quantitative regularity estimates on the flow of Sobolev vector fields, nonuniqueness of solutions via convex integration, similarity constructions, mixing, enhanced and anomalous dissipation.Such ideas have been proved useful to study nonlinear PDEs as well and we apply these results and techniques in the context of the mathematical understanding of phenomena in fluid dynamics, in particular for the Euler and Navier-Stokes equations and in relation to the Kolmogorov theory of turbulence.

Atttention : La première Leçon aura lieu à l’Ecole polytechnique, Amphithéâtre Becquerel, le 30 janvier à 15hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://www.fondation-hadamard.fr/en/events/advanced-courses/Abstract:Given a vector field in the euclidean space, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth. The theorem looses its validity as soon as the vector field is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields.The course presents a modern view, new results and open problems in the context of flows of irregular vector fields.We develop, in this framework, recent ideas and techniques such as quantitative regularity estimates on the flow of Sobolev vector fields, nonuniqueness of solutions via convex integration, similarity constructions, mixing, enhanced and anomalous dissipation.Such ideas have been proved useful to study nonlinear PDEs as well and we apply these results and techniques in the context of the mathematical understanding of phenomena in fluid dynamics, in particular for the Euler and Navier-Stokes equations and in relation to the Kolmogorov theory of turbulence.

Atttention : La première Leçon aura lieu à l’Ecole polytechnique, Amphithéâtre Becquerel, le 30 janvier à 15hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://www.fondation-hadamard.fr/en/events/advanced-courses/Abstract:Given a vector field in the euclidean space, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth. The theorem looses its validity as soon as the vector field is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields.The course presents a modern view, new results and open problems in the context of flows of irregular vector fields.We develop, in this framework, recent ideas and techniques such as quantitative regularity estimates on the flow of Sobolev vector fields, nonuniqueness of solutions via convex integration, similarity constructions, mixing, enhanced and anomalous dissipation.Such ideas have been proved useful to study nonlinear PDEs as well and we apply these results and techniques in the context of the mathematical understanding of phenomena in fluid dynamics, in particular for the Euler and Navier-Stokes equations and in relation to the Kolmogorov theory of turbulence.

Atttention : La première Leçon aura lieu à l’Ecole polytechnique, Amphithéâtre Becquerel, le 30 janvier à 15hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://www.fondation-hadamard.fr/en/events/advanced-courses/Abstract:Given a vector field in the euclidean space, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth. The theorem looses its validity as soon as the vector field is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields.The course presents a modern view, new results and open problems in the context of flows of irregular vector fields.We develop, in this framework, recent ideas and techniques such as quantitative regularity estimates on the flow of Sobolev vector fields, nonuniqueness of solutions via convex integration, similarity constructions, mixing, enhanced and anomalous dissipation.Such ideas have been proved useful to study nonlinear PDEs as well and we apply these results and techniques in the context of the mathematical understanding of phenomena in fluid dynamics, in particular for the Euler and Navier-Stokes equations and in relation to the Kolmogorov theory of turbulence.

Atttention : La première Leçon aura lieu à l’Ecole polytechnique, Amphithéâtre Becquerel, le 30 janvier à 15hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://www.fondation-hadamard.fr/en/events/advanced-courses/Abstract:Given a vector field in the euclidean space, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth. The theorem looses its validity as soon as the vector field is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields.The course presents a modern view, new results and open problems in the context of flows of irregular vector fields.We develop, in this framework, recent ideas and techniques such as quantitative regularity estimates on the flow of Sobolev vector fields, nonuniqueness of solutions via convex integration, similarity constructions, mixing, enhanced and anomalous dissipation.Such ideas have been proved useful to study nonlinear PDEs as well and we apply these results and techniques in the context of the mathematical understanding of phenomena in fluid dynamics, in particular for the Euler and Navier-Stokes equations and in relation to the Kolmogorov theory of turbulence.

The classical Weyl problem asks whether any Riemannian metric of positive curvature on the sphere S2 can be uniquely realized as the induced metric on the boundary of a convex domain in Euclidean space. In hyperbolic space, there is an analogue which was solved by Alexandrov in the 1950s, but also a dual statement describing the possible third fundamental forms of the boundaries of bounded, convex domains.We will describe those classical results, as well as some conjectural statements and partial results extending them either to convex domains in hyperbolic manifolds, or, more generally, to unbounded convex domains in H3.

I will explain and give some motivation for the notion of quantum ergodicity in an elementary manner, using wave packets. I will then describe an attempt at understanding the defect of uniqueness in this setting, first exhibited for the quantisation of the cat map on the torus. What will be discussed here is the product of a collective work under the alias JMP.

Arithmetic GeometryApril 22-26, 2024at IHES – Centre de conférence Marilyn et James SimonsOn the occasion of Hélène Esnault’s 70th birthday, Marco D’Addezio, IRMA Strasbourg, Kay Rülling, Univ. Wuppertal, and Tanya Srivastava, IIT Gandhinagar, organize a conference in her honor from April 22 to 26, 2024.This conference centers around the mathematical contributions and interests of Hélène Esnault. It aims at bringing together mathematicians with diverse backgrounds, providing a platform to exchange their ideas and foster new collaborations. Registration deadline: January 31, 2024Invited Speakers:Tomoyuki Abe, IPMU – University of Tokyo  Yves André, IMJ-PRG  Emelie Arvidsson, University of UtahBhargav Bhatt, IAS – Princeton University & University of Michigan  Ana Caraiani, Imperial College London  Dustin Clausen, IHES Johan De Jong, Columbia University  Michael Groechenig, University of Toronto  Lars Hesselholt, Nagoya University & University of Copenhagen Katharina Hübner, Goethe-Universität Frankfurt  Moritz Kerz, Universität Regensburg      Marc Levine, Universität Duisburg-Essen  Daniel Litt, University of Toronto   Alexander Petrov, Harvard University  Claude Sabbah, École polytechnique  Peter Scholze, MPIM – University of Bonn   Atsushi Shiho, University of Tokyo  Carlos Simpson, Université Nice-Sophia AntipolisVasudevan Srinivas, SUNY, BuffaloJakob Stix, Goethe-Universität Frankfurt  Scientific Committee: Marco D’Addezio (IRMA Strasbourg), Marcin Lara (Goethe Universität-Frankfurt), Simon Pépin Lehalleur (Radboud Universiteit Nijmegen), Kay Rülling (Universität Wuppertal), Annette Werner (Goethe Universität-Frankfurt), and Lei Zhang (Sun Ya-Tsen Univ. Zhuhai) Hélène Esnault is a mathematician specializing in algebraic and arithmetic geometry. She obtained her PhD in 1976 at the University of Paris VII under the direction of Lê Dũng Tráng. She then completed her habilitation at the University of Bonn in 1985. Afterwards, she was a Heisenberg scholar at the MPI in Bonn and maître de conférence in Paris VII. In 1990, she became a full professor at the Universität Duisburg-Essen. In 2012, she moved to Berlin as the first Einstein Professor at the Freie Universität Berlin, where she became emerita in 2019. She continued her mathematical work and accepted a visiting professor position in 2019 at IAS Princeton. From there, she moved back to Europe in 2020. In the fall 2022, she held the Eilenberg Chair at Columbia University and is currently a part-time professor at Copenhagen University and a Faculty Associate at Harvard University. As a mathematician, she published more than 135 research articles with 45 coauthors covering a wide range of topics. In the following, only a small extract of her influential oeuvre is mentioned. In the 1980’s, she found together with Eckart Viehweg a new method to prove vanishing theorems in algebraic geometry. At the end of the 1990s, she gave with Vasudevan Srinivas and Viehweg the first general construction of an Albanese variety for singular projective varieties over an algebraically closed field. At the beginning of the 2000s, she proved that a Fano variety over a finite field has a rational point, answering positively a conjecture of Lang-Manin. She furthermore showed that a smooth projective variety over a local field with a regular model has a rational point in the special fiber if the étale cohomology of the generic fiber has coniveau 1. With Spencer Bloch and Pierre Berthelot, she proved that Serre’s Witt vector cohomology of a singular proper variety in positive characteristic is the slope <1 part of rigid cohomology, generalizing results of Bloch and Illusie in the smooth case. A spectacular result is her proof with Vikram Mehta in 2010 of a conjecture by Gieseker, which says, there are no non-constant stratified bundles on a geometrically simply connected smooth projective variety over a perfect field of positive characteristic. She proved with Bloch and Moritz Kerz a p-adic infinitesimal version of the Fontaine-Mazur conjecture. A result that recently has drawn much attention is her joint work with Michael Groechenig on the integrality of certain rigid local systems, which was conjectured by Carlos Simpson.Hélène Esnault has mentored about 25 PhD students and even more  postdocs. In recognition of her seminal contributions, she was awarded the Doisteau-Blutet prize of the Academy of Sciences in Paris in 2001, jointly with Eckart Viehweg  the Leibniz Prize in 2003 and received the Cantor medal of the German Mathematical Society in 2019. She received multiple honorary doctorates and was a member of various significant committees, including the Fields Medal Committee of the ICM 2018, the Structure Committee ICM 2022, the Shaw Prize Committee 2021-2020 and 2021-2025, and the Infosys Prize Committee 2023. Additionally, she has served on the editorial boards of several prestigious journals, including the Duke Mathematical Journal (since 1995), Mathematische Annalen (1998–2010), Mathematical Research Letters (since 2007), Algebra and Number Theory (since 2007 as a founding editor), Memoirs of the European Mathematical Society (since 2023), and Acta Mathematica (since 2023). The conference receives partial support from the GRK 2240: Algebro-geometric Methods in Algebra, Arithmetic and Topology

Michel Gros a officiellement pris sa retraite le 1er janvier de cette année. À cette occasion, ses collègues et amis souhaitent lui rendre hommage et célébrer une carrière remarquable, principalement axée sur la géométrie arithmétique, domaine dans lequel il a obtenu des contributions importantes sur des questions variées allant des cohomologies p-adiques et modulo p à la théorie des représentations, en passant par les régulateurs syntoniques. Michel a obtenu un doctorat en 1983 à Orsay, à l’issue d’une thèse intitulée « Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique » dirigée par Luc Illusie. Après un séjour postdoctoral à l’Université de Tokyo auprès de Kazuya Kato, il a été recruté par le CNRS et affecté depuis les années 90 à l’Institut de Recherche Mathématique de Rennes où il participe à la vie scientifique du groupe initialement créé par Pierre Berthelot.Orateurs invités :- Cédric Pépin, Univ. Sorbonne Paris-Nord- Emanuel Reinecke, IHES- Simon Riche, Univ. Clermont Auvergne- Takeshi Tsuji, The University of TokyoOrganisateurs : Ahmed Abbes (CNRS & IHES), Fabrice Orgogozo (CNRS & IMJ-PRG) et Julien Sebag (Univ. de Rennes).

2024 IHES SUMMER SCHOOLOrganizing Committee: Zohar Komargodski (SCGP), Bruno Le Floch (CNRS & LPTHE), Elli Pomoni (DESY), and Masahito Yamazaki (IPMU).Scientific Committee: Anton Kapustin (Caltech), Yuji Tachikawa (IPMU), Xiao-Gang Wen (MIT).The Summer School will be held at the Institut des Hautes Études Scientifiques (IHES) from June 24 to July 05, 2024. IHES is located in Bures-sur-Yvette, south of Paris (40 minutes by train from Paris) – Access mapThis school is open to everybody but intended primarily for young participants, including Ph.D. students and postdoctoral fellows. Applications are closed. All candidates have received an email stating whether they were accepted. 2024 IHES Summer School – Symmetries and Anomalies: a Modern TakeSymmetries play an outsized role in understanding physical phenomena. In quantum systems ranging from condensed matter to high-energy particle physics, symmetries can feature different types of anomalies, which may constrain the dynamics or ruin the model’s consistency. This gives important clues on extensions to the Standard Model, or new topological phenomena in quantum materials. Anomalies have played an essential role in the modern developments of supersymmetric quantum field theories as well as string theory.  Last but not least, their study has influenced and benefitted from different areas of mathematics and in particular algebraic topology.This school will introduce students to the physical and mathematical underpinnings of anomalies including its more mathematical aspects on topological quantum field theory and characteristic classes, with a view toward recent applications to topological phases of matter and strongly coupled gauge theories.  The overarching idea is to have courses from three points of view that build upon each other: that of a mathematician (TFT, category theory, characteristic classes), a high-energy physicist (chiral anomalies and Hooft anomaly matching), and a condensed matter physicist (symmetry-protected and symmetry-enhanced topological order). The school would be suited to PhD students and postdocs coming from these three fields. We will ensure that with several tracks of exercise sessions revisiting background knowledge in math/hep-th/cond-mat as necessary.Courses will range from basic aspects of anomalies of continuous flavour symmetries to cutting-edge topics: conformal anomalies, lattice symmetries, CPT symmetries, higher-form symmetries, higher-group symmetries, as well as a categorical point of view thereon.Speakers:Clay CÓRDOVA (University of Chicago)Clément DELCAMP (IHES)Thomas DUMITRESCU (UCLA)Iñaki GARCÍA ETXEBARRIA (Durham University)Max METLITSKI (MIT)Shu-Heng SHAO (Stony Brook University)Đàm Thanh SƠN (Univ. of Chicago)Yifan WANG (New York U)

A celebrated result by Benoist in the 90s asserts that if G is a semi-simple real-algebraic group and Γ < G is a Zariski-dense semigroup, then the smallest closed cone that contains the Jordan projections {λ(γ) : γ ∈ Γ} is convex and has non-empty interior. In this talk we will focus on analogous concepts for tangent vectors to the character variety Hom(Γ,G)/G, and if time permits we will treat some applications to higher rank Teichmüller theory.