A convex real projective surface is one obtained as the quotient of a properly convex open set in the projective plane by a discrete subgroup of SL(3,R), called the holonomy group, that preserves this convex set. The most basic examples are hyperbolic surfaces, for which the convex set is an ellipse, and the holonomy group is conjugate into SO(2,1). In this case, the eigenvalues of elements of the holonomy group are symmetric. More generally, the asymmetry of the eigenvalues of the holonomy group is a natural measure of how far a convex real projective surface is from being hyperbolic. We study the problem of determining which elements (and more generally geodesic currents) may have maximal eigenvalue asymmetry. We will present some limited initial results that we hope may be suggestive of a bigger picture. Joint work with Florian Stecker.

Following C.T.C. Wall, we say that a group G is of type Fn if it admits a classifying space which is a CW complex with finite n-skeleton. For n = 2, one recovers the notion of being finitely presented. We prove that in a cocompact complex hyperbolic arithmetic lattice with positive first Betti number, deep enough finite index subgroups admit plenty of homomorphisms to Z with kernel of type Fm-1 but not of type Fm. This provides many non-hyperbolic finitely presented subgroups of hyperbolic groups and answers an old question of Brady. This is based on a joint work with C. Llosa Isenrich.

We discuss Agol’s construction of real hyperbolic lattices with arbitrarily small systole. Agol proposed his strategy in dimension 4, where the problem had theretofore been open, but Belolipetsky–Thomson and Bergeron–Haglund–Wise independently showed that this strategy goes through in all dimensions. As a byproduct, one obtains nonarithmetic real hyperbolic lattices in each dimension, and genuinely different examples from those of Gromov and Piatetski-Shapiro. Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide. 

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2023 IHES SUMMER SCHOOLOrganizing Committee: Benjamin Antieau (Northwestern University), Lars Hesselholt (University of Copenhagen / Nagoya University), and Matthew Morrow (CNRS and Université Paris-Saclay)Scientific Committee: Bhargav Bhatt (IAS and Princeton University / University of Michigan), Wiesia Niziol (CNRS and Sorbonne Université), and Akhil Mathew (University of Chicago) The Summer School will be held at the Institut des Hautes Etudes Scientifiques (IHES) from July 10 to 21, 2023. 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. Please note that there won’t be remote transmission through Zoom but mini-courses and talks will be filmed and posted on the IHES YouTube channel in the following days.Application is open until February 15, 2023. In the style of an Oberwolfach Arbeitsgemeinschaft, ten talks will be given by postdoctoral participants on the topic of syntomic and étale motivic cohomology. Once the detailed list of talks is available, postdoctoral applicants will be contacted to ask which talk they would be willing to give.2023 IHES Summer School – Recent Advances in Algebraic $K$-theoryThe last few years have witnessed an explosion of progress in algebraic $K$-theory. Derived algebraic geometry and non-commutative methods have been refined into powerful tools, especially through the theory of localizing invariants. Trace methods have brought $K$-theory and topological cyclic homology closer together than ever before. Perfectoid techniques mean that $K$-theory benefits from the recent progress in $p$-adic cohomology, such as prismatic cohomology. Condensed mathematics provides at long last a uniform approach to the $K$-theory of topological rings. Geometric foundations for motivic stable homotopy theory have been laid and new motivic filtrations have been unearthed.The goal of the Summer School will be to help bring the participants up to date on these exciting developments, via research lectures, mini-courses, and an Arbeitsgemeinschaft on the topic of syntomic and étale motivic cohomology.MINI-COURSES:Johannes Anschutz (University of Bonn) and Arthur-César Le Bras (CNRS and Université de Strasbourg)Dustin Clausen (University of Copenhagen and IHES)Elden Elmanto (Harvard University)Ryomei Iwasa (Université Paris-Saclay)Georg Tamme (University of Mainz)SPEAKERS:Kęstutis ČESNAVIČIUS (CNRS and Université Paris-Saclay)Shane KELLY (University of Tokyo)Moritz KERZ (University of Regensburg)Hana Jia KONG (Institute for Advanced Study)Achim KRAUSE (University of Münster)Thomas NIKOLAUS (University of Münster)Arpon RAKSIT (Massachusetts Institute of Technology)Charanya RAVI (Indian Statistical Institute, Bangalore)Kirsten WICKELGREN (Duke University)Maria YAKERSON (CNRS and Sorbonne Université) This is an IHES Summer School organized in partnership with the Clay Mathematical Institute and in part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 101001474).

How can mathematical results be represented visually in such a way that much information can instantly be read off the images in minimal time and easily?   How can visualization problems contribute to our understanding and progress in the theory from which these tasks arose?   Built of wedges, like « V » or « Lambda », Kontsevich’s directed graphs are a pictorial realization of non-commutative associative star-products: every such directed graph encodes an expression which is differential-polynomial with respect to the coefficients of Poisson brackets (that are placed in the internal vertices) and which is a bi-differential operator with respect to the content of the graph sinks. More general Formality graphs can contain tridents or higher out-degree vertices; Kontsevich’s Formality theorem itself suggests how, by the defition of their weights using integral formulas, the graphs in star-products want to be drawn in the upper half-plane with hyperbolic metric.   The visualization problem which we solve is how the graphs in star-product theory can be drawn — nicely! — in large quantities by using the LaTeX {picture} environment, i.e. the most economical way to draw pictures in scientific texts. In a joint work with S.Kerkhove (Utrecht) we design and implement an algorithm which, given a graph encoding, offers its several drawings in the LaTeX picture environment.   For graphs which do show up in star-products, we obtain the drawings up to order 4 in the deformation parameter. For similar graphs which never show up in the star-product (such as the vacuum diagrams) we explore their visual representations and properties.   Finally, we examine how neural networks can be deployed in two classes of problems about Kontsevich’s graphs and their weights in star-products.      We shall discuss what the Mathematics of « beauty » is in graph visualization: `nice’ is informative, `nice’ is simple, `nice’ is balanced, `nice’ is flexible, `nice’ is whole, `nice’ is unexpected. Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide. 

Séminaire Laurent Schwartz — EDP et applications 

Séminaire Laurent Schwartz — EDP et applications 

The well-known collar or Margulis lemma describes the structure of negatively curved manifolds at mesoscopic scale, in particular it allows to describe these manifolds globally through the « thick-thin decomposition ». This is not sufficient, however, to completely understand the homotopy type of the manifold, even roughly. In this talk I will describe an « arithmetic Margulis lemma » (essentially a consequence of work of E. Breuillard) which allows to describe thin parts at a macroscopic scale in certain circumstances, and how to use it to obtain sharp bounds on the volume of thin parts of arithmetic locally symmetric spaces. This is joint work with M. Frączyk and S. Hurtado. 

If a finite set S densely generates a compact, semisimple Lie group G, then how well does the set of words of length ℓ in S (and S-1) approximate G? We could ask for them words to be an ε-net of G; or, beyond an ε-net, we could ask for the words to be evenly distributed down to a scale of ε; or we could ask for an efficient algorithm to produce a word that lies within ε of any given g in G. An optimal statistical result, with ℓ = O(log 1/ε), was first established by Lubotzky, Phillips, and Sarnak when G = SU(2) for special choices of S; and later generalized by others, but still with some restrictions on S. Not long afterwards, in the context of quantum computing, Solovay and Kitaev independently established an algorithm to find a word with ℓ = O((log 1/ε)a) for any S and (initially) also G = SU(2). I will discuss the current status of different versions of this question, including versions when G might not be compact or S-1 might not be used. I will also discuss my own result, in which I improve the exponent in the (algorithmic) Solovay-Kitaev theorem from the previous best value of a = 3+δ to a = (logφ 2) + 1 + δ < 2.4405.

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Arbre de Noël du GDR »Géométrie non-commutative »La géométrie non-commutative, fondée par Alain Connes, est un domaine de recherche important des mathématiques actuelles. Le GDR « Géométrie Non-commutative » du CNRS regroupe l’ensemble des chercheurs français travaillant sur des thématiques en lien avec ce domaine. L’arbre de Noël, organisé par Amaury Freslon (LMO – Université Paris-Saclay) et co-organisé par Maria-Paula Gomez-Aparicio (Université Paris-Saclay), est la rencontre annuelle de ce GDR. La parole y est donnée aux jeunes chercheurs du domaine, doctorants et post-doctorants, qui peuvent ainsi présenter leurs travaux à la communauté. Quelques exposés de chercheurs confirmés complètent cette rencontre en offrant un panorama des évolutions actuelles.Conférenciers invités :Benjamin Anderson-Sackaney, Université de Caen-NormandieSara Azzali, Università di BariLéonard Cadilhac, Sorbonne UniversitéPurbayan Chakraborty, Université de Bourgogne-Franche-ComtéAlain Connes, IHESClément Dell’Aiera, ENS LyonAmine Marrakchi, ENS LyonPaul Meunier, KU LeuvenGilles Pisier, Sorbonne Université – Texas A&M UniversityArthur Troupel, Université Paris-CitéZhenguo Wei, Université de Bourgogne-Franche-ComtéXu Zhendong, Université de Bourgogne-Franche-ComtéKai Zeng, Université de Bourgogne-Franche-Comté