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  • Conference - Dynamics on representation varieties
    Jun 26, 2017 to Jun 30, 2017

    Rennes, from June 26th to June 30th

    Organization board: Ian Biringer, Ludovic Marquis, Juan Souto

    Scientific board: Uri Bader, Jeffrey F. Brock, Jean-Marc Schlenker

    Numerous areas of mathematics are touched by what could be called Dynamics on representation varieties. For instance one could mention ergodic theory, Riemannian geometry, low-dimensional topology, Teichmüller theory, and so on... The aim of this workshop is to bring together graduate students, recent graduates and experts in these different areas, giving everybody ample time for discussions and collaborations. Next to a number of research talks, three mini-courses by Tsachik Gelander, Francois Labourie and Julien Marché will take place.







    We the organizers of this conference affirm that scientific events must be open to everyone, regardless of race, sex, religion, national origin, sexual orientation, gender identity, disability, age, pregnancy, immigration status, or any other aspect of identity. We believe that such events must be supportive, inclusive, and safe environments for all participants. We believe that all participants are to be treated with dignity and respect. Discrimination and harassment cannot be tolerated. We are committed to ensuring that the Conference Dynamics on representation varieties follows these principles. For more information on the Statement of Inclusiveness, see this dedicated web page.

  • 5 minutes Lebesgue
    Jun 27, 2017

    Les vidéos des exposés seront mises en ligne quelques jours après l'exposé. Vidéothèque

    Prochain exposé :

    27-06-2017:  Axel Rogue
    P et NP

    Sur les 7 problèmes du millénaire de l'institut Clay, 6 sont encore sans solution. Parmi ces problèmes s'en trouve un à l'énoncé particulièrement accessible, et qui constitue la grande question de l'algorithmique : est-ce que les classes de complexité P et NP sont égales ? Le but de l'exposé sera d'expliquer ce que sont P et NP et ce que signifie la question P=NP.

    Lieu

    Rennes

    Exposés à venir:

    12-09-2017:  David Lubicz

    10-10-2017:  Rémi Coulon

    07-11-2017:  Guy Casale

  • School - Analytical aspects of hyperbolic flows
    Jul 3, 2017 to Jul 7, 2017

    Nantes, from July 3rd to July 7th

    Organization board: Sebastien Gouëzel, Laurent Guillopé, Samuel Tapie

    Scientific board: Nalini Anantharaman, Viviane Baladi, Colin Guillarmou, Masato Tsujii

    Hyperbolic flows are dynamical systems with strong chaotic properties, whose study has been started a long time ago, a crucial example being the geodesic flow on negatively curved manifolds. Whereas the qualitative properties of such flows are well understood, their fine quantitative properties (rate of mixing, spectrum...) require more sophisticated tools. They have been studied both from a dynamical point of view (Dolgopyat's techniques) and more analytically: semi-classical methods, initially introduced to study PDEs, have proven very valuable in this context.

    The purpose of this summer school is to make these different techniques accessible to PhD students and young researchers, as well as to give an opportunity for specialists in dynamical systems to learn tools from semi-classical analysis, and conversely. Therefore, the core of this summer school will consist in three introductive mini-courses, completed by a few research talks and question sessions.

    Conference brochure

  • Conference - Young researcher meeting in dynamics and geometry
    Sep 6, 2017 to Sep 8, 2017

    Rennes, from September 6th to September 8th

    Organization board: Françoise Dal'Bo, Frédéric Paulin, Barbara Schapira, Damien Thomine

    Since its creation the Platon network (GDR National Center for Scientific Research n°3341 http: // costia.free.fr / platon/) leads actions towards young researchers in ergodic geometry. The recurrent young researcher meeting is one of the highlights of the year. The goal is to allow about ten PhD students or recent doctors to expose their work and promotes discussions between young and senior researchers. The "Young researcher meeting in dynamics and geometry" follows the spirit of these recurring meetings with an international dimension brought in particular by Swiss and Senegalese networks.

    See also here

    TALKS

    Alexander Adam (UPMC) Resonances for Anosov diffeomorphism

    Kamel Belarif (Université de Bretagne Occidentale) Genericity of weak mixing in negative curvature

    Adrien Boulanger (UPMC) Cascades in affine interval exchanges

    Filippo Cerocchi (Max Planck Institute for Mathematics, Bonn) Rigidity and finiteness for compact 3-manifolds with bounded entropy

    Maria Cumplido Cabello (Université de Rennes 1) Loxodromic actions of Artin-Tits groups

    Nguyen-Bac Dang (Ecole Polytechnique) Degrees of iterates of rational maps

    Laurent Dufloux (Oulu University) Hausdorff dimension of limit sets at the boundary of the complex hyperbolic plane

    Mikolaj Fraczyk (Université Paris-Sud) Mod p homology growth of locally symmetric spaces

    Weikun He (Université Paris-Sud) Sum-product estimates and equidistribution of toral automorphisms

    Cyril Lacoste (Université de Rennes 1) Dimension rigidity of lattices in semisimple Lie groups

    Erika Pieroni (Università di Roma, Sapienza) Minimal Entropy of 3-manifolds

    Fanni M. Selley (Budapest University of Technology) Ergodicity breaking in mean-field coupled map systems

    Nasab Yassine (Université de Bretagne Occidentale) Quantitative recurrence of one-dimensional dynamical systems preserving an infinite measure

    ABSTRACTS

    • Alexander Adam Resonances for Anosov diffeomorphism

    The deterministic chaotic behavior of an invertible map T is appropriately described by the existence of expanding and contracting directions of the differential of T. A special class of such maps consist in Anosov diffeomorphisms. Every 2-by-2 hyperbolic matrix M with integer entries induces such a diffeomorphism on the 2-torus. For all pairs of real-analytic functions on the 2-torus, one defines a correlation function for T which captures the asymptotic independence of such a pair under the evolution T^n as n tends to infinity. What is the rate of convergence of the correlation as n tends to infinity, for instance what is its decay rate? The resonances for T are the poles of the Z-transform of the meromorphic continued correlation function. The decay rate is well-understood if T=M. There are no non-trivial resonances of M. In this talk, I consider small real-analytic perturbations T of M where at least one non-trivial resonance of T appears. This affects the decay rate of the correlation.

    • Kamel Belarif Genericity of weak mixing in negative curvature

    Let M be a manifold with pinched negative sectional curvature. We show that, when M is geometrically finite and the geodesic flow on T^1M is topologically mixing, the set of mixing invariant measures is dense in the set P(T^1M) of invariant probability measures. This implies that the set of weak-mixing measures which are invariant by the geodesic flow is a dense G-delta subset of P(T^1M). We also show how to extend these results to geometrically infinite manifolds with cusps or with constant negative curvature.

    • Adrien Boulanger Cascades in affine interval exchanges

    Avec un échange d'intervalle affine donné vient naturellement une famille de telles dynamiques indexées par le cercle. En effet, la pré-composition par une rotation de l'application initiale définit un autre échange d'intervalle affine. On étudiera cette famille de dynamiques dans un cas particulier à travers la géométrie de la surface affine associée et son groupe de transformation affine.

    An affine interval exchange (AIE) is a piecewise affine map from the circle to itself. Such a map defines a dynamical systems over the circle by iterating it. With an AIE comes naturally a family of AIE indexed by the circle: they are defined by pre-composing the initial AIE by a rotation. The presentation will focus on the study of possible dynamical behaviors of such a family of AIE through a peculiar example.

    • Filippo Cerocchi Rigidity and finiteness for compact 3-manifolds with bounded entropy

    We present some local topological rigidity results for the set S of non-geometric, compact -- with possibly empty boundary and no spherical boundary components --, orientable Riemannian 3-manifolds having torsionfree fundamental group, with bounded entropy and diameter. By "local", we mean that we consider S endowed with the Gromov-Hausdorff-topology. We shall provide examples to show the necessity of the assumptions and discuss some open problems. Moreover, we shall give a proof of the finiteness of the homeomorphism types of the manifolds in S. These are joint works with A. Sambusetti (Rome, Sapienza).

    • Maria Cumplido Cabello Loxodromic actions of Artin-Tits groups

    Artin-Tits groups act on a certain delta-hyperbolic complex, called the ``additional length complex". For an element of the group, acting loxodromically on this complex is a property analogous to the property of being pseudo-Anosov for elements of mapping class groups. A well-known conjecture about mapping class groups claims that "most elements" of the mapping class group of a surface are pseudo-Anosov. In fact, we can prove that a positive proportion is pseudo-Anosov.

    By analogy, we conjecture that ``most'' elements of Artin-Tits groups act loxodromically. More precisely, in the Cayley graph of a subgroup G of an Artin-Tits group, the proportion of loxodromically acting elements in a ball of large radius should tend to one as the radius tends to infinity. We will give a condition guaranteeing that this proportion stays away from zero. This condition is satisfied e.g. for Artin-Tits groups of spherical type, their pure subgroups and some of their commutator subgroups.

    • N'Guyen-Bac Dang Degrees of iterates of rational maps

    In this talk, I will explain what is a rational map, how to define its k-degrees, and I will study the k-degrees of its iterates. I will explain how the study of the growth of these sequences of numbers helps in understanding the dynamics of these maps.

    • Laurent Dufloux Hausdorff dimension of limit sets at the boundary of complex hyperbolic planes

    Consider the standard contact structure on the 3-sphere. The associated subriemannian metric has dimension 4. The Gromov comparison problem asks about how the Hausdorff dimension with respect to this subriemannian metric is related tothe Hausdorff dimension with respect to the usual (Riemannian) metric. We will look at this problem in the case of limit sets of discrete groups of complex hyperbolic isometries.

    • Mikolaj Fraczyk Mod p homology growth of locally symmetric spaces

      I will talk about the growth of the dimension of mod-p homology groups of locally symmetric spaces. Let G be a higher rank Lie group and X its symmetric space and let L be a lattice in G. Results on the rank gradient by Abert, Gelander and Nikolov imply that if L is right angled then for every sequence of subgroups (L_n) of L, the dimensions of the homology groups H_1(X/L_n,Z/pZ) grow sublinearly in the volume of X/L_n. In the special case p=2, I showed that the same statement holds for any sequence of lattices L_n with volume escaping to infinity (even if they are pairwise non-commensurable).

    • Weikun He Sum-product estimates and equidistribution of toral automorphisms

    Bourgain's sum-product theorem is a metric version of Erdős-Szemerédi sum-product theorem. It asserts that a typical set of real numbers grows fast under addition and multiplication. We will present a generalisation of Bourgain's theorem to matrix algebras and discuss how it is motivated by a ergodic problem, namely, quantitative equidistributions of orbits on the d-dimensional torus under sub-semigroups of SL(d,Z).

    • Cyril Lacoste Dimension rigidity of lattices in semisimple Lie groups

    We study actions of discrete groups on classifying spaces (or classifying spaces for proper actions). For instance the hyperbolic plane is a classifying space for proper actions of the group PSL(2,Z) (but not of minimal dimension). Such spaces can be used to compute the cohomology of the group, so we want them to have the lowest possible dimension. This leads us to the definitons of the (proper) geometric dimension and the (virtual) cohomological dimension. These two dimensions are not always equal, we will see it is the case for a lattice in the group of isometries G of a symmetric space of non-compact type without Euclidean factors (such a group is a semisimple Lie group but not necessarily connected). This result has an important consequence called "dimension rigidity", that is, the two dimensions are still equal for a group commensurable to a lattice of G.

    • Erika Pieroni Minimal Entropy of 3-manifolds

    We present the solution of the minimal entropy problem for non-geometric, closed, orientable 3-manifolds (that is, those manifolds which do not admit a com- plete metric locally isometric to one of the eight 3-dimensional model geometries). Together with the results of Besson-Courtois-Gallot for locally symmetric spaces and the work of Soma, Gromov et.al. on the simplicial volume of 3-manifolds and its relation with entropy, this gives a complete picture of the minimal entropy prob- lem for all closed, orientable 3-manifolds. Our work strongly builds on Souto's PhD work (unpublished), filling some gaps in the proof and completing the picture in the case of non-prime manifolds. In detail, we show that the minimal entropy is ad- ditive with respect to the prime decomposition and that for an irreducible manifold X it coincides with the sum of the volume entropies of all the JSJ components of hyperbolic type, each endowed with its complete, hyperbolic metric of nite volume. For the lower bound of MinEnt(X), we adapt Besson-Courtois-Gallot's barycenter method following Souto's ideas; then, we show how this lower bound is realized by producing a sequence of Riemannian metrics gk on X whose volume-entropies tend to

    • Fanni M. SelleyErgodicity breaking in mean-field coupled map systems

    Coupled map systems are simple models of a finite or infinite network of interacting units. The dynamics of the compound system is given by the composition of the (typically chaotic) individual dynamics and a coupling map representing the characteristics of the interaction. The coupling map usually includes a parameter s in [0,1], representing the strength of interaction. The main interest in such models lies in the emergence of bifurcations when s is varied. We first introduce our results for small finite systems. Then we initiate a new point of view which focuses on the evolution of distributions and allows to incorporate the investigation of a continuum of sites.

    • Nasab Yassine Quantitative recurrence of one-dimensional dynamical systems preserving an infinite measure

    We are interested in the asymptotic behaviour of the first return time of the orbits of a dynamical system into a small neighbourhood of their starting points. We study this quantity in the context of dynamical systems preserving an infinite measure. More precisely, we consider the case of Z-extensions of subshifts of finite type. We also consider a toy probabilistic model in order to enlighten the strategy of our proofs.

  • Conference - Geometric Analysis at Roscoff
    Oct 9, 2017 to Oct 13, 2017

    Roscoff, from October 9th to October 13th

    Organization board: Paul Baird, Gilles Carron, Ali Fardoun, Carl Tipler

    Scientific board: Gérard Besson (CNRS, Institut Fourier), Olivier Biquard (ENS Paris), Ahmad El Soufi (Univ. Tours)

    Geometric Analysis is the application and development of PDE tools and technics in Riemannian geometry, it is also a fundamental tool in mathematical physics. Recently, important conjectures has been solved: Poincaré's conjecture, Willmore's conjecture, Lawson's conjecture, Yau-Tian-Donaldson's conjecture and a lot of new tools has been introduced and developed : optimal transport, weak formulation of Ricci curvature, Geometric measure theory. This conference will be an opportunity for specialists from theses different areas to meet and exchange ideas, questions and knowledge.

  • Conference - Lebesgue PHD meeting 2017
    Oct 16, 2017 to Oct 18, 2017

    QR-Code

    Rennes, from October 16th to October 18th

    Organization board: Grégory Boil, Valentin Doli, Caroline Robet, Jérôme Spielmann

    Scientific board: Solène Bulteau, Clément Rouffort, Nasab Yassine

    Depuis trois ans, le Centre Henri Lebesgue soutient les Rencontres doctorales Lebesgue, initiative des doctorants du Labex. Il s'agit de trois journées de conférences durant lesquelles la parole est donnée à des doctorants de tout horizon géographique et mathématique. L'objectif est ainsi de présenter un panel le plus large possible de la recherche mathématique actuelle telle qu'elle est vue et vécue par les doctorants, mais pas seulement... Lors de ces rencontres, trois chercheurs, appelés 'parrains' de l'évènement, sont invités à exposer et ainsi à partager leur expérience personnelle de la recherche d'aujourd'hui. Cette année, les rencontres sont parrainées par :

    Jean-Marc Bardet (SAMM, Université Paris 1);

    Jasmin Raissy (Institut mathématique de Toulouse, Université Paul Sabatier);

    Gabriel Rivière (Laboratoire Paul Painlevé, Université de Lille 1).

    Bien que principalement orientée vers les doctorants, cette conférence se veut également accessible aux étudiants de M2 désireux d’avoir un aperçu des travaux auxquels une thèse en mathématiques peut mener. Il est de plus possible pour des doctorants désireux d'exposer de déposer des propositions sur l'onglet 'Proposer un Exposé'.

  • Complex dynamics and quasi-conformal geometry
    Oct 23, 2017 to Oct 25, 2017

    Our colleague Tan Lei passed away in April 2016. A conference will be held from 23/10/2017 to 25/10/2017 at the University of Angers to honour her memory.

    Scientific Committee

    Etienne Ghys (ENS Lyon)
    John Milnor (Stony Brook)
    Mitsuhiro Shishikura (Kyoto).

    Organizing Committee

    Mohammed El Amrani (Angers)
    Michel Granger(Angers)
    Jean-Jacques Loeb(Angers)
    Laurent Meersseman(Angers)
    Pascale Roesch(Toulouse).

    Provisional list of speakers

    Xavier Buff, Arnaud Cheritat, Nuria Fagella (to be confirmed), Cui Guizhen,Peter Haissinski, John Hamal Hubbard (to be confirmed), Carsten lunde Petersen, Kevin Pilgrim, Mary Rees, Pascale Roesh, Hans Henrik Rugh, Dylan Thurston, Mitsu Shishikura, Giulio Tiozzo.

    More information : page

    The registration process is already open.

School - Flows and Limits in Kähler Geometry

List of (confirmed) speakers

Mini-courses :

Talks :

Titles and abstracts

  • Bo Berndtsson (Chalmers University) : Direct image bundles and variations of complex structures

Given a smooth proper fibration $p:\mathcal X\to B$ and $L$ a line bundle over $\mathcal X$, the direct image $$ E:= p_*(L) $$ is in many cases a holomorphic vector bundle over $B$. Its fibers are the spaces of holomorphic sections of $L$ over the fibers of $p$, $X_t=p^{-1}(t)$, and they can be given various $L^2$-metrics. In case the fibration is of relative dimension $n$ so that the fibers are compact Riemann surfaces, special cases of this situation can be used to study the variation of complex structures on the fibers $X_t$. (The fibers are all diffeomorphic, but their complex structure varies with $t$, so we can view the family $X_t$ as a family of variations of complex structures on one fixed smooth manifold.) When the relative dimension is higher than one the situation is more complicated and one needs to consider also higher direct images. I will discuss the problems that arise in this connection, with previous work of Siu, Schumacher and To-Yeung and some recent joint work with Xu Wang and Mihai Paun.

  • Hans-Joachim Hein (Fordham University) : Tangent cones of Calabi-Yau varieties

It has been known for about 10 years that the classical Calabi-Yau theorem on the existence and uniqueness of Ricci-flat Kahler metrics on smooth complex manifolds with zero first Chern class can be extended to a natural setting of weak Kahler metrics on singular complex varieties. However, until relatively recently nothing was known - even in the simplest nontrivial examples - about the precise asymptotic behavior of these weak Ricci-flat metrics at the singularities of the underlying varieties. I will explain work of Donaldson-Sun, H-Naber and H-Sun that resolves this question in certain cases.

  • Valentino Tosatti (Northwestern University) : Metric Limits of Calabi-Yau Manifolds

In this mini-course I will give an introduction to the study of limits of Ricci-flat Kahler metrics on a compact Calabi-Yau manifold when the Kahler class degenerates to the boundary of the Kahler cone. Analytically, the problem is to prove suitable uniform a priori estimates for solutions of a degenerating family complex Monge-Ampère equations, away from some singular set. Geometrically, this can be used to understand the Gromov-Hausdorff limit of these metrics. And if the manifold is projective algebraic and the limiting class is rational, the limits possess an algebraic structure and are obtained from the initial manifold via contraction morphisms from Mori theory.

  • Jeff Viaclovsky (Wisconsin University) : The geometry of SFK ALE metrics

I will discuss some of the basics of scalar-flat Kaehler (SFK) metrics, and focus on the geometry of SFK metrics which are asymptotically locally Euclidean (ALE). These space arise as "bubbles" in the compactness theory of Calabi's extremal Kaehler metrics. I will also present some of the deformation theory of SFK ALE metrics.

  • Thibaut Delcroix (ENS Paris) : Kähler geometry of horospherical manifolds

Horospherical manifolds form a class of almost homogeneous manifolds whose Kähler geometry is very close to that of toric manifolds. They strictly contain homogeneous toric bundles, to which a lot of results holding for toric manifolds have been extended. I will present horospherical manifolds, trying to convince you that they are not much harder to deal with, and in particular I will present the criterion for K-stability in the Fano case that follows either from my work on spherical varieties, or from a direct, Wang-Zhu type, approach.

  • Eleonora Di Nezza (Imperial College) : Monge-Ampère energy and weak geodesic rays

The recent proof of Demailly's conjecture by Witt Nyström gives another evidence that pluripotential theory play a key role when working with complex Monge-Ampère equations in order to solve problems in differential and algebraic geometry. In this talk we investigate pluripotential tools: we characterise Monge-Ampère energy classes in terms of envelopes. And in order to do that, we develop the theory of weak geodesic rays in a big cohomogy class. We also give a positive answer to an open problem in pluripotential theory. This is a joint work with Tamas Darvas and Chinh Lu.

  • Jakob Hultgren (Chalmers University) : Coupled Kähler-Einstein Metrics

A central theme in complex geometry is to study various types of canonical metrics, for example Kähler-Einstein metrics and cscK metrics. In this talk we will introduce the notion of coupled Kähler-Einstein (cKE) metrics which are k-tuples of Kähler metrics that satisfy certain coupled Kähler-Einstein equations. We will discuss existence and uniqueness properties and elaborate on related algebraic stability conditions. (Joint work with David Witt Nyström)

  • Zakarias Sjostrom Dyrefelt (Université de Toulouse) : K-stability of constant scalar curvature Kähler manifolds

In this talk we introduce a variational/pluripotential approach to the study of K-stability of Kähler manifolds with transcendental cohomology class, extending a classical picture for polarised manifolds. Our approach is based on establishing a formula for the asymptotic slope of the K-energy along certain geodesic rays, from which we deduce that cscK manifolds are K-semistable. Combined with a recent properness result of R. Berman, T. Darvas and C. Lu we further deduce uniform K-stability of cscK manifolds with discrete automorphism group, thus confirming one direction of the YTD conjecture in this setting. If time permits we also discuss possible extensions of these results to the case of compact Kähler manifolds admitting holomorphic vector fields.

Partners

Irmar LMJL ENS Rennes LMBA LAREMA

Affiliation

ANR CNRS Rennes 1 Rennes 2 Nantes INSA Rennes INRIA ENSRennes UBO UBS Angers UBL