Séminaire HORUS

organisé par l'équipe Géométrie

  • Claudio Meneses Torres

    On the geometry of moduli spaces of stable parabolic Higgs bundles in genus 0

    27 février 2019 - 14:00Salle de séminaires IRMA

    In this talk I will describe work in progress on the construction of explicit geometric models for moduli spaces of stable parabolic Higgs bundles in genus 0 and rank 2, and explain how the nature of such construction elucidates the wall-crossing behavior of the moduli spaces in question under variations of parabolic weights. This work is motivated by some results related to the cohomology of natural Kähler forms on the moduli spaces, which I will briefly describe as well.

  • Georgios Kydonakis

    On the topology of moduli spaces of stable parabolic G-Higgs bundles in genus g

    27 février 2019 - 15:30Salle de séminaires IRMA

    For G a semisimple real Lie group, we will identify in this talk particular subspaces of the parabolic G-Higgs bundle moduli space for the cases when G is split real and when G is of Hermitian type. Using a correspondence betweenn parabolic Higgs bundles and orbifold Higgs bundles, along with a version of the Beauville-Narasimhan-Ramanan correspondence in this setting, one can show that these subspaces are actually connected components of the moduli space. A special emphasis will be given for the case when G=Sp(2n,R). Joint work with Hao Sun and Lutian Zhao.

  • Michèle Vergne

    Conditions de Horn et carquois

    27 mars 2019 - 14:00Salle de conférences IRMA

    Travail commun avec Welleda Baldoni et Michael Walter. Nous donnons des conditions inductives qui caractérisent les positions de Schubert de sous-représentations de la représentation générale d'un carquois de vecteur dimension donnée. Ce critère généralise le critère sur les conditions d'intersection de cellules de Schubert dans une grassmannienne. Nous donnons (comme dans Horn) des applications géométriques à l'image de l'application moment.

  • Andrea Seppi

    Isometric embeddings of the hyperbolic plane into Minkowski space

    27 mars 2019 - 15:30Salle de conférences IRMA

    Minkowski space of dimension 2+1 is the Lorentzian analogue of Euclidean 3-space. It is well-known that there exists an isometric embedding of the hyperbolic plane in Minkowski space, which is the analogue of the embedding of the round sphere in Euclidean space. However, differently from the Euclidean case, the embedding of the hyperbolic plane is not unique up to global isometries. In this talk I will discuss several results on the classification of these embeddings, and explain how this problem is related to Monge-Ampère equations, harmonic maps, and Teichmüller theory. This is joint work with Francesco Bonsante and Peter Smillie.

  • Clément Guérin

    Topologie des variétés de caractères du groupe libre

    24 avril 2019 - 14:00Salle de conférences IRMA

    Dans cet exposé, nous calculerons des groupes d’homotopie de variétés de caractères du groupe libre dans un groupe de Lie complexe semi-simple. Pour obtenir ces résultats, il faut décrire les singularités de variétés de caractères. On s’intéressera tout particulièrement aux sous-groupes de Borel-de-Siebenthal des groupes de Lie semi-simples qui font apparaître des singularités orbifoldes sur la variété des caractères. On donnera, en particulier, des exemples quand le groupe de Lie est simplement connexe. Ce travail est un travail commun avec Sean Lawton et Daniel Ramras.

  • Karin Melnick

    The Lorentzian Lichnerowicz Conjecture and a Conformal D'Ambra Theorem

    24 avril 2019 - 15:30Salle de séminaires IRMA

    The group of conformal transformations of the round sphere is significantly bigger than the isometries, which form a compact group. The Lichnerowicz Conjecture, proved by Ferrand and by Obata, says that, whenever the conformal group of a compact Riemannian manifold (M,g) is noncompact, then (M,g) is conformally equivalent to the round sphere. I will survey progress on the analogue of this conjecture in conformal Lorentzian geometry.

  • Federica Fanoni

    Big mapping class groups acting on homology.

    30 septembre 2019 - 14:00Salle de séminaires IRMA

    To try and understand the group of symmetries of a surface, its mapping class group, it is useful to look at its action on the first homology of the surface. For finite-type surfaces this action is fairly well understood. I will discuss joint work with Sebastian Hensel and Nick Vlamis in which we deal with infinite-type surfaces (i.e. whose fundamental group is not finitely generated).

  • Brice Loustau

    Computing equivariant harmonic maps

    30 septembre 2019 - 15:30Salle de séminaires IRMA

    I will present effective methods to compute equivariant harmonic maps, both discrete and smooth. The main setting will be equivariant maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive the convergence of the discrete heat flow to an energy minimizer. We also examine center of mass methods after showing a generalized mean value property for smooth harmonic maps. We conclude by showing convergence of our method to smooth harmonic maps as one takes finer and finer meshes. We feature a concrete illustration of these methods with Harmony, a computer software with a graphical user interface that we developed in C++, whose main functionality is to numerically compute and display equivariant harmonic maps.

  • Szilard Szabo

    P = W conjectures in the Painleve cases

    4 novembre 2019 - 14:00Salle de séminaires IRMA

    We start by stating two conjectures on non-abelian Hodge theory of a Riemann surface: the P = W conjecture of de Cataldo, Hausel and Migliorini and its geometric counterpart due to Simpson. We then explain the geometry of the real four-dimensional moduli spaces of Higgs bundles associated to the Painlevé I-VI equations and of the corresponding wild character varieties. We end by sketching the proof of the conjectures in the Painlevé VI case.

  • Jan Swoboda

    Moduli spaces of parabolic Higgs bundles: their ends structure and asymptotic geometry

    4 novembre 2019 - 15:30Salle de séminaires IRMA

    Moduli spaces of Higgs bundles are interesting mathematical objects from various point of views: as holomorphic objects, generalizing the concept of holomorphic structures on vector bundles, as topological objects, relating to surface group representations, and as analytic objects since they admit a description through a nonlinear PDE.



    In my talk, I will mostly take up this latter point of view and give an introduction to Higgs bundles on Riemann surfaces both in the smooth and the parabolic setting. In the parabolic case, i.e. in the situation where the Higgs bundles are permitted to have poles in a discrete set of points, I will discuss recent joint work with L. Fredrickson, R. Mazzeo and H. Weiss concerning the asymptotic geometric structure of their moduli spaces. Here the focus lies on the hyperkähler metric these spaces are naturally equipped with. One implication of a recent conjectural picture due to Gaiotto-Moore-Neitzke suggests that this metric is asymptotic to a so-called semiflat model metric which comes from the description of the moduli space as a completely integrable system. Building on earlier work by several groups of authors, we will present an extension of their results to this singular setting. We shall also discuss several open questions in the case where the Riemann surface is a four-punctured sphere and these moduli spaces turn out to be gravitational instantons of type ALG.

  • Anne Parreau

    Dégénérescences de représentations maximales et courants géodésiques

    2 décembre 2019 - 14:00Salle de séminaires IRMA

    Les dégénérescences de représentations maximales d'un groupe de surface dans Sp(2n,R) peuvent être vues comme des représentations maximales dans Sp(2n,F), où F est un corps réel clos non archimédien, agissant sur son immeuble de Bruhat-Tits associé. J'expliquerais comment associer à  une telle dégénérescence un courant géodésique sur la surface, et je montrerais quelques applications. Il s'agit d'un travail en commun avec Marc Burger, Alessandra Iozzi, et Beatrice Pozzetti.

  • Andrés Sambarino

    Le Hessien de la dimension de Hausdorff

    2 décembre 2019 - 15:30Salle de séminaires IRMA

    La composante de Hitchin est une composante connexe préférée de la
    variété des caractères X(\pi_1S,G)=\hom(\pi_1S,G)/G, où S est une
    surface fermée de caractéristique d'Euler négative et G est un groupe de Lie
    simple déployé. Des constructions thermodynamiques munissent cette
    composante des formes bilinéaires symétriques dites "formes de
    pression". Ces formes sont invariantes par l'action naturelle du
    groupe modulaire de la surface dans X(\pi_1S,G).

    Le but de l'exposé est d'expliquer une interprétation géométrique de
    quelques unes de ces formes, généralisant ainsi un résultat célèbre de
    Bridgeman-Taylor et McMullen concernant le Hessien de la dimension de
    Hausdorff dans l'espace des représentations quasi-Fuchsiennes. Ceci
    est un travail en commun avec M. Bridgeman, B. Pozzetti et A.
    Wienhard.