## HORUS Seminar

Funded by the European Union’s Horizon 2020 Research and Innovation Programme under Grant agreement No 795222 (HORUS Project) and the University of Strasbourg Institute of Advanced Study (USIAS).

The seminar takes place once a month at IRMA, in the afternoon. There is a coffee break between the two talks, from 3 to 3:30pm.

Please note that the style for this webpage (and, as a matter of fact, this seminar) looks suspiciously similar to this one.

## Speakers, titles and abstracts

• Monday 15 June, 2020 (Big Blue Button Online Talk: HORUS Seminar Room) I recommend using the Firefox browser. The room will be accessible 15 minutes before the talk. (^_^)
• (14:00-15:00) Beatrice Pozzetti (Heidelberg), Real spectrum compactification of character varieties.

I will discuss joint work with Burger, Iozzi and Parreau in which we investigate properties of a natural (real) algebraic compactification of character varieties, and of their semi-algebraic subsets. After describing how points at infinity in such compactification correspond to equivalence classes of actions on buildings, I will explain the relation with the Thurston-Parreau marked length spectrum compactification and mention applications to Hitchin and maximal character varieties.

• (15:30-16:30) Nguyen-Thi Dang (Heidelberg), Mélange topologique de flots hyperboliques homogènes.

Soit G un groupe de Lie semisimple de type non-compact, c'est-à-dire sans facteur compact et Γ un sous-groupe discret. Considérons un tore déployé maximal A. On s'intéresse à l'action de flots non triviaux de la forme φt ⊂ A agissant à droite sur Γ \ G. Dans le cas de SO(n,1)0, cela correspond au flot des repères agissant sur le fibré des repères de Γ \ Hn. Celui-ci, par un résultat de Maucourant-Schapira est topologiquement mélangeant lorsque Γ est Zariski-dense.

Dans cet exposé, on supposera que G est éventuellement de rang supérieur, i.e. dim A ≥ 2, et que Γ est Zariski-dense, pas forcément un réseau. Je commencerai par définir le mélange topologique, préciserai la famille de systèmes dynamiques qui m'intéresse et énoncerai un critère de flots loxodromiques lorque ZG(A) est abélien. Ensuite je définirai le cone de Benoist et les sous-ensembles invariants naturels de G pour étudier ces actions. Enfin, je donnerai les grandes lignes de la preuve.

• CANCELLED -- Monday 11 May, 2020 (IRMA Seminar Room) -- CANCELLED
• (14:00-15:00) Marco Boggi (Minas Gerais), TBA.

TBA.

• (15:30-16:30) Joao Pedro Dos Santos (Paris 6), TBA.

TBA.

• CANCELLED -- ¡¡¡Special Session!!! --> Tuesday 14 April, 2020 (IRMA Conference Room) -- CANCELLED
• (14:00-15:00) Yohan Brunebarbe (CNRS & Université de Bordeaux), TBA.

TBA.

• (15:30-16:30) Olivier Benoist (CNRS & ENS Paris), TBA.

TBA.

• Monday 9 March, 2020 (IRMA Seminar Room)
• (14:00-15:00) Gabriele Rembado (Genève), Hyper-Kaehler geometric quantisation.

We will review the geometric quantisation of moduli spaces of unitary flat connections on surfaces. This was used to construct compact quantum Chern-Simons theory.

Then we will pose the problem of quantising moduli spaces of flat connections for complex reductive groups, with the aim of constructing complex quantum Chern-Simons theory. If time allows we will explain how to solve this problem in genus one.

• (15:30-16:30) Louis-Clément Lefèvre (Essen), Déformation des représentations des groupes fondamentaux des variétés algébriques complexes non-compactes.

Nous étudions localement les variétés de représentations de groupes fondamentaux de variétés algébriques complexes lisses. Il s'agit de schémas dont les points complexes paramètrent de telles représentations à valeur dans un groupe algébrique linéaire. En une représentation donnée, la structure de l'anneau local complété à la variété des représentations contient des informations sur les déformations formelles de cette représentations et, ultimement, peut donner des obstructions sur la topologie des variétés considérées.

Ceci a d'abord été décrit par Goldman-Millson dans le cas des variétés compactes, avec des méthodes de déformations formelles et d'algèbres de Lie différentielles graduées. Pour traiter du cas non-compact nous revoyons ceci avec des méthodes de déformations dérivées et de théorie de Hodge mixte.

• ¡¡¡Special session!!! --> Tuesday 18 February, 2020 (IRMA Conference Room)
• (11:00-12:00) Sebastián Hurtado Salazar (University of Chicago), Random walks by homeomorphisms on the line and orderability of lattices (Joint work with Bertrand Deroin).

The standard random walk in the integers is known to be recurrent, it passes through any integer infinitely many times. We will discuss a generalization of this theorem for random walks given by homeomorphisms of the line due to Deroin-Navas-Kleptsyn-Parwani and discuss some applications to the theory of left-orderable groups. Using this Theorem we will show that cocompact lattices in simple Lie groups of higher rank are not left-orderable groups (as a consequence, do not act by homeomorphisms in the line or the circle), a conjecture due to Dave Witte-Morris. This geometrically can be interpreted as saying that every (topological) line bundle over a compact locally symmetric space of higher rank is a trivial bundle. We will try to explain the ideas from a geometric standpoint and the talk should be accessible to people with no background in dynamics.

• Monday 17 February, 2020 (IRMA Seminar Room)
• (14:00-15:00) Nermin Salepci (Université de Lyon), Pavages et estimations des nombres de Betti de sous-complexes aléatoires dans un complexe simplicial fini.

On va introduire d'abord des sous-complexes aléatoires dans un complexe simplicial fini et donner des bornes supérieures pour les nombres de Betti de ces sous-complexes. On va améliorer ces bornes à l’aide d’empilement qu’on construit en utilisant un pavage sur les complexes simpliciaux fini qu’on va définir. Il s'agit d'un travail en commun avec Jean-Yves Welschinger.

• (15:30-16:30) Martin Deraux (Université Grenoble-Alpes), Réseaux non-arithmétiques dans PU(n,1).

On s'intéressera aux réseaux (sous-groupes discrets de covolume fini) dans les groupes d'isométries d'espaces symétriques à courbure négative. Grâce à des travaux importants de super-rigidité dûs à Margulis, Corlette, Gromov-Schoen, on sait que pour la plupart des espaces symétriques, tous les réseaux sont arithmétiques (en gros donnés par l'ensemble des matrices entières dans un groupe défini sur les rationnels).

Les seuls espaces qui échappent ces résultats de super-rigidité sont les espaces hyperboliques réels et complexes, pour lesquel le problème de la classification des réseaux (à commensurabilité près) reste ouverte. J'expliquerai les rares constructions connues, et quelques résultats récents permettant d'établir des ponts entre ces constructions.

• Monday 2 December, 2019 (IRMA Seminar Room)
• (14:00-15:00) Anne Parreau (Université Grenoble Alpes), Dégénérescences de représentations maximales et courants géodésiques.

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.

• (15:30-16:30) Andrés Sambarino (CNRS & Université Paris 6), Le Hessien de la dimension de Hausdorff.

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.

• Monday 4 November, 2019 (IRMA Seminar Room)
• (14:00-15:00) Szilárd Szabó (TU Budapest), P = W conjectures in the Painlevé cases.

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.

• (15:30-16:30) Jan Swoboda (Ruprecht Karls Universität Heidelberg), Moduli spaces of parabolic Higgs bundles: their ends structure and asymptotic geometry.

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.

• Monday 30 September, 2019 (IRMA Seminar Room)
• (14:00-15:00) Federica Fanoni (CNRS & Université de Strasbourg), Big mapping class groups acting on homology.

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).

• (15:30-16:30) Brice Loustau (TU Darmstadt), Computing equivariant harmonic maps.

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.

• Wednesday 24 April, 2019 (IRMA Conference Room)
• (14:00-15:00) Clément Guérin (University of Luxembourg), Topologie des variétés de caractères du groupe libre.

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.

• (15:30-16:30) Karin Melnick (University of Maryland and MPIM-Bonn), The Lorentzian Lichnerowicz Conjecture and a Conformal D'Ambra Theorem.

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.

• Wednesday 27 March, 2019 (IRMA Conference Room)
• (14:00-15:00) Michèle Vergne (Université Paris 7), Conditions de Horn et carquois.

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.

• (15:30-16:30) Andrea Seppi (CNRS & Université Joseph Fourier Grenoble), Isometric embeddings of the hyperbolic plane into Minkowski space.

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.

• Wednesday 27 February, 2019 (IRMA Conference Room)
• (14:00-15:00) Claudio Meneses Torres (Christian-Albrechts-Universität zu Kiel), On the geometry of moduli spaces of stable parabolic Higgs bundles in genus 0.

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.

• (15:50-16:30) Georgios Kydonakis (Université de Strasbourg), On the topology of moduli spaces of stable parabolic G-Higgs bundles in genus g>1.

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.

• Wednesday 19 December, 2018 (IRMA Seminar Room)
• (14:00-15:00) Claudio Llosa Isenrich (Université Paris-Sud), Complex hypersurfaces in direct products of Riemann surfaces.

I will discuss smooth complex hypersurfaces in direct products of Riemann surfaces and present a classification in terms of their fundamental groups. This answers a question of Delzant and Gromov on subvarieties of products of Riemann surfaces for the smooth codimension one case. I will then proceed to explaining how the techniques developed in the proof can be applied to answer the three factor case of Delzant and Gromov's question which subgroups of a product of surface groups are Kähler.

• (15:30-16:30) Lucas Branco (MPIM-Bonn), Low rank orthogonal Higgs bundles and singular Hitchin fibres.

According to mirror symmetry, complex Lagrangians in the Higgs bundle moduli space for a complex group are related to hyperkahler subvarieties of the Higgs bundle moduli space for the Langlands dual group. After discussing some general constructions, we focus on this duality for complex Lagrangians arising from two real forms of SO(4,C) and explain how our results relate to the conjectural picture.

• Wednesday 28 November, 2018 (IRMA Seminar Room)
• (14:30-15:30) Nicolas Tholozan (CNRS & ENS Paris), Compact relative components in Hermitian character varieties.

Let (\Gamma) be the fundamental group of a sphere with $$n\geq 3$$ holes and $$G$$ be the Hermitian Lie group $$\mathrm{SU}(p,q)$$. We call \emph{relative component} of the character variety $$\mathrm{Hom}(\Gamma,G)/G$$ a subset consisting of representations with fixed conjugacy classes on the peripheral elements of $$\Gamma$$ and fixed Toledo invariant. We will see that, perhaps surprisingly, an open subset of the character variety $$\mathrm{Hom}(\Gamma,G)/G$$ is foliated by \emph{compact} relative components. When $$G = \mathrm{SU}(1,1) \simeq \mathrm{SL}(2,\mathbb R)$$, we gave with Bertrand Deroin a nice geometric description of these components and of the representations therein. In higher rank, the construction is much less explicit and transits via the non-Abelian Hodge correspondence (this is a joint work with Jérémy Toulisse).

• (16:00-17:00) Andy Sanders (Ruprecht Karls Universität Heidelberg), An invitation to opers.

In classical Riemann surface theory, complex projective structures play a significant role through their relation to quadratic differentials, ordinary differential equations, and function theory. In the 1990's, Beilinson-Drinfeld gave a generalization of complex projective structures (called opers), which makes sense for a general complex simple Lie group G: to wit, opers for the special linear group in two dimensions recovers the case of complex projective structures. The Beilinson-Drinfeld definition depends, in particular, on the notion of a Borel subgroup inside of G. As a Borel subgroup is a particular case of a parabolic subgroup P, this begs the question of extending the definition of opers to allow more general parabolic subgroups. In this talk, we will explain recent work with Brian Collier which gives this generalization yielding the notion of a (G,P)-oper, and in the course of doing so, give a survey of the history outlined above. After this, we will explain the structure theory of (G,P) opers in the particular case of the special linear group in even complex dimension with P being the stabilizer of a half dimensional plane. If there is time, we will describe the situation for general pairs (G,P) and the relationship to maximal variations of Hodge structure.

• Friday 26 October, 2018 (IRMA Seminar Room)
• (14:30-15:30) Daniele Alessandrini (Ruprecht Karls Universtät Heidelberg), Classification of real and complex projective structures with fixed holonomy.

Consider the following problem: given a fixed subgroup of PGL(n,R) or PGL(n,C), we want to classify all real and complex projective structures on some closed manifold whose holonomy is in the given subgroup. In this problem, the topology of the closed manifolds is not fixed in advance, and all possible topologies need to be determined. We can answer this question in some special cases. For example, consider a Fuchsian subgroup of SL(2,R), embedded diagonally in SL(2n,R). We can classify all the RP^{2n-1} and the CP^{2n-1}-structures on closed manifolds with holonomy contained there. Some obvious ones are the quotients the domains of discontinuity. We can construct more via grafting, and we prove that all of them are of this form. This is joint work with Bill Goldman and Qiongling Li.

• (16:00-17:00) Gye-Seon Lee (Ruprecht Karls Universtät Heidelberg), Convex real projective Dehn filling.

Thurston's hyperbolic Dehn filling theorem states that if the interior of a compact 3-manifold M with toral boundary admits a complete finite volume hyperbolic structure, then all but finitely many Dehn fillings on each boundary component of M yield 3-manifolds which admit hyperbolic structures. In this talk, I will explain that although Dehn filling is not possible in d-dimensional hyperbolic geometry for d > 3, it is possible in the category of convex real projective d-orbifolds for d = 4, 5, 6. Joint work with Suhyoung Choi and Ludovic Marquis.