### Le 04-11-2019 : Szilard Szabo (TU Budapest)

**P = W conjectures in the Painleve 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.*### Le 04-11-2019 : Jan Swoboda (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.<br /><br /><br /> <br /><br /><br /> 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.*### Le 30-09-2019 : 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.*### Le 30-09-2019 : 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).*### Le 24-04-2019 : Clément Guérin (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.*### Le 24-04-2019 : Karin Melnick (Maryland & 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.*### Le 27-03-2019 : Michèle Vergne (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.*### Le 27-03-2019 : 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.*### Le 27-02-2019 : Claudio Meneses Torres (U. 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 KÃƒÂ¤hler forms on the moduli spaces, which I will briefly describe as well.*### Le 27-02-2019 : Georgios Kydonakis (U. Strasbourg)

**On the topology of moduli spaces of stable parabolic G-Higgs bundles in genus g***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.*### Le 19-12-2018 : 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 KÃ¤hler.*### Le 19-12-2018 : 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.*### Le 28-11-2018 : Andy Sanders (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. <br /> <br /> 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.*### Le 28-11-2018 : 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.<br /> <br /> 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).*### Le 26-10-2018 : Gye-Seon Lee (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.*### Le 26-10-2018 : Daniele Alessandrini (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.*