Séminaire HORUS

organisé par l'équipe Géométrie

  • Daniele Alessandrini

    Classification of real and complex projective structures with fixed holonomy

    26 octobre 2018 - 14:30Salle de séminaires IRMA

    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.

  • Gye-Seon Lee

    Convex real projective Dehn filling

    26 octobre 2018 - 16:00Salle de séminaires IRMA

    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.

  • Nicolas Tholozan

    Compact relative components in Hermitian character varieties

    28 novembre 2018 - 14:30Salle de séminaires IRMA

    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).

  • Andy Sanders

    An invitation to opers

    28 novembre 2018 - 16:00Salle de séminaires IRMA

    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.

  • Claudio Llosa Isenrich

    Complex hypersurfaces in direct products of Riemann surfaces.

    19 décembre 2018 - 14:00Salle de séminaires IRMA

    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.

  • Lucas Branco

    Low rank orthogonal Higgs bundles and singular Hitchin fibres

    19 décembre 2018 - 15:30Salle de séminaires IRMA

    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.