Heidelberg Geometry Seminar

Heidelberg Geometry Seminar & RTG Colloquium schedule, Winter 2016-17

Heidelberg Geometry Seminar, 13:00 - 14:00 (Seminarraum 4).

RTG colloquium, 1:30 - 2:30 (Lecture in 5.104 Konferenzraum, Coffee break in 5.303 Common Room).

5

Date	Speaker	Title	Arrival/Departure	Talk type
October 2016
18	Pierre Will (Grenoble)	On representations of the Whitehead link group in PU(2,1).	Oct 17 -19	Seminar, Heidelberg
25	Bruno Duchesne			RTG, Karlsruhe
November
1	Holiday!
8	Jean-Marc Schlenker (Luxembourg)	The renormalized volume of quasifuchsian manifolds.	Nov 8 - 8	RTG, Heidelberg
15	Léo Brunswic (Avignon)	Decorated Teichmüller spaces, singular space-times and polyhedral surfaces.	Nov 14 - 17	Seminar, Heidelberg
22	Arie Levit			RTG, Karlsruhe
24 11-12 pm (SR8)	Xin Nie (Seoul)	Affine deformations of SL(3,R)-Hitchin representations and invariant surfaces.	Dec 21-25	Seminar, Heidelberg
29	Lotte Hollands (Oxford)	Spectral networks, abelianization and the T[3] theory.	Nov 28 - 30	Seminar, Heidelberg
December
6	Julien Marché (Paris)	A Jones polynomial for curves on surfaces.	Dec 5-6	RTG, Heidelberg
13				RTG, Karlsruhe
14 12-13 pm (SR3)	Elena Frenkel (Strasbourg)	Euler formula and Lexell problem for hyperbolic triangles	Dec 13-14	Seminar, Heidelberg
January 2017
10	Stephan Tillmann (Sydney)	Moduli spaces of projective structures on manifolds	Jan 9-12	RTG, Heidelberg
17	Clara Löh (Regensburg)	The uniform boundary condition without bounded cohomology	Jan 17-18	Seminar, Heidelberg
24				RTG, Karlsruhe
31	Camilla Felisetti (Bologna)	A desingularization of the moduli space of Higgs bundles on a smooth projective curve	Jan 30- Feb 1	Seminar, Heidelberg
February
2. 11 pm SR8	Anja Randecker (Toronto)	A class of infinite translation surfaces where almost every direction is uniquely ergodic	Jan 30-Feb 3	Seminar, Heidelberg
7	Karin Melnick (Maryland)	Limits of local automorphisms of geometric structures		RTG, Heidelberg

On representations of the Whitehead link group in PU(2,1).

Pierre Will

In this talk, I will discuss representations of the fundamental group of the Whitehead link complement in PU(2,1), the isometry group of the complex hyperbolic plane. My goal is to describe an example of a spherical CR uniformisation of the Whitehead link complement. This is an example of an (X,G)-structure on the Whithead link complement, where X is the 3-sphere, seen as the boundary at infinity of the complex hyperbolic plane, and G is PU(2,1). These structures appear quite naturally when studying discrete subgroups of PU(2,1), but only very few examples of hyperbolic 3-manifolds admitting such a uniformisation are known. This talk will combine results of two joint works with Antonin Guilloux and John Parker.

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The renormalized volume of quasifuchsian manifolds.

Jean-Marc Schlenker

Quasifuchsian hyperbolic manifolds have infinite volume, but a well-defined "renormalized" volume, obtained through a regularization procedure. This renormalized volume has remarkable properties, for instance it provides a Kähler potential for the Weil-Petersson metric on Teichmüller space. We will recall one construction and some key properties of the renormalized volume, and then explain how recent results on its relation to other viewpoints on quasifuchsian manifolds has led to progress in understanding the topology of 3-manifolds or the Weil-Petersson geometry of Teichmüller space.

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Decorated Teichm üller spaces, singular space-times and polyhedral surfaces.

Léo Brunswic

The Teichmüller space parametrizes the complete hyperbolic metrics on a given surface up to equivalence. In 1987, Penner introduced a so-called decorated Teichmüller space (a fiber bundle over the usual Teichmüller space) in order to obtain a stratification of the Teichmüller space. His construction can be re-explained using flat space-times and BTZ-extensions of flat space-times. This leads to correspondences between complete hyperbolic surfaces, Cauchy-compact space-times with BTZ and compact singular Euclidean surfaces. As by products, we obtain a parametrization of Cauchy-compact flat space-times with BTZ and a the existence of polyhedral Cauchy-surfaces in every such space-time.

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Affine deformations of SL(3,R)-Hitchin representations and invariant surfaces.

Xin Nie

Consider representations of a surface fundamental group into the affine transformation group of R³. The moduli space of such representations with Fuchsian (resp. Hitchin) linear part was first studied by Geoffrey Mess (resp. François Labourie). We shall first review the known results, especially those about constant mean/Gaussian curvature invariant surfaces. We then discuss the more general problem of finding a constant affine Gaussian curvature surface in certain convex domains of R³ without assuming any group action. This is work in progress with Francesco Bonsante and Andrea Seppi.

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Spectral networks, abelianization and the T[3] theory

Lotte Hollands

A spectral network is a collection of trajectories on a (punctured) Riemann surface. Given a spectral network we can define a notion of "abelianization", which relates flat SL(K) connections on the Riemann surface to flat C^* connections on a covering. For any spectral network abelianization gives a construction of a local Darboux coordinate system on the moduli space of flat SL(K) connections. In this seminar we focus on a particularly rich example for rank K=3, with interesting applications to WKB analysis and to quantum physics. This is based on work (in progress) with Andy Neitzke.

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A Jones polynomial for curves on surfaces.

Julien Marché

In a recent work, T. Koberda et R.Santharoubane have studied representations of surface groups with the following property: every simple curve has an image of finite order. These representations come from topological quantum field theory (TQFT) and depend on an integer k called level. I will explain that when k goes to infinity, these representations are controlled by a sort of Jones polynomial which may distinguish simple curves from the others. This is joint work with R. Santharoubane.

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Euler formula and Lexell problem for hyperbolic triangles

Elena Frenkel

My talk will be about an area formula in terms of side lengths for triangles in plane hyperbolic geometry and its geometrical interpretation. The proof of this formula is analogous to a proof given by Leonhard Euler in the spherical case. I will speak in particular about the Lexell problem, that is, the problem of finding the locus of vertices of triangle of fixed area and fixed base.

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Moduli spaces of projective structures on manifolds

Stephan Tillmann

Strictly convex projective manifolds of finite volume generally behave like hyperbolic manifolds. Hyperbolic manifolds of dimension at least 3 have a unique hyperbolic structure of finite volume. In contrast, there are examples of strictly convex projective structures of finite volume on 3- manifolds that can be deformed. I will describe some facts, examples and questions concerning the moduli spaces of these structures. This will touch on joint work with Cooper, Long, Porti, Wong and Haraway.

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The uniform boundary condition without bounded cohomology

Clara Löh

(Joint work with Daniel Fauser.) The uniform boundary condition on a normed chain complex requires the existence of controlled fillings for all cycles. The uniform boundary condition naturally comes up in the context of glueing results for simplicial volume. Matsumoto and Morita showed that the singular chain complex of spaces with amenable fundamental group satisfies the uniform boundary condition, using bounded cohomology. We give a direct geometric proof of this fact in the aspherical case. This proof admits generalisations to integral variations and hence has applications to integral variations of simplicial volume.

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A desingularization of the moduli space of Higgs bundles on a smooth projective curve

Camilla Felisetti

Let X be a smooth projective curve of genus g over C. The character variety M_B parametrizing conjugacy classes of representations from the fundamental group of X into SL(2,C) is an affine irreducible singular projective variety. The famous Non Abelian Hodge theorem, proved in this case by S.Donaldson and N.Hitchin, states that there is a real analytic isomorphism between M_B and the quasi projective singular variety M_Dol which parametrizes semistable Higgs bundles of rank 2 and degree 0 on X. During the seminar I will present a desingularization of this moduli space, explaining some possible applications to the so called P=W conjecture, stated and proved by De Cataldo, Hausel and Migliorini in the case of rank 2 and degree 1.

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A class of infinite translation surfaces where almost every direction is uniquely ergodic

Anja Randecker

Translation surfaces can be obtained from gluing finitely many polygons along parallel edges of the same length. In recent years, people have asked what happens when you glue infinitely instead of finitely many polygons. From that question the field of infinite translation surfaces has evolved.
It turns out that the behaviour of infinite translation surfaces is in many regards very different and more diverse than in the finite case. For instance, Kerckhoff, Masur, and Smillie showed in 1986 that on a finite translation surface the geodesic flow is uniquely ergodic in almost every direction. This is not at all true for infinite translation surfaces in general.
However, in this talk, I will introduce a class of infinite translation surfaces where the statement remains true. I will recall the ideas of the proof by Kerckhoff, Masur, and Smillie and show how the proof has to be adapted and why that class of infinite translation surfaces was chosen. The presented work is joint with Kasra Rafi.

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Limits of local automorphisms of geometric structures

Karin Melnik

The automorphism group of a rigid geometric structure is a Lie group. In fact, the local automorphisms form a Lie pseudogroup; this property is often taken as an informal definition of rigid geometric structure. In which topology is this the case? The classical theorems of Myers and Steenrod say that C⁰ convergence of local isometries of a smooth Riemannian metric implies C^∞ convergence; in particular, the compact-open and C^∞ topologies coincide on the isometry group. I will present joint results with C. Frances in which we prove the same result for local automorphisms of smooth parabolic geometries, a rich class of geometric structures including conformal and projective structures.