# Geometry and Topology seminar. Academic year 2018-2019.

This is the webpage of the seminar of the Research Cluster in Geometry at the Mathematics Department of the University of Luxembourg.

Unless otherwise specified, the seminar will be on Monday, 4-5pm and will take place in Room S-6A (MNO).

Here is a link to the seminars of former years.

Organizers: Clément Guérin, Alan McLeay, Vincent Pecastaing.

## Next seminar

• December 10, 2018: Thibaut Dumont (University of Jyväskylä)

On the volume cocycle of a product of trees

A Euclidean building is the $p$-adic analog of the symmetric space of a semisimple Lie group. Klingler introduced a volume cocycle for Euclidean buildings and used it to prove rigidity statements about group action on buildings. The volume cocycle conjecturally behaves like as a polynomial of degree the dimension of the building but no satisfactory estimate exists at moment. In this seminar, I will present the general construction of Klingler and the solution for the case of a finite product of trees.

## Upcoming seminars

• December 17, 2018: Maxime Fortier Bourque (University of Glasgow)

Low index singularities of the systole function

The systole of a hyperbolic surface is the length of any of its shortest geodesics. Akrout showed that this defines a topological Morse function on the Teichmuller space of the surface. As such, the critical points of the systole function carry information about the topology of moduli space. Schmutz Schaller found a critical point of index $2g-1$ in every genus $g>1$ and conjectured that this was the smallest index possible, because of the virtual cohomological dimension of moduli space calculated by Harer. I will describe a family of counterexamples: for every $c>0$, there exists a closed hyperbolic surface of genus $g$ which is a critical point of index at most $cg$.

## Previous seminars

• December 3, 2018: Celeste Damiani (Leeds University)

A journey through loop braid groups

The study of loop braid groups has been widely developed during the last twenty years, in different domains of mathematics and mathematical physics. They have been called with several names such as motion groups, groups of permutation-conjugacy automorphisms, braid-permutation groups, welded braid groups, untwisted ring groups,...and others! We will give a glance on how this richness of formulations carries open questions in different areas.

• November 26, 2018: Christian El Emam (Università di Pavia)

On immersions of surfaces into $SL(2, \mathbb{C})$ and on a generalization of embedding data.

The aim of the talk is to present complex orthogonal structures on surfaces $S$, which can be seen as complex quadratic forms on $\mathbb{C}TS$ with some conditions of non-degeneracy, and to show some results on their immersions in $SL(2,\mathbb{C} )$ endowed with its Killing form. Totally geodesic immersions correspond to immersions into the space of geodesics of $\mathbb{H}^3$: this observation leads to a correspondence between orthogonal structures with constant curvature and some pairs of projective structures. Using this correspondence, we prove an analogue of the Uniformization Theorem for orthogonal structures as an application of Bers’ Simultaneous Uniformization Theorem.

This is joint work with Francesco Bonsante.

• November 20 and 21, 2018 : SOS meeting
Caution: No Geometry and Topology seminar this week

• November 12, 2018: Florent Schaffhauser (Université de Strasbourg and Universidad de Los Andes)

Higher Teichmüller spaces of orbifolds

The Teichmüller space of a compact 2-orbifold $X$ can be defined as the space of faithful and discrete representations of the fundamental group of $X$ into $PGL(2,\mathbb{R})$. It is a contractible space. For closed orientable surfaces, "Higher analogues" of the Teichmüller space are, by definition, (unions of) connected components of representation varieties of $\pi_1(X)$ that consist entirely of discrete and faithful representations. There are two known families of such spaces, namely Hitchin representations and maximal representations, and conjectures on how to find others. In joint work with Daniele Alessandrini and Gye-Seon Lee, we show that the natural generalisation of Hitchin components to the orbifold case yield new examples of Higher Teichmüller spaces: Hitchin representations of orbifold fundamental groups are discrete and faithful, and share many other properties of Hitchin representations of surface groups. However, we also uncover new phenomena, which are specific to the orbifold case.

• November 5, 2018: Jean Raimbault (Université de Toulouse)

Profinite invariants of arithmetic lattices

Two finitely generated, residually finite groups are said to be profinitely isomorphic to each other if their profinite completions are isomorphic as topological groups. This does not imply that they are isomorphic to each other, and the profinite isomorphism class of a given group is very badly understood in general. This motivates the search for profinite invariants, that is group invariants which depend only on the profinite completion. In this talk I will present a joint work with Holger Kammeyer, Steffen Kionke and Roman Sauer where we prove that the sign of Euler characteristic and $L^2$-torsion is a profinite invariant among arithmetic groups which have the congruence subgroup property.

• October 29, 2018 Miguel Acosta (Université du Luxembourg)

Fundamental domains in the complex hyperbolic plane and its boundary at infinity

After recalling the construction of the complex hyperbolic plane and describing the hypersurfaces called "bisectors", we will study some fundamental domains for subgroups of $PU(2,1)$, namely the ones constructed by Parker-Will and Deraux-Falbel. These domains can be used to prove the discreteness of the groups and to obtain geometric structures on manifolds.

• October 22, 2018: Joan Porti (Universitat Autònoma de Barcelona)

A local characterization of Anosov representations

Let $X=G/K$ be a symmetric space of noncompact type and $\Gamma$ a hyperbolic group. For a representation of $\Gamma$ in $G$, I give a characterization for this representation to be Anosov in terms of the image of finitely many elements of $\Gamma$. I will not assume that the audience is familiar with Anosov representations. This is joint work with M. Kapovich and B. Leeb

• October 17, 2018: Jérôme Betrand (Université de Toulouse)
Caution: Seminar on Wednesday. Time 16:00, Room S6A

Prescribing the Gauss curvature of convex bodies in the hyperbolic space

I will state and prove an hyperbolic analogue of a classical theorem on Euclidean convex bodies due to Alexandrov. It consists in prescribing the shape of a (pointed) convex body given its Gaussian curvature measure (viewed as a measure on the unit sphere). The existence of such a convex body is based on the study of a non-linear analogue of Kantorovitch's dual problem, a standard tool in optimal mass transport. This is joint work with Philippe Castillon.

• October 9, 2018: Zheng Huang (Graduate Center and College of Staten Island, CUNY)
Caution: Seminar on Tuesday. Time 09:45, Room S6A.

On (non)existence of minimal fibrations on mapping tori

Thurston asked whether a mapping torus admits a fibration of closed minimal surfaces and he speculated that the answer is no for mapping tori with short curves. We confirm this with a quantified theorem for C^2 minimal fibrations. This is a computer assisted proof: we prove necessary conditions, then use programs SNAPPY and TWISTER to find examples of mapping tori which satisfy these necessary conditions. This is joint work with B. Wang.

• October 3, 2018: Gilles Courtois (Sorbonne Université)
Caution: Seminar on Wednesday. Time 16:00, Room S6A.

Entropy of manifolds and their fundamental groups

A lemma of Milnor-Svarc says that the entropy of a closed riemannian manifold does not vanish if and only if its fundamental group has exponential growth but it does not provide bounds of the minimal entropy of the manifold in term of the fundamental group. The aim of this talk is to give such bounds when the fundamental group is a hyperbolic group.

• September 17, 2018: Roberto Frigerio (University of Pisa)

Ideal simplicial volume of manifolds with boundary (joint work with M. Moraschini)

The simplicial volume is a homotopy invariant of manifolds defined by Gromov in 1982. Roughly speaking, the simplicial volume of M measures the minimal size of real fundamental cycles for M. It is related to many topological and geometric invariants (most notably, to the Riemannian volume), and it is usually surprisingly difficult to compute.
We define a variation of the simplicial volume which still enjoys many features of its classical counterpart, while being easier to compute. Our new invariant is defined for manifolds with boundary, and differs from the ordinary simplicial volume because ideal simplices are now allowed to appear in fundamental cycles. We discuss some applications of the ideal simplicial volume to the study of maps between hyperbolic 3-manifolds with geodesic boundary.

• September 10, 2018: Julien Keller (Aix-Marseille University)
Caution: Seminar on Monday. Time 15:00, Room S6A.

Constant scalar curvature Kähler cone metrics

We will explain what we think is a good definition of a constant scalar curvature Kähler metric with conic singularities. A construction of such special metrics over certain ruled manifolds will be given. To do so, we present a Fredholm alternative result for the Lichnerowicz operator. This also requires to study Hermitian-Einstein metrics on parabolic vector bundles with good regularity properties. Eventually, we will explain how these results fit in the logarithmic version of Yau-Tian-Donaldson conjecture that relates differential geometry to algebraic geometry. This is a joint work with Kai Zheng (Univ. of Warwick).

• September 5, 2018: Amir Vig (University of California, Irvine)
Caution: Seminar on Wednesday. Time 14:00, Room S6A.

Spectral theory of the ellipse

A famous inverse problem posed by Kac in the 1960s is to determine the shape of a drum from the set of resonant frequencies at which it vibrates. In this talk, I will discuss recent results in this direction pertaining to elliptical drumheads. In particular, I will outline a proof of “spectral rigidity” using a new parametrix for the wave propagator, which makes an interesting connection to billiards and Birkhoff s conjecture.