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.
Following ideas of Iriyeh and Shibata I'll explain a proof of the three dimensional centrally symmetric Mahler conjecture, namely that for every convex set $K$ in three space, such that $K=-K$, if $K^*$ denotes the projective dual, then the product volume: $vol(K)vol(K^*)$ is maximized when $K$ is a cube or a octahedron.
We use a construction of Epstein to prove a conjecture of Labourie regarding constant curvature surfaces in quasi-Fuchsian manifolds.
Path geometry and CR structures on real 3 manifolds were studied by E. Cartan. There is an interesting local geometry with curvature invariants and an interesting global geometry of those structures which are flat. We will review these geometries and discuss a notion of FLAG STRUCTURE which includes both geometries, its curvature invariants and the associated flat manifolds which may be thought as totally real embeddings into flag manifolds.
In this talk, we consider the rigidity problem of the hexagonal triangulated plane under the piecewise linear conformal changes introduced by Luo. In 2013, Wu, Gu and Sun showed the rigidity under the assumption all angles lying in $[\delta,\pi/2-\delta]$, for some $\delta>0$. In this talk, we improve their results by releasing the angle restriction to Delaunay condition. This is a joint work with Huabin Ge and Shiguang Ma.
Finding lattices in $PU(n,1)$ has been one of the major challenges of the last decades. One way of constructing a lattice is to give a fundamental domain for its action on the complex hyperbolic space.
One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle.
In this talk we will see how this construction can be used to build fundamental polyhedra for all Deligne-Mostow lattices in $PU(2,1)$.
The program aiming to understand group actions preserving a rigid geometric structure has developped significantly since the pioneering works of R. Zimmer and M. Gromov, almost four decades ago.
We will present in this talk, how tools coming from coarse geometry can bring new perspectives in the subject, especially in studying actions of discrete groups.
For positive integers $p$ and $q$ consider a quadratic form on $\mathbb{R}^{p+q}$ of signature $(p,q)$ and let $O(p,q)$ be its group of linear isometries. We study counting problems in the Riemannian symmetric space of $PSO(p,q)$ and in the pseudo-Riemannian hyperbolic space of signature $(p,q-1)$. The space $X$ of $q$-dimensional subspaces of $\mathbb{R}^{p+q}$ on which the quadratic form is negative definite is the Riemannian symmetric space of $PSO(p,q)$. Let $S$ be a totally geodesic copy of the Riemannian symmetric space of $PSO(p,q-1)$ inside $X$. We look at the orbit of $S$ under the action of a projective Anosov subgroup of $PSO(p,q)$. For certain choices of such a subgroup we show that the number of points in this orbit which are at distance at most $t$ from $S$ is asymptotically purely exponential as $t$ goes to infinity. We provide an interpretation of this result in the pseudo-Riemannian hyperbolic space of signature $(p,q-1)$, as the counting of lengths of space-like geodesic segments in the orbit of a point.
Let $M$ be a hyperbolic finite-volume $3$-manifold. To any representation of its fundamental group is associated a volume, extending the hyperbolic volume of $M$. So we get a function "volume" defined on the representation variety. This function has fascinating properties. After presenting the construction of this function, I will speak about maximal volume rigidity, the differential of this function and links with the Mahler measure of some bivariate polynomials. (This is in part a joint work with J. Marché)
In this talk, we will consider generalizations of Minkowski's second theorem to Riemannian and Finsler manifolds. For example we will explain why graphs, Finsler tori or Finsler surfaces with normalized volume always admit a $\mathbb{Z}_2$-homology basis induced by closed geodesics whose length product is bounded from above by some constant depending only on their topology. Based on joint work with S. Karam and H. Parlier.
In 1942 P. Alexandrov proved that every Euclidean metric on the 2-sphere with conical singularities of positive curvature can be uniquely realized (up to isometry) as the induced metric on the boundary of a convex 3-dimensional polytope. It provided a complete inner description of such metrics and was used in the development of a general theory of metrics with nonnegative curvature.
Various authors gave several generalizations of this result. In particular, Jean-Marc Schlenker proved a similar statement about hyperbolic cusp-metrics on surfaces of genus > 1 (realized in Fuchsian manifolds). Another proof was obtained by François Fillastre. Both of them used the non-constructive "deformation method".
In our talk we introduce the discrete curvature functional for polytopal manifolds and use it to give a new variational proof in the ideal Fuchsian case.
We also devote some time to discuss the perspectives of this approach towards other problems. If the time permits, we also mention the relation with discrete uniformization theory.
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 $cg$.
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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).
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.