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

### Meetings

The seminar usually takes place on Monday, 4-5pm in Room S-6A (MNO)

### Next seminar

• March 16th, 2020 POSTPONED -- Room S6A
Andy Sanders (Heidelberg university), "TBA"

### Upcoming sessions

• March 23rd, 2020 POSTPONED -- Room S6A
Francesco Bonsante (Università degli Studi di Pavia), "TBA"
• April 6th, 2020 -- Room S6A
Zili Wang (University of Paris-Est Marne-la-Vallée), "TBA"
• April 13th, 2020 -- Room S6A
Julie Déserti (Université de Nice-Sophia Antipolis), "TBA"
• April 27th, 2020 -- Room S6A
Mélanie Theillière (Institut Camille Jordan, Lyon), "TBA"

### Previous seminars

• March 9th, 2020 -- Room S6A
Janko Latschev (University of Hamburg), "The algebra of symplectic field theory"
Abstract: The goal of symplectic field theory is to associate invariants to contact manifolds and symplectic cobordisms between them. In this talk, I will give a gentle introduction to the subject, concentrating on those aspects that help explain the algebraic form of the theory.
• February 24th, 2020 -- Room S6A
Thierry Barbot (Université d'Avignon), "Orbital equivalence classes of Anosov flows in circle bundles over surfaces"
Abstract: E. Ghys proved in 1981 that every Anosov flow on a circle bundle $M$ over a closed surface $S$ is orbitaly equivalent to a finite covering of the geodesic flow of $S$. I will present a recent result showing that on $M$, the number of orbital equivalence classes on $M$ is $1$ if the degree $n$ of the covering is odd, and $2$ if $n$ is even.
The key fact is the determination of the number of orbits of the action of the mapping class group of $S$ on the subgroups of the fundamental group of $T^1S$ of index $n$.
• February 17th, 2020 -- Room S6A
Nariya Kawazumi (University of Tokyo), "The mapping class group orbits in the framings of compact surfaces"
Abstract: We compute the mapping class group orbits in the homotopy set of framings of a compact connected oriented surface with non-empty boundary. If the genus of the surface is greater than 1, it is a slight modification of Johnson's computation for spin structures. There is a new phenomenon if the genus is 1. In this talk, I would like to explain some reasons why we need to consider framings. This work is already published in Quart. J. Math. 69 (2018), 1287–1302
• December 18th, 2019 -- Room S6A
CAUTION: Seminar on Wednesday at 10am
Indira Chatterji (Université de Nice), "Introduction to median spaces."
Abstract: Median metric spaces are a class of geometric subspaces of $L^1$ spaces, generalizing trees. I will give definitions and the first examples, and show some use of those spaces in geometric group theory.
• December 16th, 2019 -- Room S6A
Hongming Nie (Hebrew University of Jerusalem), "Bounded hyperbolic components for Newton maps."
Abstract: In context of Kleinian groups and the quotient manifolds, there is characterization of precompact “hyperbolic component". According to Sullivan’s dictionary between Kleinian groups and complex dynamics, it is of interest to study the boundedness (precompactness) of hyperbolic components in the moduli space of complex rational maps. In this talk, I will focus on the moduli space of degree $d \ge 2$ Newton maps, which has complex dimension d-2, and give a complete description for the bounded hyperbolic components when d=4. It is based on the work with K. Pilgrim and with Y. Gao, respectively.
• December 9th, 2019 -- Room S6A
John Parker (Durham University), "Non-arithmetic lattices"
Abstract: A lattice in a semi-simple Lie group is a discrete subgroup with finite co-volume. An arithmetic group is a subgroup of a linear algebraic group corresponding to integral points. By fundamental work of Borel and Harish-Chandra, all arithmetic groups are lattices. By work of Margulis, Corlette, Gromov and Schoen any lattice that is not arithmetic is (up to commensurability) contained in SO(n,1) or SU(n,1). Gromov and Piatetski-Shapiro constructed examples of non-arithmetic lattices in SO(n,1) for all n at least 2. In this talk I will survey what is known about SU(n,1): There are many lattices in SU(1,1); nothing is known about SU(n,1) for n at least 4; a few examples are known for n=2 and n=3. These examples may be described from several rather different points of view.
• November 27th, 2019 -- Room S6A
CAUTION: Seminar on Wednesday at 2pm
Andrés Sambarino (Sorbonne Université, IMJ-PRG), "Pressure forms on pure imaginary directions"
Abstract: Anosov groups are a class of discrete subgroups of semi-simple algebraic groups analogue to what is known as convex-co-compact groups in negative curvature. Thermodynamical constructions equip the (regular points of the) moduli space of Anosov representations from $\Gamma$ to $G$ with natural positive semi-definite bi-linear forms, known as pressure forms. Determining whether such a pressure form is Riemannian requires non-trivial work. The purpose of the lecture is to explain some geometrical meaning of these forms, via a higher rank version of a celebrated result for quasi-Fuchsian space by Bridgeman-Taylor and McMullen on the Hessian of Hausdorff dimension on pure bending directions. This is work in collaboration with M. Bridgeman, B. Pozzetti and A. Wienhard.
• November 25th, 2019 -- Room S6A
Javier Aramayona (Instituto de Ciencias Matemáticas), "The first cohomology group of pure mapping class groups"
Abstract: We will start by recalling the proof of a classical result of Powell, which asserts that mapping class groups of finite-type surfaces of genus at least three have trivial abelianization. In stark contrast, we will show that this is no longer the case if the surface is allowed to have infinite type; more concretely, we will explain how to construct non-trivial integer-valued homomorphisms from mapping class groups of infinite-genus surfaces. Further, we will give a description the first integral cohomology group of pure mapping class groups in terms of the first homology of the underlying surface. This is joint work with Priyam Patel and Nick Vlamis.
• November 18th, 2019 -- Room S6A
Wai Yeung Lam (Internal seminar), "Holomorphic quadratic differentials on graphs"
Abstract: In the classical theory, holomorphic quadratic differentials are tied to a wide range of objects, e.g. harmonic functions, minimal surfaces and Teichmüller space. We present a discretization of holomorphic quadratic differentials that preserves such a rich theory. We introduce discrete holomorphic quadratic differentials and discuss their connections to the surface theory and Teichmüller theory. On one hand, holomorphic quadratic differentials are related to discrete minimal surfaces via a Weierstrass representation. On the other hand, they arise from deformations of circle packings on surfaces with complex projective structures.
• November 11th, 2019 -- Room S6A
Vincent Pecastaing (Internal seminar), "Actions of higher-rank lattices on conformal and projective structures"
Abstract: The main idea of Zimmer's program is that in real-rank at least 2, the rigidity of lattices of semi-simple Lie groups makes that their actions on closed manifolds are understandable. After a short survey giving a more precise idea of Zimmer's conjectures and their context, I will give recent results about conformal and projective actions of cocompact lattices. The fact that these geometric structures do not carry a natural invariant volume is a major motivation. We will see that the real-rank is bounded above like when the ambient Lie group is acting, and that at the critical value, the manifold is globally isomorphic to a model homogeneous space. The proofs rely in part on an "invariance principle" recently introduced by Brown, Rodriguez-Hertz and Wang, which guarantees the existence of finite invariant measures in some dynamical context.
• November 4th, 2019 -- Room S6A
Sourav Ghosh (Internal seminar), "Affine Anosov representations"
Abstract: In this talk I will define affine Anosov representations and explain their relation to proper affine actions of a word hyperbolic group. Moreover, I will also explain how affine Anosov representations capture infinitesimal versions of certain special Anosov representations.
• October 21st, 2019 -- Room S6A
Christian El Emam (Università di Pavia), "On immersions of surfaces into $PSL(2, \mathbb C)$ and on a tool for constructing holomorphic maps into its character variety."
Abstract: We will discuss immersions of surfaces into $PSL(2, \mathbb C)$ equipped with its complex killing form: in a sense, the formalism provided by the study of such immersions is able to unify the theories about immersions of surfaces into $\mathbb H^3$, $AdS^3$, $\mathbb S^3$ and $dS^3$. We will also show that a holomorphic variation of the immersion data into $PSL(2, \mathbb C)$ provides a holomorphic variation of the holonomy. Time permitting, we will provide an example of this result concerning landslide flow and smooth grafting. This is joint work with Francesco Bonsante.
• October 14th, 2019 -- Room S6A
Martin Leguil (Université Paris-Sud), "Spectral determination of open dispersing billiards"
Abstract: In an ongoing project with P. Bálint, J. De Simoi and V. Kaloshin, we have been studying the inverse problem for a class of open dispersing billiards obtained by removing from the plane a finite number of smooth strictly convex scatterers satisfying a non-eclipse condition. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift of finite type that provides a natural labeling of all periodic orbits. We show that the Marked Length Spectrum determines the curvatures of the scatterers at the base points of 2-periodic orbits, and the Lyapunov exponents of each periodic orbit. Besides, we show that it is generically possible, in the analytic category and for billiard tables with two (partial) axial symmetries, to determine completely the geometry of those billiards from the purely dynamical data encoded in their Marked Length Spectrum.
• September 30th, 2019 -- Room S6A
Funda Gültepe (University of Toledo), "Space filling curves, Cannon-Thurston maps and boundaries of curve complexes"
Abstract: Given a hyperbolic 3-manifold which fibers over the circle with hyperbolic surface fiber, the inclusion map between the fiber and the manifold can be extended continuously to a map, resulting in a space-filling Peano curve. Such continuous extension of a map, in particular extension to a map between corresponding boundaries is called a 'Cannon-Thurston map' . In this talk we will discuss existence of Cannon-Thurston maps in different settings. In particular, we will explain how to construct a Cannon-Thurston map for the boundary of 'surviving' curve complex of a surface with punctures. Joint work with Christopher Leininger.
• September 23rd, 2019 -- Room S6A
Nariya Kawazumi (University of Tokyo), "Gate double derivatives"
Abstract: Recently Turaev introduced the notion of a gate derivative on the group ring of the fundamental group of an oriented surface. Its double version gives a topological interpretation of a double divergence, which connects the homotopy intersection form and the Turaev cobracket. We will explain the definition of a gate double derivative and some of its properties including a topological proof of the formula connecting the double divergence and the Turaev cobracket. This is a joint work with Anton Alekseev, Yusuke Kuno and Florian Naef.
• September 16th, 2019 -- Room S6A
Tian Yang (Texas A&M University), "Recent progress on the volume conjecture for the Turaev-Viro invariants"
Abstract: In 2015, Qingtao Chen and I conjectured that at the root of unity $\exp(2\pi \sqrt{-1}/r)$ instead of the usually considered root $\exp(\pi \sqrt{-1}/r)$, the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic 3-manifold grow exponentially with growth rates respectively the hyperbolic and the complex volume of the manifold. In this talk, I will recall known results about this conjecture and present a recent joint work with Giulio Belletti, Renaud Detcherry and Effie Kalfagianni on an infinite family of cusped hyperbolic 3-manifolds, the fundamental shadow links complement, for which the conjecture is true.
• September 9th, 2019 -- Room S6A
Masashi Yasumoto (Osaka City University), "Discrete Weierstrass-type representations"
Abstract: In this talk we consider discrete surfaces with Weierstrass-type representations. In the smooth case, these representations for surfaces are powerful tools for constructing surfaces and analyzing their global behaviors. By the same reason, Weierstrass-type representations for discrete surfaces are important both for investigating the theory itself and for expanding our knowledge of global behaviors. We introduce how to derive the formulae in terms of transformation theory for discrete Omega surfaces, and introduce how these are related to a discrete version of holomorphic functions. This talk is partly based on joint work with Mason Pember and Denis Polly (TU Wien).