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

### 2020-2021

• 12 July, 2021--Hybrid mode, also in MSA 4.520
Fathi Ben Aribi (Université Catholique de Louvain), "The Teichmüller TQFT volume conjecture for twist knots"
Abstract: In 2011, Andersen and Kashaev defined an infinite-dimensional TQFT from quantum Teichmüller theory. This Teichmüller TQFT yields an invariant of triangulated 3-manifolds, in particular knot complements. The associated volume conjecture states that the Teichmüller TQFT of an hyperbolic knot complement contains the hyperbolic volume of the knot as a certain asymptotical coefficient, and Andersen–Kashaev proved this conjecture for the first two hyperbolic knots. In this talk, after a brief history of quantum knot invariants and volume conjectures, I will present the construction of the Teichmüller TQFT and how we proved its volume conjecture for the infinite family of twist knots, by constructing new geometric triangulations of the knot complements. No prerequisites in quantum topology or hyperbolic geometry are needed. (joint project with E. Piguet–Nakazawa and F. Guéritaud)
• 7 June, 2021--Hybrid mode, also in MNO SALLE DE REUNION
Tommaso Cremaschi (University of Southern California), "Geometry of infinite-type hyperbolic 3-manifolds"
Abstract: In this talk I will give some purely topological construction for hyperbolic 3-manifolds with infinitely generated fundamental group, this will let us constructs LOTS of infinite-type hyperbolic 3-manifolds. Then, I will say something about their deformation space, that is the set of hyperbolic structures that they admit.
• 20 April, 2021
Tengren Zhang (National University of Singapore), "Cusped Hitchin representations and Anosov representations for geometrically finite Fuchsian groups"
Abstract: In this work, we develop a theory of Anosov representation of geometrically finite Fuchsian groups in SL(d,R) and show that cusped Hitchin representations are Borel Anosov in this sense. We establish analogues of many properties of traditional Anosov representations. In particular, we show that our Anosov representations are stable under type-preserving deformations and that their limit maps vary analytically. We also observe that our Anosov representations fit into the previous frameworks of relatively Anosov and relatively dominated representations developed by Kapovich-Leeb and Zhu. This is joint work with Richard Canary and Andrew Zimmer.
• 14 December, 2020
Mohammad Ghomi (Georgia Tech), "Total curvature and the isoperimetric inequality"
Abstract: The classical isoperimetric inequality states that in Euclidean space spheres provide enclosures of least perimeter for any given volume. According to the Cartan-Hadamard conjecture, this inequality may be generalized to spaces of nonpositive curvature. In this talk we discuss an approach to proving this conjecture via a comparison formula for the total curvature of level sets of functions on manifolds. This is joint work with Joel Spruck.
• 23 November, 2020
Andrea Tamburelli (Rice University), "Length spectrum compactification of the SL(3,R)-Hitchin component"
Abstract: Higher Teichmuller theory studies geometric and dynamical properties of surface groups representations into higher rank Lie groups. One of these higher Teichmuller spaces is the SL(3,R)-Hitchin component, a connected component in the SL(3,R)-character variety that entirely consists of faithful and discrete representations that are the holonomies of convex real projective structures on a surface. In a joint work with Charles Ouyang, inspired by Bonahon's interpretation of Thurston's compactification of Teichmuller space by means of geodesic currents, we describe the length spectrum compactification of the SL(3,R)-Hitchin component. We interpret the boundary points as hybrid geometric structures on a surface that are in part flat and in part laminar.
• 20 October, 2020
Sourav Ghosh (Internal Seminar), "Isospectral rigidity"
Abstract: In this talk we will discuss recent results which show that under certain conditions, representations giving rise to isospectral Margulis spacetimes are conjugate with each other via an automorphism of the ambient affine Lie group.
• 15 October, 2020
Zhe Sun (Internal Seminar), "SL_3-webs and tropical A points on surfaces"
Abstract: The Fock and Goncharov duality for SL_2 starts from the relation among the regular function ring of SL_2 representation variety, the Thurston's measured laminations, and the tropical points of the A moduli space (decorated Teichmuller space). Given an ideal triangulation of the decorated surface, we define a mapping class group equivariant bijection from the space of Kuperberg's non-elliptic SL_3-webs up to equivalence to the set A_L of certain tropical rational points of the A_{SL_3,S} moduli space. As a consequence, we provide an explicit mapping class group equivariant bijection between the irreducible linear basis of the regular function ring of SL_3 representation variety and A_L using the SL_3-webs. This is a joint work with Daniel Douglas.

### 2019-2020

• June 8th, 2020 -- Virtual meeting: You can follow the Zoom link here
Brice Loustau (TU Darmstadt), "The hyper-Kähler geometry of minimal hyperbolic germs"
Abstract: We prove that Taubes's moduli space of germs of minimal surfaces in hyperbolic 3-manifolds is hyper-Kähler outside the singular locus of the Jacobi operator, extending known results by Donaldson and others on the almost-Fuchsian defornation space. We interpret this hyper-Kähler structure as a simultaneous unfolding and stitching of the complex symplectic structures on the moduli space of flat SO(3,C)-connections and the cotangent bundle of Teichmüller space. This hyper-Kähler structure has particularly attractive properties, for instance: it is invariant under the mapping class group action, admits the area functional as a Kähler potential, is invariant by the U(1)-action, and coincides with the Feix-Kaledin hyper-Kähler extension of the Weil-Petersson metric. This is joint work with F. Bonsante, A. Sanders, and A. Seppi.
• May 25th, 2020 -- Virtual meeting: Please contact the organizers if you want to attend the session
Hanh Vo (Internal seminar), "Short closed geodesics on cusped hyperbolic surfaces."
Abstract: This talk is about the set of closed geodesics on complete finite type hyperbolic surfaces. For any non-negative integer k, we consider the set of closed geodesics that self-intersect at least k times, and investigate those of minimal length. We show that, if the surface has at least one cusp, their self-intersection numbers are exactly k for large enough k.
• May 4th, 2020 -- Virtual meeting: Please contact the organizers if you want to attend the session
Filippo Mazzoli (Internal seminar), "Para-hyperKähler structures on the space of GHMC AdS-manifolds"
Abstract: A celebrated result by Donaldson asserts that the space of almost-Fuchsian manifolds admits a natural hyperKähler structure, invariant under the action of the mapping class group. Donaldson's construction proceeds by applying an infinite-dimensional version of the symplectic reduction procedure, originally due to Marsden and Weinstein, to the space of smooth sections of a particular vector bundle over a closed surface of genus g. In this talk I will recall the main ingredients of Donaldson’s original work and I will describe how we expect to adapt this construction for the derivation of a "para-hyperKahler structure” on the space of GHMC AdS3 manifolds, where a pseudo-Riemannian metric and 3 symplectic structures coexist with a integrable complex structure and two para-complex structures, satisfying the relations of para-quaternionic numbers.
The topic of this talk is a joint and ongoing project with Andrea Seppi (Institut Fourier, Grenoble).
• April 27th, 2020 -- Virtual meeting: Please contact the organizers if you want to attend the session
Mélanie Theillière (Institut Camille Jordan, Lyon), "Convex Integration Theory with Corrugations"
Abstract: The Convex Integration Theory was developped in the 70s by Gromov. This theory allows to solve differential problems seen as subsets of the jet space and called Differential Relations. In the case of a relation of order $1$, it allows to build a solution $F$ from a section $(x, f(x), L(x))$ of the bundle $J^1(M, W) \to M$ whose image lies in the relation using an iteration of suitable integrations called “Convex Integrations”. Recently this theory led to explicit constructions of $C^1$-isometric embeddings. In this talk, we will propose an alternative formula to the Convex Integrations called Corrugation Process and we will introduce the notion of Kuiper relations. For these relations, the formula is greatly simplified. As an application of this result, we will give an idea of the construction of a new immersion of $RP^2$ and we will state a Nash-Kuiper $C^1-$isometric embedding theorem in the case of totally real maps.
• April 20th, 2020 -- Virtual meeting : Please contact the organizers if you want to attend the session
Alan McLeay (Internal seminar), "Homeomorphic subsurfaces and the omnipresent arcs."
Abstract: No surface S of finite-type admits a subsurface homeomorphic to S, unless the inclusion map is homotopic to the identity. To prove this fact, one only needs to count the genus or number of punctures on the surface. For surfaces of infinite-type, we will show that more "interesting" homeomorphic subsurfaces always occur. This then leads naturally to a subclass of arcs and a new graph on which many big mapping class groups act.
• March 16th, 2020 POSTPONED -- Room S6A
Andy Sanders (Heidelberg university), "TBA"
• March 23rd, 2020 POSTPONED -- Room S6A
Francesco Bonsante (Università degli Studi di Pavia), "TBA"
• April 6th, 2020 POSTPONED -- Room S6A
Zili Wang (University of Paris-Est Marne-la-Vallée), "TBA"
• April 13th, 2020 POSTPONED -- Room S6A
Julie Déserti (Université de Nice-Sophia Antipolis), "TBA"
• 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).