The G&T seminar
Welcome to the webpage of the seminar of the Research Cluster in Geometry at the Department of Mathematicsof the University of Luxembourg.
Organizers: Mingkun Liu, Nathaniel Sagman, Danyu Zhang.
20232024
Next talk

Monday 17 June 2024  4pm, chalk room MNO 1st floorAlex Moriani (Université Côte d'Azur)Title: Polygonal surfaces in pseudohyperbolic spacesAbstract: A polygonal surface in the pseudohyperbolic space is a complete maximal surface bounded by a lightlike polygon with finitely many vertices. These surfaces admit several characterizations : being asymptotically flat or having finite total curvature. In this talk we will explain some constructions coming from nonpositive curvature geometry to prove the equivalence, for a maximal surface, between being polygonal and having finite total curvature.
Future talks

Monday 1 July 2024  4pmHomin Lee (Northwestern University)Title: TBAAbstract: TBA
Previous talks

Wednesday 12 June 2024  4pm, chalk room MNO 1st floorGeorg Gruetzner (University of Luxembourg)Title: Representation theory of hyperbolic groupsAbstract: Does every hyperbolic group admit a uniformly bounded representation with a proper cocycle of "optimal" growth? In this talk I will take on the challenge to convince you, why this question is interesting and why one may (or may not) believe in it. I will assume minimal background knowledge in representation theory. Instead, I will take you on a ride through almost 150 years of representation theory, starting with Frobenius, and Weyl on compact groups, over HarishChandra's work on semisimple Lie groups to modern research on Kazhdan's property (T) and Haagerup's property.

Monday 3 June 2024  4pm, MNO 1.020Carl Lutz (TU Berlin)Title: The discrete uniformization mapAbstract: In the last decades a rich theory of discrete conformal equivalence has been developed, which is closely related to circle patterns and hyperbolic polyhedra. The corresponding discrete uniformization theorems provide a map from the space of discrete conformal classes to the Teichmüller space of the underlying surface. In this talk I will give an overview of different approaches to discrete conformal equivalence and discuss what is known about the discrete uniformization map.

Monday 27 May 2024  4pm, chalk room MNO 1st floorDamien Gayet (Institut Fourier)Title: Local topology of random projective complex hypersurfacesAbstract: The generic complex projective hypersurfaces benefit a magical property: their topology depends only on the degree of a defining polynomial. If we fix a small ball and look at the trace of the hypersurface in it, then of course this property no longer holds. However, if the hypersurface is taken at random, I will explain that for large degrees, the average local topology remembers the global one.

Friday 24 May 2024  2:30pm, chalk room MNO 1st floorRym Smaï (University of Strasbourg)Title: An explicit construction of the maximal extension of a globally hyperbolic conformally flat spacetimeAbstract: The physical theory of general relativity suggests that our universe is modelized by a four dimensional manifold equipped with a metric of signature (,+,+,+), called Lorentzian metric, which satisfies Einstein equations. In 1969, ChoquetBruhat and Geroch established the existence of a unique maximal development of a given initial data for the Einstein equations. These solutions fit within the general framework of globally hyperbolic spacetimes. There is a partial order relation on globally hyperbolic spacetimes. Following the work of ChoquetBruhat and Geroch, the questions of the existence and the uniqueness of a maximal extension of a globally hyperbolic spacetime arise naturally. In this talk, I will discuss these questions in the context of globally hyperbolic conformally flat spacetimes. In 2013, C. Rossi positively answered both questions in this specific context. However, her proof has the unsatisfactory feature that it does not provide any description of the maximal extension. I will present an alternative, constructive proof of this result. This approach is based on the concept of enveloping space, within which the maximal extension will be realized. After defining the enveloping space, I will illustrate this concept with some examples.

Monday 13 May 2024  4pm, MNO 1.020Peter Buser (EPFL)Title: Algorithms around simple closed geodesics on hyperbolic surfacesAbstract: Let S be a compact hyperbolic surface. In contrast to the general case the simple closed geodesics on S are not dense. Hence there are—in general quite small—“forbidden regions” where no simple closed geodesic can enter. How big are these regions, how do they look like, can we visualise them? The talk shall be about algorithms that enumerate geodesics and algorithms that draw them. An obstacle is numerical accuracy: is what we see in the outcoming figures really what there is? We present an approach to accuracy based on length decreasing algorithms. With the help of an auxiliary triangle group we hope to extend the method to the computation of the matrices of the Fuchsian group that represents S. The research is jointly with Antonio Costa, Hugo Parlier and KlausDieter Semmler.

Monday 6 May 2024  4pm chalk room MNO 1st floorBruno Dular (University of Luxembourg)Title: Convex cocompact hyperbolic threemanifolds are determined by their pleating laminationAbstract: At the center of a convex cocompact hyperbolic threemanifold lies its convex core, which is responsible for most of its geometry. The boundary of the convex core consists of disjoint pleated surfaces: hyperbolic surfaces that are pleated along a geodesic lamination, with the amount of bending recorded by a transverse measure. Thurston conjectured that the geometry of the threemanifold is uniquely determined by this induced pleating measured lamination. In a joint work with JeanMarc Schlenker, we prove that the pleating lamination indeed determines the geometry of the threemanifold. This provides a parametrization of the deformation spaces of convex cocompact hyperbolic threemanifolds by internal geometric data, in contrast with the AhlforsBers parametrization by the conformal structure at infinity.

Monday 29 April 2024  4pm chalk room MNO 1st floorMartin MionMouton (Max Planck Institute for Mathematics in the Sciences, Leipzig)Title: Lorentzian metrics with conical singularities and bifoliations of the torusAbstract: The constant curvature Lorentzian metrics having a finite number of conical singularities offer new examples of geometric structures on the torus, naturally generalizing the analogous Riemannian case. In the latter, works of Troyanov show that the data of the conformal structure and of the angles at the singularities entirely classify the metrics with conical singularities. In this talk, we will introduce the Lorentzian metrics with conical singularities and construct some examples, and we will present a work in progress concerning their classification. This classification is linked with topological equivalences between pairs of transverse foliations, and with the dynamics of interval exchange maps.

Monday 22 April 2024  4pm chalk room MNO 1st floorQiongling Li 李琼玲 (Nankai University)Title: Index and total curvature of minimal surfaces in nonpositively curved symmetric spacesAbstract: We prove two main theorems about equivariant minimal surfaces in an arbitrary nonpositively curved symmetric spaces extending classical results on minimal surfaces in Euclidean space. First, we show that a complete equivariant branched immersed minimal surface in a nonpositively curved symmetric space of finite total curvature must be of finite Morse index. It is a generalization of the theorem by FischerColbrie, GulliverLawson, and Nayatani for complete minimal surfaces in Euclidean space. Secondly, we show that a complete equivariant minimal surface in a nonpositively curved symmetric space is of finite total curvature if and only if it arises from a wild harmonic bundle over a compact Riemann surface with finite punctures. Moreover, we deduce the JorgeMeeks type formula of the total curvature and show it is an integer multiple of $2\pi/N$ for $N$ only depending on the symmetric space. It is a generalization of the theorem by ChernOsserman for complete minimal surfaces in Euclidean nspace. This is joint work with Takuro Mochizuki (RIMS).

Monday 8 April 2024  4pm chalk room MNO 1st floorLyosha Balitskiy Лёша Балицкий (University of Luxembourg)Title: Systolic freedom and rigidity modulo 2Abstract: Given a closed $n$dimensional Riemannian manifold $M$, let us define its systole as the length of a shortest loop that does not bound a surface in $M$; and its cosystole as the smallest area of an $(n1)$dimensional submanifold that does not bound an $n$dimensional domain (with mod 2 coefficients). Answering a question of Gromov, in 1998 Freedman exhibited the first examples of manifolds in which the product of the systole and the cosystole cannot be bounded from above by the volume of $M$; this manifests the phenomenon of systolic freedom. In a joint work with Hannah Alpert and Larry Guth, we showed that Freedman's examples are almost as "free" as possible, by bounding the systolic product by the volume raised to the power of $1+\epsilon$. I will give an overview of the systolic freedom phenomenon (including its connection to quantum error correcting codes), and say a couple of words about proofs.

Monday 25 March 2024  4pm, MNO 1.020Wayne Lam 林偉揚 (University of Luxembourg)Title: Hyperbolic discrete LaplacianAbstract: The Laplace operator on a Riemannian manifold is a fundamental tool to study the geometry of the manifold. Inspired by electric networks, Laplacians on graphs are defined with edge weights playing the role of conductance. When the edge weights are constant, the graph Laplacian becomes the combinatorial Laplacian and is known to reveal rich combinatorial information of the graph. Given a graph embedded on a surface, it is natural to consider a geometric Laplacian, where edge weights are adapted to the geometry. For the 1skeleton graph of a geodesic triangulation on a Euclidean surface, there is a “cotangent formula” relating the edge weights to the Euclidean metric. It is known to connect with various problems, e.g. deformations of circle patterns, Delaunay decomposition, discrete harmonic maps and the YDelta transform in graphs. In the talk, we introduce the analogue for hyperbolic surfaces.

Monday 11 March 2024  4pmFilippo Mazzoli (Max Planck Institute for Mathematics in the Sciences, Leipzig)Title: Volume, entropy, and diameter in $\mathrm{SO}(p,q+1)$higher Teichmüller spacesAbstract: The notion of $\mathbb{H}^{p,q}$convex cocompact representations was introduced by Danciger, Guéritaud, and Kassel and provides a unifying framework for several interesting classes of discrete subgroups of the orthogonal groups $\mathrm{SO}(p,q+1)$, such as convex cocompact hyperbolic manifolds and maximal globally hyperbolic antide Sitter spacetimes of negative Euler characteristic. By recent works of SeppiSmithToulisse and BeyrerKassel, we now know that any such representation admits a unique invariant maximal spacelike $p$dimensional manifold inside the pseudoRiemannian hyperbolic space $\mathbb{H}^{p,q}$, and that the space of $\mathbb{H}^{p,q}$convex cocompact representations of a group $\Gamma$ consists of a union of connected components of the associated $\mathrm{SO}(p,q+1)$character variety. In this talk, I will describe a recent joint work with Gabriele Viaggi in which we provide various applications for the existence of invariant maximal spacelike submanifolds. These include a rigidity result for the pseudoRiemannian critical exponent (which answers affirmatively to a question of GlorieuxMonclair), a comparison between entropy and volume, and several compactness and finiteness criteria in this framework.

Monday 26 February 2024  4pm chalk room MNO 1st floorThiziri Moulla (Université de Montpellier & EPFL)Title: On the minimale volume entropy of finitely generated groupsAbstract: In this talk I'll speak about the minimal volume entropy of finitely presented groups and I'll give some properties. After this, we will see a nother concept of minimal volume entropy given here for geometrically finite groups, and then I'll present a comparative analysis of these two notions. Finally, I'll introduce a class of groups called Soft Groups and its particularities in the study.

Monday 19 February 2024  4pm, MNO 1.020Baptiste Louf (CNRS & Université de Bordeaux)Title: Combinatorial maps and hyperbolic surfaces in high genusAbstract: In this talk, we consider two models of random geometry of surfaces in high genus: combinatorial maps and hyperbolic surfaces. The geometric properties of random hyperbolic surfaces under the Weil–Petersson measure as the genus tends to infinity have been studied for around 15 years now, building notably on enumerative results of Mirzakhani. On the other hand, random combinatorial maps were first studied in the planar/finite genus case, and then in the high genus regime starting 10 years ago with unicellular (i.e. onefaced) maps. In a joint work with Svante Janson, we noticed some numerical coincidence regarding the counting of short closed curves in unicellular maps/hyperbolic surfaces in high genus (comparing it to results of Mirzakhani and Petri). This leads us to conjecture some similarities between the two models in the limit, and raises several other open questions.

Friday 26 January 2024  10:30, 6A MNO 6st floorJavier MartínezAguinaga (Universidad Complutense de Madrid)Title: Distributions on manifolds and convex integrationAbstract: In many contexts, the resolution of a geometric problem can be reduced to studying a condition coming form the underlying Algebraic Topology. If this condition satisfies the "hprinciple", then there exist techniques to solve this problem, such as Convex Integration. In this talk we will provide a historical overview of this branch of Mathematics, motivating it through visual examples with special emphasis on Convex Integration. Then we will use this method to tackle classification questions about maximal growth distributions on smooth manifolds. This is a joint work with Álvaro del Pino (Utrecht University).

Monday 22 April 2023  4 pm, chalk room MNO 1st floorMarie Trin (IRMAR, University of Rennes)Title: Counting arcs of the same typeAbstract: Two closed curves on a hyperbolic surface are said to be of the same type if they differ from a mapping class. The question of counting curves of the same type with bounded length has been studied by M.Mirzakhani who showed that the counting is polynomial into the length. M.Mirzakhani's results were recovered and extended by ErlandssonSouto, proving convergence theorems for certain sequences of measures. In 2022, N.Bell obtained results similar to those of M.Mirzakhani for counting arcs (of the same type) in surfaces with boundary. We will introduce the method based on the convergence of measures for curves counting and then look at how to adapt these methods to the case of arcs.

Monday, 27 November 2023  4 pm, chalk room MNO 1st floorOgnjen Tošić (University of Oxford)Title: Harmonic projections in negative curvatureAbstract: A classical question in the theory of harmonic maps is the SchoenLiWang conjecture: given a quasiisometry f between rank one symmetric spaces, does there exist a harmonic map at a finite distance from f. This question was ultimately resolved by BenoistHulin, who in fact proved this existence for arbitrary quasiisometries between pinched Hadamard manifolds. A natural class of maps to study outside the scope of their work are the nearestpoint retractions to convex sets. In this talk we show the existence of harmonic maps at a finite distance from nearestpoint retractions under suitable conditions on the convex set.

Tuesday, 21 November 2023  1 pm, chalk room MNO 1st floor(Zeno) Zheng Huang (CUNY, Graduate Center)Title: On the CMC foliation problem in a class of almost Fuchsian manifoldsAbstract: An almost Fuchsian manifold is a special type of quasiFuchsian hyperbolic threemanifolds: it admits a closed minimal surface of principal curvature less than one in absolute value. Thurston conjectured that any almost Fuchsian manifold is foliated by closed CMC surfaces. Recently Choudhury, Mazzoli and Seppi proved the existence of an open neighborhood of the Fuchsian locus (size of the neighborhood depends on the Fuchsian point) such that any point in the neighborhood is foliated by monotone CMC surfaces. I will discuss, in joint work with L. Lin (Santa Cruz) and Z. Zhang (Sydney), we confirm the conjecture for a subclass (which forms a uniform neighborhood of the Fuchsian locus) of almost Fuchsian manifolds.

Monday, 13 November 2023  4 pm, MNO 1.010Clément LegrandDuchesne (LaBRI, University of Bordeaux)Title: Reconfiguration of squaretiled surfacesAbstract: A squaretiled surface is a special case of a quadrangulation of a surface, that can be encoded as a pair of permutations in $S_n \times S_n$ that generates a transitive subgroup of $S_n$. Squaretiled surfaces can be classified into different strata according to the total angles around their conical singularities. Among other parameters, strata fix the genus and the size of the quadrangulation. Generating a random squaretiled surface in a fixed stratum is a widely open question. We propose a Markov chain approach using "shearing moves" (natural reconfiguration operation preserving the stratum of a squaretiled surface). In a subset of strata, we prove that this Markov chain is irreducible and has diameter $\mathrm{O}(n^2)$, where n is the number of squares in the quadrangulation.

Monday, 30 October 2023  4 pm, chalk room MNO 1st floorEduardo Reyes (MPI, Bonn)Title: Approximating hyperbolic lattices by cubulationsAbstract: The fundamental group of an ndimensional closed hyperbolic manifold admits a natural isometric action on the hyperbolic space H^n by Deck transformations. If n is at most 3 or the manifold has infinitely many totally geodesic codimension1 immersed hypersurfaces, then the group also acts isometrically on CAT(0) cube complexes, which are spaces of combinatorial nature. I will talk about a joint work with Nic Brody in which we approximate the asymptotic geometry of the action on H^n by the actions on these complexes, solving a conjecture of Futer and Wise. In the 3dimensional case, some ingredients are a recent result of Al Assal about limits of measures on the 2plane Grassmannian of the manifold induced by immersed minimal surfaces, and the work of Seppi about minimal disks in H^3 with prescribed quasicircles as limit sets.

Monday, 23 October 2023  4 pm, chalk room MNO 1st floorKarin Melnick (University of Luxembourg)Title: Compact, Lorentzian conformally flat manifoldsAbstract: Any closed, flat Riemannian manifold is finitely covered by the torus, by Bieberbach’s classical theorem. Similar classifications have been pursued for closed, Riemannian conformally flat manifolds, notably by Fried and Goldman, as well as for closed, flat Lorentzian manifolds, notably by Carrière and D’Albo. I will discuss current work with Nakyung Lee and Goldman towards classifying closed, Lorentzian conformally flat manifolds when they have nilpotent holonomy.

Wednesday, 18 October 2023  1 pm, chalk room MNO 1st floorDanyu Zhang (University of Luxembourg)Title: Classification problems of (fibrewise) Anosov diffeomorphismsAbstract: A fibrewise Anosov diffeomorphism is a fibrepreserving self diffeomorphism on a fibre bundle that preserves an invariant hyperbolic (contracting and expanding) splitting along the fibres. When the base is a point, it reduces to a usual Anosov diffeomorphism. We give a generalization of the FranksManning classification of Anosov diffeomorphisms on tori, that is, every fibrewise Anosov diffeomorphism on a principal torus bundle is topologically conjugate to a map that is linear in the fibres. If time permits, we will also discuss manifolds that admit Anosov diffeomorphisms.

Monday, 9 October 2023  4 pm, MNO 1.030Ivan Yakovlev (LaBRI, University of Bordeaux)Title: Studying squaretiled surfaces combinatoriallyAbstract: Squaretiled surfaces are surfaces glued from a finite number of squares respecting some global monodromy constraint. Their asymptotic enumeration is equivalent to computing the volumes of the moduli spaces of translation surfaces, which are necessary for the dynamics applications. Interestingly, these volumes have been computed using an algebraic approach (intersection theory). I will present an alternative, purely combinatorial approach to this problem. This approach relies on the study of metric ribbon graphs and gives a refined count of squaretiled surfaces according to their number of cylinders. It also produces some neat questions at the intersection of combinatorics, geometry and topology.

Monday, 2 October 2023  4 pm, chalk room MNO 1st floorNiklas Affolter (Technical University Berlin)Title: TCD mapsAbstract: A TCD map is a map from a triple crossing diagram to projective space, satisfying an incidence requirement. We introduce projective invariants and dynamics on TCD maps. The invariants allow for several existence and uniqueness theorems. We explain how the invariants relate to cluster algebras, and how dynamics relate to mutation in cluster algebra theory. Conveniently, TCD maps include as special cases a large number of known maps, including various objects from discrete differential geometry, discrete integrable systems, statistical mechanics and even projective flag configurations considered in higher Teichmüller theory. We also introduce a second class of invariants for TCD maps, which are certain ratios of dimer partition functions. These are not just projective invariants, but also invariants of motion. We will look a bit more at example of flagconfigurations from higher Teichmüller theory, and present some related (projective) geometric questions.

Friday, 29 September 2023  9:30 am, MNO 1.030Abderrahim Mesbah (University of Luxembourg)Title: The induced metric and the bending lamination of the boundary of the convex core of quasiFuchsian manifoldsAbstract: Let $S$ be a closed hyperbolic surface and $M=S \times (0,1)$. Suppose $h$ is a Riemannian metric on $S$ with curvature strictly greater than $−1$, $h_*$ is a Riemannian metric on $S$ with curvature strictly less than $1$, and every contractible closed geodesic with respect to $h_∗$ has length strictly greater than $2\pi$. Let $L$ be a measured lamination on $S$ such that every closed leaf has weight strictly less than $\pi$. Then, we prove the existence of a convex hyperbolic metric $g$ on the interior of $M$ that induces the Riemannian metric $h$ (respectively $h_∗$) as the first (respectively third) fundamental form on $S \times \{ 0 \}$ and induces a pleated surface structure on $S \times \{ 1 \}$ with bending lamination $L$. This statement remains valid even in limiting cases where the curvature of $h$ is constant and equal to $−1$. Additionally, when considering a conformal class $c$ on $S$, we show that there exists a convex hyperbolic metric $g$ on the interior of $M$ that induces $c$ on $S \times \{ 0 \}$, which is viewed as one component of the ideal boundary at infinity of $(M,g)$, and induces a pleated surface structure on $S \times \{ 1 \}$ with bending lamination $L$. Our proof differs from previous work by Lecuire for these two last cases. Moreover, when we consider a lamination which is small enough, in a sense that we will define, and a hyperbolic metric, we show that the metric on the interior of $M$ that realizes these data is unique.

21 September 2023  10 am, chalk room MNO 1st floorAlex Nolte (Rice University / Georgia Tech)Title: Leaves of properly convex foliated projective structuresAbstract: In one of the few cases where qualitative descriptions of geometric structures giving rise to a higher Teichmüller space is known, Guichard and Wienhard showed in 2008 that PSL(4,R) Hitchin representations can be understood in terms of certain projective structures on unit tangent bundles of surfaces. These projective structures come with natural equivariant foliations of their developed images by properly convex domains in projective lines and planes. The family of leaves of the codimension1 foliation gives an interesting and mysterious invariant of these Hitchin representations. I will discuss some recent results on these leaves. For instance, except in the Fuchsian case, there must be many projectively nonequivalent leaves and there is a strong sense in which no leaf can have many symmetries.

Tuesday, 4 July 2023  11 am, chalk room MNO 1st floorClara Aldana (Universidad del Norte)Title: Isospectral and quasiisospectral Schrodinger operatorsAbstract: On this talk I will first briefly talk about the isospectral problem in geometry and about some known results of isospectral StrumLiouville operators on a finite interval in the simplified form of a Schroedinger operator. I will introduce quasiisospectrality as a generalization of isospectrality. I will mention the history of the problem and how to construct quasiisospectral potentials. I will present what we know so far about them. The work presented here is still ongoing joint work with Camilo Perez.

Tuesday, 27 June 2023  11 am, chalk room MNO 1st floorJeffrey Danciger (University of Texas at Austin)Title: Eigenvalue asymmetry for convex real projective surfacesAbstract: A convex real projective surface is one obtained as the quotient of a properly convex open set in the projective plane by a discrete subgroup of SL(3,R), called the holonomy group, that preserves this convex set. The most basic examples are hyperbolic surfaces, for which the convex set is bounded by an ellipse, and the holonomy group is conjugate into SO(2,1). In this case, the eigenvalues of elements of the holonomy group are symmetric. More generally, the asymmetry of the eigenvalues of the holonomy group is a natural measure of how far a convex real projective surface is from being hyperbolic. We study the problem of determining which elements (or generally geodesic currents) may have maximal eigenvalue asymmetry. We will present some limited initial results that we hope may be suggestive of a bigger picture. Joint work with Florian Stecker.

Monday, 19 June 2023  3 pm, chalk room MNO 1st floorJing Tao (University of Oklahoma)Title: Classification of isometries of the Thurston metricAbstract: In 1986, Thurston introduced an asymmetric metric on Teichmuller space using Lipschitz maps between marked hyperbolic surfaces. It is known that the mapping class group is the isometry group of Teichmuller space equipped with this metric. We give a classification of the isometries of the Thurston metric in terms of the NielsenThurston type. Unlike the Teichmuller metric, partial pseudoAnosov maps are hyperbolic isometries of the Thurston metric, which we demonstrate by constructing geodesic axes for such maps. This is joint with Camille Horbez.

Tuesday, 13 June 2023  11 am, chalk room MNO 1st floorMartin Bobb (University of Heidelberg)Title: Simple elliptic elements in many euler class zero PSL(2,R) surface group representationsAbstract: The character variety of a genus g surface group representations into PSL(2,R) has connected components distinguished by the euler class, an integer in the interval [22g, 2g2]. Goldman conjectured that on each of these components (with the exception of the maximal two), the mapping class group should act ergodically. We focus on those representations of euler class zero, and use new techniques to show that on a neighborhood of the reducible representations (excepting the identity), every representation sends a simple closed curve to an elliptic element. An observation of Bowditch and Goldman relates the existence of such an element to the dynamical question of ergodicity. Along the way we find and describe a rich action of the Torelli group on a real vector space of dimension 2g2. Joint work with James Farre and Peter Smillie.

Tuesday, 6 June 2023  11 am, chalk room MNO 1st floorYan Mary He (University of Oklahoma)Title: Nielsen realization for some big mapping class groupsAbstract: In this talk, we show that most compactly supported big mapping class groups cannot be realized as a subgroup of the homeomorphism group. Time permitting, we will also prove the nonrealizability of the mapping class group of the plane minus a Cantor set or the sphere minus a Cantor set. This is joint work with Lei Chen.

Tuesday, 30 May 2023  11 am, chalk room MNO 1st floorStefano Riolo (Università di Bologna)Title: Hyperbolic 4manifolds of low volumeAbstract: There is a natural interest in hyperbolic manifolds of low volume, and this talk addresses dimension four. As opposite to dimension n = 3 where Thurston's hyperbolic Dehn filling holds, for n > 3 the volume spectrum is discrete, and there is at most a finite number of hyperbolic nmanifolds with bounded volume (Wang's finiteness). Computing the number of hyperbolic 4manifolds of given small (even minimal) volume appears nowadays far from reach. Counting such manifolds up to commensurability seems less unrealistic, at least by restricting the count to arithmetic manifolds. We will give an overview of the known examples of lowvolume hyperbolic 4manifolds, with particular attention to the construction of some cusped manifolds by means of a remarkable family of polytopes discovered in 2010 by Kerckhoff and Storm. This will include some results obtained in joint works with Martelli and Slavich.

Tuesday, 16 May 2023  11 am, MNO 1.030Christian El Emam (University of Luxembourg)Title: The holomorphic extension of WeilPetersson metric to the QuasiFuchsian spaceAbstract: QuasiFuchsian representations extend the notion of Fuchsian representations in the character variety of PSL(2,C), and Bers theorem provides a milestone in understanding the space of these representations. In this talk, I will present a "metric" approach for quasiFuchsian space, which leads to a "metric" model of its holomorphic tangent bundle extending the usual metric model for the tangent bundle for the Teichmüller space (seen as the space of hyperbolic metrics up to isotopy). This formalism comes with some interesting applications. For instance, this allows giving an alternative proof of the existence of a unique holomorphic Riemannian metric on quasiFuchsian space extending the WeilPetersson metric: moreover, this approach gives a new description of this metric, showing some connections with the renormalized volume and McMullen's reciprocity theorem. Time permitting, I will show how this approach can help understand what quadratic differentials arise as Schwarzian derivatives of Bers' projective structures.

Tuesday, 25 April 2023  11 am, chalk room MNO 1st floorRichard Wentworth (University of Maryland)Title: Compactifications of Hitchin's moduli spaceAbstract: The moduli space of rank 2 Higgs bundles has two compactifications, one from the algebraic geometry of the Cstar action, and another from the analytic "limiting configurations" of solutions to the Hitchin equations. In this talk, I will discuss how the nonabelian Hodge correspondence extends as a map between these compactifications. Somewhat surprisingly, the extension is not continuous.

Wednesday, 19 April 2023  2:30 pm, chalk room MNO 1st floorAndrea Seppi (CNRS  Université Grenoble Alpes)Title: Maximal submanifolds in pseudohyperbolic space and applicationsAbstract: The Asymptotic Plateau Problem is the problem of existence of submanifolds of vanishing mean curvature with prescribed boundary “at infinity”. It has been studied in the hyperbolic space, in the Antide Sitter space, and in several other contexts. In this talk, I will present the solution of the APP for complete spacelike maximal pdimensional submanifolds in the pseudohyperbolic space of signature (p,q). In the second part of the talk, I will discuss applications of this result in Teichmüller theory and for the study of Anosov representations. This is joint work with Graham Smith and Jérémy Toulisse.

Tuesday, 11 April 2023  10 am, chalk room MNO 1st floorJacques Audibert (Sorbonne University)Title: Zariskidense surface subgroups in latticesAbstract: Lattices are a well studied family of discrete subgroups of Lie groups. Recently, there have been a good deal of interest in constructing subgroups of lattices. When such subgroups are Zariskidense and infinite index, they are called "thin". Although thin subgroups are not themselves lattices, they share many properties with them and have been an active field of research in the last decade. In this talk we will construct Zariskidense surface groups in some lattices of split real Lie groups. The construction relies on special representations of surface groups in split real Lie groups, so called Hitchin representations. These are discrete and faithful representations of a surface group that form a whole connected component of the character variety. Our goal is to prove the existence of Zariskidense Hitchin representations that have image in a lattice. To do so, we investigate arithmetic properties of lattices.

Tuesday, 28 March 2023  11 am, chalk room MNO 1st floorAnna Roig Sanchis (Sorbonne University)Title: Random hyperbolic 3manifoldsAbstract: We are interested in studying the behavior of geometric invariants of hyperbolic 3manifolds, such as the length of their geodesics. A way to do so is by using probabilistic methods. That is, we consider a set of hyperbolic manifolds, put a probability measure on it, and ask what is the probability that a random manifold has a certain property. There are several models of construction of random manifolds. In this talk, I will explain one of the principal probabilistic models for 3 dimensions and I will present a result concerning the length spectrum  the multiset of lengths of all closed geodesics  of a 3manifold constructed under this model.

Tuesday, 21 March 2023  11 am, chalk room MNO 1st floorPeter Smillie (Heidelberg University)Title: Index of equivariant minimal surfaces in R^3 and symmetric spacesAbstract: In joint work with Nathaniel Sagman, we gave a lower bound for the index of any minimal surfaces in rank n symmetric spaces in terms of the equivariant index of a related minimal surface in R^n. In this talk, I will explain what I mean by equivariant index, and prove lower bounds for equivariant index for certain minimal surfaces in R^3. As a consequence, we'll conclude that the Labourie conjecture cannot hold for any Hitchin component with genus at least 2 and rank at least 3.

Tuesday, 7 March 2023  11 am, MNO 1.030Francesco Bonsante (University of Pavia)Title: Circle packing on projective surfacesAbstract: Circle packings over the plane or the hyperbolic plane are widely investigated, and have been shown to be rich and interesting objects. Observing that the notion of disk in $\mathbb{C}P^1$ is invariant under projective transformations, Kojima, Mizushima and Tan proposed the study of circle packings on surfaces $S$ equipped with complex projective structures. The main observation is that the combinatoric of a circle packing is determined by a triangulation of the surface $S$, said the nerve of the triangulation. They proposed to fix a triangulation $T$ on a surface $S$ of genus $g>1$ and study the moduli space of pairs $(P,C)$, where $P$ is a projective structure on $S$ and $C$ is a circle packing with nerve equal to $T$. Indeed they showed that this moduli space can be identified to a real semialgebraic set of dimension equal to $6g6$, where $g$ is the genus of $S$, and asked whether it is not singular and the forgetful map sending $(P,C)$ to $P$ is an embedding in the space of projective structures. Even more, they asked whether the map sending $(P,C)$ to the underlying Riemann surface $X(P)$ realises a homeomorphism. In the talk I will show that the moduli space is indeed a smooth manifold of dimension $6g6$ and that the forgetful map $(P,C) \to P$ is an immersion. If time remains I will also illustrate some partial results on the injectivity of the map sending $(P,C)$ to $X(P)$. Results presented in the talk are part of a collaboration with Mike Wolf.

Tuesday, 28 February 2023  11 am, room MNO 1.030Katie Vokes (University of Luxembourg)Title: Thickness and relative hyperbolicity for graphs of multicurvesAbstract: Various graphs associated to surfaces have proved to be important tools for studying the large scale geometry of mapping class groups of surfaces, among other applications. A seminal paper of Masur and Minsky proved that perhaps the most well known example, the curve graph, has the property of Gromov hyperbolicity, a powerful notion of negative curvature. However, this is not the case for every naturally defined graph associated to a surface. We will present joint work with Jacob Russell classifying a wide family of graphs associated to surfaces according to whether the graph is Gromov hyperbolic, relatively hyperbolic or not relatively hyperbolic.

Tuesday, 21 February 2023  11 am, room MNO 1.030Davide Spriano (Oxford University)Title: Hyperbolic models for CAT(0) spacesAbstract: A very successful approach in geometric group theory is to construct "hyperbolic models" for interesting groups, namely a hyperbolic space on which a (nonhyperbolic) group acts in a nice enough way. The earliest example of this philosophy is the BassSerre, and other more recent examples include the curve graph for mapping class groups, contact graph for cubical groups, free factor/free splitting/cyclic splitting complex for $\mathrm{Out}(F_n)$ and so on.

Friday, 10 February 2023  3 pm, room MNO 1.050Adele Jackson (Oxford University)Title: Triangulations of Seifert fibered spacesAbstract: If a 3manifold has a nontrivial JSJ decomposition, the resulting pieces are either hyperbolic or Seifert fibered. When $M$ is a Seifert fibered manifold with boundary, I will describe a triangulation of $M$ that has at most a multiplicative constant more tetrahedra than the minimal triangulation of $M$. This result relies on the technical proposition that for any triangulation of a Seifert fibered space with boundary, all singular fibres (aside from those of multiplicity two) of the manifold are simplicial in its 79th barycentric subdivision.

Friday, 3 February 2023  3 pm, room MNO 1.020(Preceded by an informal talk at 11 am, room MNO 1.050)Parker Evans (Rice University)Title: Polynomial almostcomplex curves in $S^{2,4}$Abstract: In this talk, no $G_2$ background is assumed and all relevant terminology will be defined. We discuss the nonabelian Hodge theory on the punctured sphere for the split real Lie group $G_2'$. We study almostcomplex curves $v_q: C \to S^{2,4}$ in the pseudosphere $S^{2,4}$ associated to polynomial sextic differential $q$. Focusing on the asymptotic geometry, we detect stable regions and critical lines where the limits of $v$ along rays change. Moreover, we find such polynomial almostcomplex curves have polygonal boundaries in $\mathrm{Ein}^{2,3}$ satisfying a condition we call the annihilator property. Time permitting, we discuss a conjectural homeomorphism from a moduli space of sextic differentials to a moduli space of annihilator polygons.

Monday, 16 January 2023  3 pm, chalk roomSourav Ghosh (Ashoka University)Title: Margulis spacetimesAbstract: In this talk I will discuss the construction of Margulis spacetimes. Moreover, I will sketch a few interesting properties of these geometric objects.

Monday, 9 January 2023  11 am, chalk roomHao Chen (ShanghaiTech University)Title: Triply Periodic Minimal Surfaces, an interdisciplinary topicAbstract: I will summarize how my recent works on minimal surfaces have been motivated or inspired by natural sciences, including material sciences, biomembranes, fluid dynamics, etc.

Monday, 12 December 2022  3 pm, room MNO 1.050Suzanne Schlich (University of Strasbourg)Title: Bowditch and primitive stable actions on hyperbolic spaceAbstract: In this talk, we will introduce Bowditch representations of the free group of rank two (defined by Bowditch in 1998) along with primitive stable representations (defined by Minsky in 2010). Recently, Series on one hand, and Lee and Xu on an other hand, proved that Bowditch and primitive stable representations with value in $\mathrm{PSL}(2,\mathbb{C})$ are equivalent. This result can be generalised to representations with value in the isometry group of an arbitrary Gromov hyperbolic space.

Monday, 5 December 2022  3 pm, room MNO 1.030Valentina Disarlo (University of Heidelberg)Title: Stretch lines for surfaces with boundaryAbstract: In 1986 William Thurston introduced a new distance for the Teichmuller space of closed surface. In collaboration with Daniele Alessandrini (Columbia) we extend this theory to the space of Teichmuller surfaces with geodesic boundary. We will construct a large family of geodesics for the Teichmüller space of surfaces with boundary with respect to its "arc metric": we will call them "generalized stretch lines". We will prove that the Teichmüller space with the arc metric is a geodesic metric space, and that it is a Finsler space. This generalizes a result by Thurston on punctured surfaces. This is joint work with Daniele Alessandrini (University of Heidelberg).

Monday, 28 November 2022  3 pm, room MNO 1.010Gianluca Faraco (University of Milano Bicocca)Title: Period realisation of meromorphic differentialsAbstract: Let $S$ be an oriented surface of genus $g$ and $n$ punctures. The periods of any meromorphic differential on $S$, with respect to a choice of complex structure, determine a representation $\chi:\Gamma_{g,n} \to\mathbb C$, where $\Gamma_{g,n}$ denotes the first homology group of $S$. ChenakkodF.Gupta characterised the representations that thus arise, that is, lie in the image of the period map $\textsf{Per}:\Omega\mathcal{M}_{g,n}\to \textsf{Hom}(\Gamma_{g,n},\Bbb C)$. This generalises a classical result of Haupt in the holomorphic case. Moreover, we determine the image of this period map when restricted to any stratum of meromorphic differentials, having prescribed orders of zeros and poles. Strata generally fail to be connected and in fact they may exhibits connected components parametrised by some additional invariants. In collaboration with D. Chen we extend the earlier result by ChenakkodF.Gupta to connected components of strata.

Thursday, 24 November 2022  3:45 pm, MNO 1.040joint with the "Algebra, geometry and graph complexes" seminarAnton Alekseev (University of Geneva)Title: Hamiltonian actions and miracles of hyperbolic geometryAbstract: We consider Hamiltonian action of the (central extension of) the group of diffeomorphisms of the circle. One class of interesting examples is given by second order differential operators on the circle. We recall the classification by LazutkinPankratova, Kirillov, Segal, Witten (and others), and we give a new point of view on this result. Another class of interesting examples are moduli spaces of conformally compact hyperbolic metrics on two dimensional surfaces. In this case, the moment map is given by a surprising formula which involves the metric near the boundary and the geodesic curvature of certain curves on the surface. The talk is based on a joint work in progress with Eckhard Meinrenken.

Tuesday, 22 November 2022  10:45 am, room MNO 1.020Ludovico Battista (University of Bologna)Title: Hyperbolic 4manifolds, Perfect CircleValued Morse Functions, and Infinitesimal RigidityAbstract: An intriguing 3dimensional phenomenon is the existence of hyperbolic manifolds that fiber over the circle. Such manifolds cannot exist in dimension 4, due to a constraint given by the Euler Characteristic and the GaussBonnet formula. We will introduce the notion of "perfect circlevalued Morse function", which appears to be the natural generalization of "fibration over $S^1$", and we will state some consequences of the existence of hyperbolic 4manifolds that admit such a function. Then, we will introduce the notion of infinitesimal rigidity for the holonomy of a hyperbolic manifold, and we will provide two examples of infinitesimally rigid and geometrically infinite hyperbolic 4 and 5manifolds. The example in dimension 4 (resp. 5) is obtained using the perfect circlevalued Morse function (resp. a fibration over $S^1$ built by Italiano, Martelli, and Migliorini). Time permitting, we will introduce the tools used to build a hyperbolic 4manifold that admits a perfect circlevalued Morse function and how we proved the infinitesimal rigidity for the geometrically infinite manifolds we talked about. These results were obtained during my PhD in collaboration with prof. Bruno Martelli.

Thursday, 17 November 2022  1:45 pm, room MNO 1.020Alex Nolte (Rice University / Georgia Tech)Title: Plateau problems and fundamental groups of hyperbolic manifoldsAbstract: Earlier this year, Antoine Song introduced and studied a variant of the Plateau problem that produces distinguished metric spaces out of purely grouptheoretic data. These spaces have remarkable properties, which suggest that their study should lead to applications in geometric group theory. I'll discuss one such application: an equivalent reformulation of Cannon's conjecture from geometric group theory. Viewing Cannon's conjecture through Song's framework qualitatively changes the way in which it is difficult, and seems to open up the possibility of counterexamples. I will end the talk by proposing at least 6 related open problems, which I expect to be more feasibly tractable than Cannon's conjecture. This is joint work with Tam CheethamWest.

Monday, 24 October 2022  3 pm, room MNO 1.050Roman Prosanov (University of Vienna)Title: On hyperbolic 3manifolds with polyhedral boundaryAbstract: It is known that convex bodies in the model 3spaces of constant curvature are rigid with respect to the induced intrinsic metric on the boundary. This story has two classical chapters: the rigidity of convex polyhedra and the rigidity of smooth convex bodies, though there is also a common generalization obtained by Pogorelov. Similarly to this, JeanMarc Schlenker proved that hyperbolic metrics with smooth strictly convex boundary on a compact hyperbolizable 3manifold M are rigid with respect to the induced metric on the boundary (and also with respect to the dual metric). It is reasonable to expect that similar results should hold also for polyhedral boundaries, and eventually for general convex boundaries. Curiously enough, no polyhedral counterparts were proven up to now. One of the reasons is that convex hyperbolic conemetrics on the boundary of M (which is a standard intrinsic description of what we expect to be the induced metric on a polyhedral boundary) might admit not so polyhedral realizations, which are hard to handle or to exclude. A prototypical example is the boundary of a convex core bent along an irrational lamination. I will present a recent work proving the rigidity (and the dual rigidity) of hyperbolic metrics on M with convex polyhedral boundary under mild additional assumptions. As another outcome, it follows that convex cocompact hyperbolic metrics on the interior on M with the convex cores that are "almost polyhedral" are globally rigid with respect to the induced metric on the boundary of the convex core, and are infinitesimally rigid with respect to the bending lamination.

Monday, 17 October 2022  3 pm, room MNO 1.050Samuel Bronstein (ENS Paris)Title: Almostfuchsian disks in hyperbolic 3spaceAbstract: Almostfuchsian disks are immersed disks whose normal bundles is diffeomorphic to the hyperbolic 3space via the exponential map. In this talk we describe a possible parametrization of almostfuchsian disks via quadratic differentials on the disk. Applying these results to equivariant immersions under a surface group, one gets back Uhlenbeck's notion of almostfuchsian representation, and we build a Finsler metric on the space of hyperbolic metric on a surface.

Monday, 10 October 2022  3 pm, Meeting room 6BNathaniel Sagman (University of Luxembourg)Title: Hitchin representations and minimal surfaces in symmetric spacesAbstract: Labourie proved that every Hitchin representation into $\mathrm{PSL}(n,\mathbb{R})$ gives rise to an equivariant minimal surface in the corresponding symmetric space. He conjectured that uniqueness holds as well (this was known for $n=2,3$), and explained that if true, then the Hitchin component admits a mapping class group equivariant parametrization as a holomorphic vector bundle over Teichmüller space.
In this talk, we will define Hitchin representations, Higgs bundles, and minimal surfaces, and give the background for the Labourie conjecture. We will then explain that the conjecture fails for $n$ at least $4$, and point to some future questions and conjectures.

Monday, 3 October 2022  3 pm, room MNO 1.050Yassin Chandran (City University of New York)Title: Space of marked hyperbolic structures on infinite type surfaces.Abstract: Bers' gave a proof of the celebrated NielsenThurston classification that organizes elements of the mapping class group of finite type surfaces in terms of their action on Teichmüller space. Inspired by this perspective, we define a space of marked hyperbolic structures associated to an infinite type surface. We'll discuss various connectivity properties of this space and organize elements of the mapping class group into three classes based on their action on this space. This is work in progress joint with Ara Basmajian.

Monday, 26 September 2022  3 pm, room MNO 1.050Mélanie Theillière (University of Luxembourg)Title: Convex Integration and isometric embeddingsAbstract: Convex Integration is a theory developped by Gromov in the 1970's. This theory allows to make the link between the sphere eversion of Smale and the NashKuiper $C^1$isometric embeddings. In this talk, we will present the Convex Integration Theory. As illustrations, we will use it to remove the singular point of a cone. Then we will use it to build explicitly a $C^1$ isometric embedding of the hyperbolic plane $\mathbb{H}^2$ in $\mathbb{E}^3$. This last construction is a joint work with the Hevea team.

Tuesday, 20 September 2022  11:15 am, Meeting room 6AMahnTien Nguyen (Université Libre de Bruxelles)Title: Monotonicity theorems and how to compare themAbstract: The classical monotonicity theorem dictates how minimal submanifolds of $\mathbb{R}^n$ distribute their volume among spheres of different radii. I will show that in the hyperbolic space, each Minkowskian coordinate yields a monotonicity theorem. Such theorems concern the volume distribution of the submanifold among level sets of the coordinate function and can be used to prove nonexistence or uniqueness results for minimal surfaces. If time permits, I will explain a version of the isoperimetric inequality for complete minimal surfaces of the hyperbolic space. The classical isoperimetric inequality is a relation between area and perimeter of a minimal surface in $\mathbb{R}^n$. In $\mathbb{H^n}$, the area of such surface is necessarily infinite and so this will be a statement about its renormalisation, as defined by Graham and Witten with strong motivation from String Theory.