### The G&T seminar

Welcome to the webpage of the seminar of the Research Cluster in Geometry at the Mathematics Department of the University of Luxembourg.
Organizers: Christian El Emam, Mingkun Liu, Nathaniel Sagman.

### Next talk

• Tuesday, 21 March 2023 -- 11 am, chalk room MNO 1st floor
Peter Smillie (Heidelberg University)
Title: Index of equivariant minimal surfaces in R^3 and symmetric spaces
Abstract: 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.

### Future talks

• Tuesday, 28 March 2023 -- 11 am, room TBA
Anna Roig Sanchis (Sorbonne University)
Title: TBA
Abstract: TBA
• Tuesday, 11 April 2023 -- 11 am, room TBA
Jacques Audibert (Sorbonne University)
Title: TBA
Abstract: TBA
• Wednesday, 19 April 2023 -- time TBA, room TBA
Andrea Seppi (CNRS - Université Grenoble Alpes)
Title: TBA
Abstract: TBA
• Tuesday, 25 April 2023 -- 11 am, room TBA
Richard Wentworth (University of Maryland)
Title: TBA
Abstract: TBA
• Tuesday, 30 May 2023 -- 11 am, room TBA
Stefano Riolo (Università di Bologna)
Title: TBA
Abstract: TBA

### Previous talks

• Tuesday, 7 March 2023 -- 11 am, MNO 1.030
Francesco Bonsante (University of Pavia)
Title: Circle packing on projective surfaces
Abstract: 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 $6g-6$, 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 $6g-6$ 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 1.030
Katie Vokes (University of Luxembourg)
Title: Thickness and relative hyperbolicity for graphs of multicurves
Abstract: 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 1.030
Davide Spriano (Oxford University)
Title: Hyperbolic models for CAT(0) spaces
Abstract: A very successful approach in geometric group theory is to construct "hyperbolic models" for interesting groups, namely a hyperbolic space on which a (non-hyperbolic) group acts in a nice enough way. The earliest example of this philosophy is the Bass-Serre, 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 1.050
Title: Triangulations of Seifert fibered spaces
Abstract: If a 3-manifold has a non-trivial 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 almost-complex 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 non-abelian Hodge theory on the punctured sphere for the split real Lie group $G_2'$. We study almost-complex 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 almost-complex 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 room
Sourav Ghosh (Ashoka University)
Title: Margulis space-times
Abstract: In this talk I will discuss the construction of Margulis space-times. Moreover, I will sketch a few interesting properties of these geometric objects.
• Monday, 9 January 2023 -- 11 am, chalk room
Hao Chen (ShanghaiTech University)
Title: Triply Periodic Minimal Surfaces, an interdisciplinary topic
Abstract: I will summarize how my recent works on minimal surfaces have been motivated or inspired by natural sciences, including material sciences, bio-membranes, fluid dynamics, etc.
• Monday, 12 December 2022 -- 3 pm, room MNO 1.050
Suzanne Schlich (University of Strasbourg)
Title: Bowditch and primitive stable actions on hyperbolic space
Abstract: 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 1.030
Valentina Disarlo (University of Heidelberg)
Title: Stretch lines for surfaces with boundary
Abstract: 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 1.010
Gianluca Faraco (University of Milano Bicocca)
Title: Period realisation of meromorphic differentials
Abstract: 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 $S$. Chenakkod-F.-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 Chenakkod-F.-Gupta to connected components of strata.
• Thursday, 24 November 2022 -- 3:45 pm, MNO 1.040
joint with the "Algebra, geometry and graph complexes" seminar
Anton Alekseev (University of Geneva)
Title: Hamiltonian actions and miracles of hyperbolic geometry
Abstract: 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 Lazutkin-Pankratova, 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 1.020
Ludovico Battista (University of Bologna)
Title: Hyperbolic 4-manifolds, Perfect Circle-Valued Morse Functions, and Infinitesimal Rigidity
Abstract: An intriguing 3-dimensional 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 Gauss-Bonnet formula. We will introduce the notion of "perfect circle-valued 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 4-manifolds 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 5-manifolds. The example in dimension 4 (resp. 5) is obtained using the perfect circle-valued 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 4-manifold that admits a perfect circle-valued 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 1.020
Alex Nolte (Rice University / Georgia Tech)
Title: Plateau problems and fundamental groups of hyperbolic manifolds
Abstract: Earlier this year, Antoine Song introduced and studied a variant of the Plateau problem that produces distinguished metric spaces out of purely group-theoretic 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 Cheetham-West.
• Monday, 24 October 2022 -- 3 pm, room MNO 1.050
Roman Prosanov (University of Vienna)
Title: On hyperbolic 3-manifolds with polyhedral boundary
Abstract: It is known that convex bodies in the model 3-spaces 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, Jean-Marc Schlenker proved that hyperbolic metrics with smooth strictly convex boundary on a compact hyperbolizable 3-manifold 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 cone-metrics 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 1.050
Samuel Bronstein (ENS Paris)
Title: Almost-fuchsian disks in hyperbolic 3-space
Abstract: Almost-fuchsian disks are immersed disks whose normal bundles is diffeomorphic to the hyperbolic 3-space via the exponential map. In this talk we describe a possible parametrization of almost-fuchsian disks via quadratic differentials on the disk. Applying these results to equivariant immersions under a surface group, one gets back Uhlenbeck's notion of almost-fuchsian representation, and we build a Finsler metric on the space of hyperbolic metric on a surface.
• Monday, 10 October 2022 -- 3 pm, Meeting room 6B
Nathaniel Sagman (University of Luxembourg)
Title: Hitchin representations and minimal surfaces in symmetric spaces
Abstract: 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 1.050
Yassin Chandran (City University of New York)
Title: Space of marked hyperbolic structures on infinite type surfaces.
Abstract: Bers' gave a proof of the celebrated Nielsen-Thurston 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 1.050
Mélanie Theillière (University of Luxembourg)
Title: Convex Integration and isometric embeddings
Abstract: 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 Nash-Kuiper $C^1$-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 6A
Mahn-Tien Nguyen (Université Libre de Bruxelles)
Title: Monotonicity theorems and how to compare them
Abstract: 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 non-existence 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.

### Archive

Here is a link to the seminars of former years.