Geometry and Topology seminar

Academic year 2023-2024

The G&T seminar

2022-2023

Next talk

• 21 September 2023 -- 10 am, chalk room MNO 1st floor
  Alex Nolte (Rice University / Georgia Tech)
  Title: Leaves of properly convex foliated projective structures

  Abstract: 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 codimension-1 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 non-equivalent leaves and there is a strong sense in which no leaf can have many symmetries.

Future talks

• Monday, 2 October 2023 -- 4 pm, room TBA
  Niklas Affolter (Technical University Berlin)
  Title: TBA

  Abstract: TBA

• Monday, 9 October 2023 -- 4 pm, room TBA
  Ivan Yakovlev (LaBRI, Bordeaux)
  Title: TBA

  Abstract: TBA

• Monday, 16 October 2023 -- 4 pm, room TBA
  Danyu Zhang (University of Luxembourg)
  Title: TBA

  Abstract: TBA

• Monday, 23 October 2023 -- 4 pm, room TBA
  Karin Melnick (University of Luxembourg)
  Title: TBA

  Abstract: TBA

• Monday, 30 October 2023 -- 4 pm, room TBA
  Eduardo Reyes (University of California, Berkeley)
  Title: TBA

  Abstract: TBA

• Monday, 13 November 2023 -- 4 pm, room TBA
  Clément Legrand-Duchesne (LaBRI, Bordeaux)
  Title: TBA

  Abstract: TBA

• Monday, 20 November 2023 -- 4 pm, room TBA
  (Zeno) Zheng Huang (CUNY, Graduate Center)
  Title: TBA

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• Monday, 27 November 2023 -- 4 pm, room TBA
  Ognjen Tosic (University of Oxford)
  Title: TBA

  Abstract: TBA

• Monday, 11 December 2023 -- 11 pm, room TBA
  Marie Trin (IRMAR, Rennes)
  Title: TBA

  Abstract: TBA

• Monday, 18 December 2023 -- 4 pm, room TBA
  Martin Mion-Mouton (IRMA, University of Strasbourg)
  Title: TBA

  Abstract: TBA

Previous talks

• Tuesday, 4 July 2023 -- 11 am, chalk room MNO 1st floor
  Clara Aldana (Universidad del Norte)
  Title: Isospectral and quasi-isospectral Schrodinger operators

  Abstract: On this talk I will first briefly talk about the isospectral problem in geometry and about some known results of isospectral Strum-Liouville operators on a finite interval in the simplified form of a Schroedinger operator. I will introduce quasi-isospectrality as a generalization of isospectrality. I will mention the history of the problem and how to construct quasi-isospectral potentials. I will present what we know so far about them. The work presented here is still on-going joint work with Camilo Perez.

• Tuesday, 27 June 2023 -- 11 am, chalk room MNO 1st floor
  Jeffrey Danciger (University of Texas at Austin)
  Title: Eigenvalue asymmetry for convex real projective surfaces

  Abstract: 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 floor
  Jing Tao (University of Oklahoma)
  Title: Classification of isometries of the Thurston metric

  Abstract: 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 Nielsen-Thurston type. Unlike the Teichmuller metric, partial pseudo-Anosov 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 floor
  Martin Bobb (University of Heidelberg)
  Title: Simple elliptic elements in many euler class zero PSL(2,R) surface group representations

  Abstract: 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 [2-2g, 2g-2]. 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 2g-2. Joint work with James Farre and Peter Smillie.

• Tuesday, 6 June 2023 -- 11 am, chalk room MNO 1st floor
  Yan Mary He (University of Oklahoma)
  Title: Nielsen realization for some big mapping class groups

  Abstract: 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 non-realizability 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 floor
  Stefano Riolo (Università di Bologna)
  Title: Hyperbolic 4-manifolds of low volume

  Abstract: 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 n-manifolds with bounded volume (Wang's finiteness). Computing the number of hyperbolic 4-manifolds 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 low-volume hyperbolic 4-manifolds, 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 1.030
  Christian El Emam (University of Luxembourg)
  Title: The holomorphic extension of Weil-Petersson metric to the Quasi-Fuchsian space

  Abstract: Quasi-Fuchsian 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 quasi-Fuchsian 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 quasi-Fuchsian space extending the Weil-Petersson 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 floor
  Richard Wentworth (University of Maryland)
  Title: Compactifications of Hitchin's moduli space

  Abstract: The moduli space of rank 2 Higgs bundles has two compactifications, one from the algebraic geometry of the C-star 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 floor
  Andrea Seppi (CNRS - Université Grenoble Alpes)
  Title: Maximal submanifolds in pseudo-hyperbolic space and applications

  Abstract: 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 Anti-de Sitter space, and in several other contexts. In this talk, I will present the solution of the APP for complete spacelike maximal p-dimensional submanifolds in the pseudo-hyperbolic 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 floor
  Jacques Audibert (Sorbonne University)
  Title: Zariski-dense surface subgroups in lattices

  Abstract: 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 Zariski-dense 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 Zariski-dense 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 Zariski-dense 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 floor
  Anna Roig Sanchis (Sorbonne University)
  Title: Random hyperbolic 3-manifolds

  Abstract: We are interested in studying the behavior of geometric invariants of hyperbolic 3-manifolds, 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 3-manifold constructed under this model.

• 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.

• 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
  Adele Jackson (Oxford University)
  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.

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