Organizers: Vincent Pecastaing, Clara Aldana, Sergei Merkulov.
Unless otherwise specified, the seminar will be on Monday,
3-4pm.
We introduce some deformations of complete finite-volume hyperbolic four-manifolds that may be interpreted as four-dimensional analogues of Thurston's hyperbolic Dehn filling. More precisely, we exhibit an analytic path of complete finite-volume cone four-manifolds that interpolates between two non-isometric hyperbolic four-manifolds by drilling and filling some of their cusps. The cone four-manifolds have singularities along a geodesically immersed surface, with varying cone angles. This is joint work with Stefano Riolo.
Abstract.
Abstract.
The topic of the talk is spectral geometry. In 1966 Mark Kac asked in his paper "Can one hear the shape of a drum?". Mathematically the question is formulated as follows. A drum is a domain in Euclidean space that is held along its boundary. When we play a drum we hear an infinite sequence of frequencies. The question is if one can determine geometry of a domain using only information on this sequence. Nowadays the question is generalised to Riemannian manifolds and even to spaces with singularities. One of the main tools in spectral geometry is the heat trace. On any closed smooth Riemannian manifold the heat trace expansion gives some geometrical information such as dimension, volume and total scalar curvature of the manifold. On a manifold with conic singularities we derive a detailed asymptotic expansion of the heat trace. Then we investigate how the terms in the expansion reflect the geometry of the manifold.
The goal of this talk is to investigate the following question: is there a natural mapping class group invariant Kahler structure on character varieties of closed surfaces enriching the natural symplectic structure. The Weil-Petersson Kahler metric on Teichmuller space is the archetypal example of a Hodge theoretic L^{2}-pairing which gives rise to a Riemannian metric on a moduli space of geometric structures. Given the character variety of conjugacy classes of homomorphisms of a closed, oriented surface group into a semisimple Lie group, we will give a general construction of L^{2}-Riemannian metrics; this construction depends on a choice of complex structure on the surface for each homomorphism. For any such choice, these metrics carry orthogonal almost complex structures, and combine with the symplectic structure to form an almost Kahler structure. In closing, we will discuss some intriguing relationships between minimal surfaces in symmetric spaces and Riemannian metrics on character varieties.
There are various graphs associated to surfaces (of finite topological type), constructed using curves or arcs, which have been very useful in the study of Teichmueller space (the space of hyperbolic structures on a surface) and of the mapping class group. If the surface has infinite topological type (e.g. it has infinite genus), these graphs turn out to be not as interesting. I will discuss why and present an alternative construction which gives graphs with better properties. Joint work with Matt Durham and Nick Vlamis.
In order to study geometric structures on manifolds, we often consider the space of representations of its fundamental group \Gamma with values in a given Lie group G up to conjugation. When the Lie group is SL(n,C), we can use the character variety, which is an algebraic object allowing to understand representations up to conjugation. After giving the definition an some properties of the SL(n,C)-character variety, we propose a definition for a "G-character variety" when G is a real form of SL(n,C), and we will verify that this object is useful for studying the representations of \Gamma with values in G up to conjugation.
Abstract: In many physical situations divergence-free fields, such as the vorticity field in hydrodynamics and the magnetic field in magnetohydrodynamics, show a high degree of topological complexity, that is the field lines of these fields are braided or even knotted. Topological measures, such as the magnetic helicity, has been shown to be useful in quantifying this type of complexity but it is defined for unbounded domain, while typical observations or measurements yield information only about bounded domain. In this talk, we are going to discuss some of the topological invariants such as linking (winding) numbers which can be used as criteria to measure different aspects of topological complexity in the structures of the vector fields, and discuss how we could introduce the helicity in terms of the high-order winding numbers.
The quantum Teichmüller space is an algebraic object associated with a punctured surface admitting an ideal triangulation. Two somewhat different versions of it have been introduced, as a quantization by deformation of the Teichmüller space of a surface, independently by Chekhov and Fock [CF99] and by Kashaev [Kas95]. As in the article [BL07], we follow the exponential version of the Chekhov-Fock approach, whose setting has been established in [Liu09]. In this way, the study is focused on non-commutative algebras and their finite-dimensional representations, instead of Lie algebras and self-adjoint operators on Hilbert spaces, as in [CF99] and [Kas95]. In this talk I will briefly describe the construction of the quantum Teichmüller space and I will talk about its theory of representations in a finite dimensional vector space. In particular, I will focus my attention on the so-called “local representations” and their classifications results described in [BBL07], which make a link between the algebraic theory and some more familiar objects in low-dimensional geometry, as representations of the fundamental group and pleated surfaces. In the same paper Bai, Bonahon and Liu showed a procedure to select one distinguished operator (up to scalar multiplication) for every choice of a surface S, of a couple of ideal triangulations and of a couple of isomorphic local representations, requiring that the whole family of operators verifies certain Fusion and Composition properties. This selection was also used to produce invariants for pseudo-Anosov diffeomorphisms and their hyperbolic mapping tori (extending to local representations what had been done in [BL07] for irreducible ones). However, by analyzing the constructions of [BBL07], during my master thesis work under the supervision of Prof. Riccardo Benedetti, I found a difficulty that I eventually fix by a slightly weaker (but actually optimal) selection procedure. I will conclude the talk by quickly describing how we can restate that result and what we can recover from it.
ReferencesAbstract: Jacob Steiner asked in 1832: Is every polyhedron inscribable? That is, can every polyhedron be realized with all its vertices on a sphere? People do not have any clue until Steinitz constructed the first counterexamples in 1928. In 1992 Rivin et al. gave a complete characterization for inscribable polyhedra under the disguise of ideal hyperbolic polyhedra. This was considered as the final solution of Steiner’s problem. Nevertheless, in a parenthesis of Steiner’s original text, he also asked about polyhedra inscribed to other quadratic surfaces. These bonus questions were overlooked until very recently, a strong version was answered by Danciger, Maloni and Schlenker. In this talk, I will present the ongoing joint work with Schlenker that aims at a complete solution to Steiner’s problem.
A very well known result in Riemannian geometry, the Obata-Lichnerowicz theorem, relates the Ricci curvature and the spectrum of the Laplacian: for a compact Riemannian manifold of dimension n, if the Ricci tensor is bounded below by (n-1) , then the first non-zero eigenvalue of the Laplacian is larger than or equal to the dimension of the manifold. Equality holds if and only if the manifold is isometric to the sphere. We will show how an analogue result holds in the singular setting of stratified spaces, which are metric spaces generalizing the notion of isolated conical singularities. The last part of the talk is devoted to the consequences of this result on the existence of a constant scalar curvature metric.
To a hyperbolic surface and a finite-dimensional representation of its fundamental group, we associate a Selberg zeta function. The main goal of the talk is to show that under certain conditions, the Selberg zeta function admits a meromorphic extension to the whole complex plane. Our main tool is the use of transfer operators. This is a joint work with Anke Pohl.
The coefficients in Maxim Kontsevich's universal formula for the quantization of Poisson brackets are given by high-dimensional period integrals that are difficult to compute. In forthcoming joint work with Peter Banks and Erik Panzer, we adapt Francis Brown's approach to the periods of the moduli space of genus zero curves to give an algorithm for the exact evaluation of these integrals. It proves that the integrals are always rational linear combinations of special transcendental numbers called multiple zeta values (related to the Riemann zeta function). I will discuss our approach and some of the concrete problems in deformation quantization that motivated it.
We will talk about an ergodic theorem for free group actions. We will prove the convergence in Lˆ1 of spherical averages for a free group. The standard definition will be generalized in such a way that the elements of the sphere in a free group will be taken with weights defined by a Markov chain. Under mild conditions on the stochastic matrix that defines a chain, we will prove the convergence of spherical averages. This convergence was previously known only for symmetric Markov chains, now it is established for the open set in the space of stochastic matrices. This is a joint work with Lewis Bowen and Alexander Bufetov.
It is well known that the automorphism group of a rigid geometric structure is a Lie group. In fact, as there are multiple notions of rigid geometric structures, the property that the local automorphisms form a Lie pseudogroup is sometimes taken as an informal definition of rigidity for a geometric structure. In which topology is this the case? The classical theorems of Myers and Steenrod say that C^0 convergence of isometries of a smooth Riemannian metric implies C^\infty convergence; in particular, the compact-open and C^\infty topologies coincide on the isometry group. I will present joint results with C. Frances in which we prove the same result for local automorphisms of smooth parabolic geometries, a rich class of geometric structures including conformal and projective structures. As a consequence, the automorphism group admits the structure of a Lie group in the compact-open topology.
Software packages for the computation of Delaunay triangulations of the flat torus of genus one in two and three dimensions are available in the Computational Geometry Algorithms Library (CGAL). The simplest possible extension to surfaces of genus two is the Bolza surface, and as far as we know there is no available software for this case. In this talk, I will present our implementation, based on the theoretical results and the incremental algorithm proposed last year at SoCG by Bogdanov, Teillaud, and Vegter [1]. I will describe the representation of the triangulation, and the different steps of the algorithm will be given in detail. In conclusion, we will discuss the algebraic complexity of predicates, and experimental results. [1] Mikhail Bogdanov, Monique Teillaud, and Gert Vegter. Delaunay triangulations on orientable surfaces of low genus. In Proceedings of the Thirty-second International Symposium on Computational Geometry, pages 20:1â€“20:15, 2016. URL: https://hal.inria.fr/ hal-01276386, doi:10.4230/LIPIcs.SoCG.2016.20.
Spherical CR structures on 3-manifolds are natural complex hyperbolic counterparts to flat conformal structures. They correspond to (X,G)-structures, where X is the boundary at infinity of complex hyperbolic plane, and G is PU(2,1), the group of holomorphic isometries of the same space. In this talk, I will consider the case where the manifold is the Whitehead link complement. I will describe a component of the corresponding SL(3,C) character variety, and show that certain points in this variety give spherical CR structures on the Whitehead link complement. This talk will combine results obtained in collaboration with John Parker, and Antonin Guilloux.
Let Y be a compact connected 2-orbifold of negative Euler characteristic and let \Pi be its orbifold fundamental group. For n > 1, we denote by R(\Pi,n) the space of representations of \Pi into PGL(n,R). The purpose of the talk is to show that R(\Pi,n) possesses connected components homeomorphic to an open ball whose dimension we can compute explicitly (for n=2 and 3, we find again formulae due to Thurston and to Choi and Goldman, respectively). We then give applications of the result to the study of rigidity properties of hyperbolic Coxeter groups. This is joint work with Daniele Alessandrini and Gye-Seon Lee (Heidelberg).
There has been a recent surge in studying surfaces of infinite type, i.e. surfaces with infinitely-generated fundamental groups. In this talk, we will focus on their mapping class groups, often called big mapping class groups. In contrast to the finite-type case, there are many open questions regarding the basic algebraic and topological properties of big mapping class groups. I will discuss several such questions and provide some answers. In particular, I will focus on automorphisms of pure mapping class groups. This work is joint with Priyam Patel.
The Hitchin component is a (special) connected component of the space of homomorphisms of a surface group into PSL(d,R). This component is a higher rank analogue of the Teichmuller space of the surface. The purpose of the talk is to show that the critical exponent of a Hitchin representation has a rigid upper bound. This is a joint work with Rafael Potrie.
The main goal of the talk shall be to explain a few ideas from the classical "thermodynamical formalism", which makes it possible to describe many invariant measures of certain maps, such as dilating maps. This formalism is based on the "transfer operator", which depends on the choice of a function on the phase space called a "potential". When this operator has a spectral gap, the invariant measure has very good statistical properties (exponential mixing, Central Limit Theorem, etc.) We shall end the talk with a result showing that some non- or weakly dilating maps exhibit a spectral gap in the "high temperature" regime; the novelty is a very explicit bound for this regime.
A branched complex projective structure is a geometric structure on a surface which is locally modelled on (PSL(2,C),CP^1), possibly with integral conical singularities; motivating examples come from the study of constant curvature metrics and linear ODEs on Riemann surfaces. We investigate the interactions between some geometric surgeries which can be performed on a given structure without changing its holonomy; we show in particular that bubbling (i.e. taking a connected sum with a copy of the Riemann sphere) is enough to describe almost every structure with quasi-Fuchsian holonomy and at most two branch points.
Hyperbolic earthquakes on a surface S with negative Euler characteristic constitute a well known class of deformations of hyperbolic metrics on S. They consist in shearing the surface along complete disjoint geodesics on S, so they are associated with the space of measured geodesic laminations on S. Moving from F. Bonsante and J.-M. Schlenker's results on the association with couples of hyperbolic metrics of couples of measured geodesic laminations on compact surfaces, using constructions in a Lorentzian environment, I will consider a generalization to the case of hyperbolic surfaces with closed geodesic boundary.
A triangulation of the 2-sphere is combinatorially non-negatively curved if each vertex is shared by no more than six triangles. Thurston showed that non-negatively curved triangulations of the 2-sphere correspond to orbits of vectors of positive norm in a lattice in C^(1,9) under the action of a group of isometries. We show that an appropriately weighted count of triangulations of the 2-sphere with 2n triangles are the coefficients of a modular form, and specifically that the number is 809/2612138803200 sigma_9(n), where sigma_9(n) is the 9th divisor function. This is joint work with Philip Engel.
The handlebody group is the subgroup of the mapping class group of a surface formed by those mapping classes which extend to a handlebody. We will study rigidity phenomena of this group as a subgroup of the mapping class group. On the one hand, the group is rigid: any inclusion (of a finite index subgroup of) the genus g handlebody group in the genus g mapping class group is simply a conjugation. On the other hand, once we consider inclusions in other mapping class groups, rigidity ceases to hold.
We will present in this talk a 1-parameter family of affine interval exchange transformations and we will try to show that it displays rich and various dynamical behavior.
An affine interval exchange transformation is a piecewise affine bijection of the interval [0,1[. Few things are known about the possible behaviors of such dynamical systems, despite a lovely piece of work of Marmi, Moussa and Yoccoz clearly indicating that they present interesting features.
We will articulate our talk around numerical simulations that motivated conjectures/results that we will expose along the way. We will also try to highlight links with TeichmÃ¼ller theory.
This is joint work with Adrien Boulanger and Charles Fougeron.
The theorem of Alexandrov on the realisation of a Euclidean surface with conical singularities of angles lesser than 2π has been the object of many generalisations since the 1950's. Our attempt to generalise a variational method of Bobenko used to prove Alexandrov-like theorem leads naturally to the study of polyhedral surfaces in 3-dimensionnal spacetimes with BTZ-like singularities. We present some results obtained toward a classification of Cauchy-compact spacetimes with BTZ and some results on embeddings of singular Euclidean surfaces in such spacetimes.
This is based on a joint work with A. Alekseev, N. Kawazumi and F. Naef. Due to results of Goldman and Turaev, there is a natural Lie bialgebra structure on the vector space spanned by the (homotopy classes of) free loops on an oriented surface. In this talk, we address the formality question for this Lie bialgebra: is the completion of the Goldman-Turaev Lie bialgebra isomorphic to its associated graded? We show that in the genus 0 case, any solution of the Kashiwara-Vergne problem gives a formality isomorphism for the Goldman-Turaev Lie bialgebra, thus solving the question affirmatively. Motivated by this result, we then introduce a generalization of the Kashiwara-Vergne problem associated to any compact oriented surface with boundary. We also discuss the existence and uniqueness of solutions to our generalization.
The Auslander Conjecture states that all discrete groups acting properly and cocompactly on R^n by affine transformations should be virtually solvable. In 1983, Margulis constructed the first examples of proper (but not cocompact) affine actions of nonabelian free groups. It seems that until now all known examples of irreducible proper affine actions were by virtually solvable or virtually free groups. I will explain that any right-angled Coxeter group on k generators admits a proper affine action on R^{k(k-1)/2}. This is joint work with J. Danciger and F. Guéritaud.
In this talk I will describe Margulis spacetimes and show their inter relationship with Anosov representations. Moreover, if time permits I will define the Pressure metric on the moduli space of Margulis spacetimes without "cusps" and describe its properties.
Spacelike-Zoll surfaces are Lorentzian surfaces all of whose spacelike geodesics are simple, closed and have the same length. The basic example is de Sitter surface, the Lorentzian analogue of the round sphere. We will construct three explicit families of such surfaces. We will see that a spacelike-Zoll surface with a Killing field is always C0-conformal to a finite cover of de Sitter space but not always C2-conformal.
The study of Anti-de Sitter space in dimension (2+1) has grown interest in recent times, motivated by its relation with Teichmüller theory of hyperbolic surfaces and quasiconformal mappings. In particular, maximal surfaces (i.e. of vanishing mean curvature) are related to minimal Lagrangian extensions of a quasisymmetric homeomorphism.
In this talk, we will discuss geometric properties of maximal surfaces in Anti-de Sitter space. We will thus show that, if
In this talk, we look for a compactification of the space of Riemannian metrics with conical singularities on a fixed compact surface. The accumulation of cone points (along a curve, or along a more complicated set) naturally leads to the study of metrics with Bounded Integral Curvature (B.I.C.). This theory of singular surfaces was developed in Leningrad, between the 40's and the 70's, by Alexandrov and many others. These are intrisic metrics, for which there is a natural notion of curvature, which is a Radon measure. This includes Riemannian metrics (possibly with conical singularities), as well as Alexandrov spaces of curvature bounded by above or by below. In this talk, by analogy with the classical Cheeger-Gromov's compactness theorem, we prove a compactness theorem for metrics with B.I.C. ; as a corollary, we obtain a compactification of the space of Riemannian metrics with conical singularities."
I will review the functorial approach to 2-dimensional topological quantum field theory, and its algebraic formulation in terms of Frobenius algebras. One source of examples, called state sum models, is of a combinatorial nature and involves triangulations of surfaces. Another source, called the orbifold construction, draws from group actions. I will highlight the similarities between these two constructions and explain how "state sum models are orbifolds of trivial theories". Systematically pursuing such reasoning leads to a general theory for pivotal 2-categories, which in turn has interesting applications to the geometry of isolated singularities.
Guichard and Wienhard constructed compact manifolds admitting projective structures with holonomy in (Quasi)-Hitchin representations of a surface group into the Lie group SL(2n,R) or SL(2n,C). In this talk, we study the topology of such manifolds using Higgs bundle techniques. In particular, we focus on representations in a small neighborhood of the Fuchsian locus inside the representation variety of SL(2n,R) or SL(2n,C). This is joint work with Daniele Alessandrini.
In most complex simple Lie groups, counter-examples to Schur's lemma exist : we can find an irreducible subgroup whose centralizer strictly contains the center of the complex simple Lie group. We will explain how to classify those non-trivial centralizers of irreducible subgroups when the Lie group is PSL(n,C). It naturally involves finite Abelian groups equipped with an alternate bilinear form, a.k.a. alternate modules. Finally, we will see how we can use this classification in order to parametrize the orbifold locus of some character varieties.
The past fifteen years have seen a great deal of progress towards a complete picture of hyperbolic manifolds of low volume. The volume of a hyperbolic manifold is a topological invariant and can be viewed as a measure of complexity. In fact, this invariant is finite-to-one. For manifolds with cusps, one can also consider the volume of the maximal horoball neighborhood of a cusp. In this talk, we will present preliminary results for classifying the infinite families of hyperbolic 3-manifolds of cusp volume <2.62 and the implications of this classification. These families are of particular interest as they exhibit the largest number of exceptional Dehn fillings. As in some other results on hyperbolic 3-manifolds of low volume, our technique utilizes a rigorous computer assisted search. The talk will focus on providing sufficient background to explain our approach and describe our conclusions. This work is joint with David Gabai, Robert Meyerhoff, Nathaniel Thurston, and Robert Haraway.
In the talk I will discuss the geometry of surfaces of constant curvature in anti-de Sitter space. In particular I will prove that any acausal meridian in the boundary of AdS space is the border of a surface of constant curvature K<-1. In the proof the classical correspondence between space-like surfaces in AdS space and area-preserving maps of the hyperbolic plane will be used. In particular an explicit way to recover the surface from the area preserving map will be presented. Using those methods we will construct some explicit solutions of the problem which will be used as barriers in the general case. Results presented in the talk are part of a joint work with Andrea Seppi.
In 1982, Maxwell defined the notion of "level" for Coxeter groups, and proved that those of level 2 correspond to infinite ball packings, generalizing the famous Apollonian packing. Recent studies on "limit roots" lead Labbé and the speaker to revisit Maxwell's work, and we enumerated all the 326 Coxeter groups of level 2. In the talk, I will present a new definition for the notion of "level", which is more suitable for geometric studies, yields many more infinite ball packings.
We introduce a classical dynamical invariant for certain geometric actions of a hyperbolic group on the anti-de Sitter space, and prove a rigidity theorem that echoes the famous theorem of Bowen for quasi-Fuchsian groups.
A Clifford-Klein form is a quotient of a homogeneous space G/H by a discrete subgroup Gamma of G acting properly and freely on G/H. It admits a natural structure of a manifold locally modelled on G/H. There is a natural homomorphism from relative Lie algebra cohomology to de Rham cohomology of a compact Clifford-Klein form. Relating this homomorphism with an upper-bound estimate for cohomological dimensions of discontinuous groups, we give a new obstruction to the existence of compact Clifford-Klein forms of a given homogeneous space. We obtain some examples of homogeneous spaces that do not have a compact Clifford-Klein form, such as the "pseudo-Riemannian sphere" SO_0(p+1, q)/SO_0(p, q) with p, q > 0, q: odd.
I will present the construction of a (hyper-)Kähler metric on the character variety associated to a closed surface group and a reductive Lie group. This metric generalizes both the Weil-Petersson metric on Teichmüller space and the Hitchin metric on the moduli space of Higgs bundles.
We prove that a Kleinian surface groups is determined, up to conjugacy in the isometry group of H^3, by its simple marked length spectrum. As a first application, we show that a discrete faithful representation of the fundamental group of a compact, acylindrical, hyperbolizable 3-manifold M is similarly determined by the translation lengths of images of elements of pi_1(M) represented by simple curves on the boundary of M. As a second application, we show the group of diffeomorphisms of quasifuchsian space which preserve the renormalized intersection number is generated by the (extended) mapping class group and complex conjugation. This is joint work with R. Canary.
The group of polynomial automorphisms of C^n which preserve the Markoff-Hurwitz equation has an interesting and complicated domain of discontinuity in C^n. We will describe some interesting identities which are satisfied by the orbits of points in the domain of discontinuity, and give a sketch of the proof. This is joint work with Hengnan Hu and Ying Zhang.
A Riemannian manifold having an open, dense, locally homogeneous subset is everywhere locally homogeneous; this is a consquence of the classical theorems of Myers and Steenrod. The same is not necessarily true for a pseudo-Riemannian manifold. I will present a theorem with S. Dumitrescu which says, if an analytic three-dimensional Lorentzian manifold has an open, dense, locally homogeneous subset, then it is everywhere locally homogeneous. We have a proof using the Cartan connection associated to a Lorentzian metric, with arguments combining dynamics and representation theory.
A classical result of Myers and Steenrod states that the isometry group of a compact Riemannian manifold is a compact Lie transformation group. It is also classical that this compactness property fails for general pseudo-Riemannian manifolds. Nevertheless, the noncompactness of the isometry group generally imposes strong restrictions, especially on the topology of the underlying manifold. In this talk, we will focus on the case of closed 3-dimensional Lorentz manifolds. We will in particular classify all the topologies compatible with the existence of a noncompact isometry group.
Motivated by the work of Kin-Kojima-Takasawa, Schlenker and Kojima-McShane, I shall study quasi-isometric constants between pants complex and Weil-Peterson distance, and between convex core volume and pants complex. I shall also classify geometric limits of hyperbolic surface bundles with fixed genus, and interpret meanings of some specific sequence appearing the graph of volume/topological entropy of Kin-Kojima-Takasawa.
A result of R. Zimmer going back to the 1980's asserts that up to local isomorphism, PSL(2,R) is the only non-compact simple Lie group that can act by isometries on a Lorentz manifold of finite volume. He presented it as a corollary of another strong result, known as "Zimmer's embedding theorem". The latter, based on ergodic theory, gives strong algebraic constrainst on Lie groups acting on a G-structure by preserving a finite measure and is well adapted to the study of isometric actions in any signature (they preserve the volume form). If we relax the assumption and only consider conformal dynamics of Lie groups, we loose the existence of an invariant finite measure, even when the manifold is compact. However, conformal structures are rigid in dimension at least 3, so that it seems possible to describe conformal Lie group actions. I will present a result that extends Zimmer's theorem and classifies semi-simple Lie groups without compact factors acting by conformal transformations on a compact Lorentz manifolds. This work is in the continuation of a result of U. Bader and A. Nevo (2002). I will also discuss the local conformal geometry of the Lorentz manifolds where such dynamics occur, especially when the group that acts is locally isomorphic to PSL(2,R).
Like a hyperbolic isometry, each pseudo-Anosov automorphism of Teichmüller space determines an axis, which descends to a closed loop on the moduli space of Riemann surfaces. This feature carries over to the Weil-Petersson metric, where little is known about the shortest loop, or "systole", length spectrum, and synthetic geometry of moduli space.
In this talk we will discuss how, using the notion of "renormalized volume", Schlenker found an effective version of bounds due to the speaker on the volume of the convex core of quasi-Fuchsian manifolds in terms the Weil-Petersson distance between the two components of the conformal boundary. These estimates, in turn, lead to an effective improvement of estimates by the speaker relating volumes of hyperbolic mapping tori to Weil-Petersson translation length, obtained independently by Kojima and MacShane in the closed case. These give the first explicit estimates for Weil-Petersson systoles of moduli space, as well as many interesting implications for its synthetic and combinatorial geometry. We will give an elementary overview of the history and setting and focus in on the Weil-Petersson geometry of moduli space. This is joint work with Ken Bromberg.
Roughly speaking, spinors could be seen as square roots of differential forms and the Dirac operator as square root of the Laplacian. We start by introducing some key ingredients of Spin Geometry and give two applications to illustrate the role of spinors in the study of the geometry of submanifolds and in Witten's proof of the positive mass theorem.
A flat vector bundle over a manifold can be encoded in terms of its holonomy representation. I will explain how this picture generalizes to flat superconnections, i.e. after one replaces vector bundles by complexes of vector bundles. Time permitted, I will outline yet another point of view in terms of categorified sheaf cohomology.
This talk deals with the classification of compact complex manifolds bearing holomorphic geometric structures. One can think at the following interesting examples of geometric structures: holomorphic affine connections (in the holomorphic tangent bundle of the manifold), holomorphic Riemannian metrics or holomorphic conformal structures. Under general hypothesis, those geometric structures must be locally modelled on complex homogeneous spaces.
In particular, we will present a recent joint work with Benjamin McKay (University College Cork) proving that compact simply connected complex manifolds of algebraic dimension zero (meaning that all meromorphic functions must be constant) do not admit holomorphic affine connections and holomorphic conformal structures.
I will describe an analog of the Fenchel-Nielsen coordinates on the Hitchin component, and then use these coordinates to define a large family of deformations in the Hitchin component called "internal sequences". Then, I will explain some geometric properties of these internal sequences, which allow us to conclude some structural similarities and differences between the higher Hitchin components and Teichmüller space.
We study the topology of the configuration space of a device with d legs ("centipede") under some constraints, such as the impossibility to have more than k legs off the ground. We construct feedback controls stabilizing the system on a periodic gait and defined on a "maximal" subset of the configuration space. The talk is supposed to be self-contained.
We show that the Gravity operad, formed by the homologies of the open moduli spaces of curves of genus zero and with composition dual to PoincarÃ© residue, is the linear hull of a free nonsymmetric operad of sets. This has a number of consequences. First of all, it implies that the mixed Hodge structure on the degree k cohomology of Francis Brown's partial compactification of the moduli space is pure of weight 2k. Secondly, it implies that any (formal) period integral on the moduli space has a "renormalization" to an honest integral, converging to a linear combination of multiple zeta values. The results are part of joint work with Dan Petersen.
In this talk we will define a more sophisticated version of the classical simplicial volume, a homotopy invariant introduced by Gromov for his proof of Mostow's rigidity theorem. We will then explain how it is possible to use it for the study of a conjecture by Gromov, which relates the vanishing of the simplicial volume with the vanishing of the Euler characteristic of aspherical manifolds.
We will study groups that can act isometrically on globally hyperbolic Lorentzian surfaces with non trivial dynamics. The main tool is the introduction of two representations into the group of circle diffeomorphisms. After classifying the groups up to isomorphism, we will investigate the classification of these representations up to several notions of conjugacy.
We will describe some estimation of the length of a geodesic arc along Thurston's stretch lines or pinched deformations. Applications include Thurston's compactification and the translation length of the mapping class group under Thurston's metric.
We consider contact structures that appear in connection with singularities of polynomial mappings through the notion of open book decompositions. For real singularities, we analyze the range of possible contact structures, from the perspective of the dichotomy tight versus overtwisted, highlighting the tightness restrictions of the holomorphic case.
Asymptotically hyperbolic manifolds arise in general relativity to describe isolated gravitational systems. They are non-compact manifolds with one end on which the Riemannian metric decays at infinity towards the reference hyperbolic metric, at a suitable rate. I will explain the notion of geometric invariance at infinity, and introduce a method based on representation theory for the group of hyperbolic isometries SO(n,1) to classify the invariants at infinity, depending on the decay rate of the metric. In particular, the classical Wang-Chrusciel-Herzlich's mass is recovered. This talk is based on a joint work with Mattias Dahl and Romain Gicquaud.
There is a classical result first due to Keen known as the collar lemma for hyperbolic surfaces. A consequence of the collar lemma is that if two closed curves A and B on a closed orientable hyperbolizable surface have non-zero geometric intersection number, then there is an explicit lower bound for the length of A in terms of the length of B, which holds for any hyperbolic structure one can choose on the surface. By slightly weakening this lower bound, we generalize this statement to hold for all Hitchin representations. This is joint work with Tengren Zhang.
A flat structure on a closed orientable surface of genus g is a flat Riemannian metric with a finite number of singular points, which have a neighborhood isometric to a euclidean cone. Troyanov proved that the set of all such structures (up to isometry) with $n$ singular points and prescribed angle at those singular points is isomorphic in a very natural way to the moduli space of Riemann surfaces with $n$ marked points $M_{g,n}$. In a quite unknown and beautiful paper, Veech defines foliations on those moduli spaces whose leaves are flat surfaces with the same parallel transport, and proves that those leaves are endowed with natural homogenous geometric structures (precisely (PU(p,q), CP^{p+q-1})-structures).
The case of flat spheres has already been extensively studied by Picard, Deligne and Mostow in the framework of hypergeometric fucntions, and Thurston gave a much geometrically flavored interpretation of their work relating it directly to Veech's work. In that case, studying the metric completion of the geometric structures on the leaves leads to a construction of complex hyperbolic orbifolds and therefore construction of lattices in PU(1,n).
I will give an insight of the case where g=1 and n=2, namely flat tori with 2 conical singularities, where the geometric structure on the leaves happen to be complex hyperbolic of dimension 1 (or real hyperbolic of dimension 2). I will explain how the metric completion of the leaves can be geometricaly understood and address a few open questions on this foliation, which is far from being well understood. This is a joint work with Luc Pirio.
In this talk, we study the action of the group of polynomial automorphisms of C^n ( where n >2 ) which preserve the Markoff-Hurwitz polynomial. Our main results include the determination of the group, the description of a non-empty open subset on which the group acts properly discontinuously, and identities for the orbits of points in the domain of discontinuity. This is a joint work with Ser Peow Tan of National University of Singapore and Ying Zhang of Soochow University.
The critical exponent is a dynamical invariant which have been intensively studied in negatively curved manifolds. In this talk, i will define the basic notions about critical exponent and then present critical exponent for the diagonal action of two TeichmÃ¼ller representations of surface groups. C. Bishop and T. Steger have shown how to characterize that two representations are conjugated with this critical exponent. We will explain how this result admits a generalized version by an isolation theorem. During the talk we will explain the different notions used along the proof, in particular earthquakes and a theorem of large deviation of geodesic flow.
A group manifold is the quotient of a Lie group G by a subgroup Gamma of GxG (acting by left and right multiplication). When G has real rank one, e.g. G = SO(1,n), and when Gamma is undistorted in GxG, we relate this with the notion of Anosov representations, a generalization of convex cocompact subgroup due to François Labourie. Nice properties (sharpness, openness) of those rank-one group manifolds follow as corollaries. This is joint work with François Guéritaud, Fanny Kassel and Anna Wienhard.
A complex hyperbolic lattice is a lattice $\Gamma$ in the Lie group ${\rm SU}(n,1)$, $n\geq 2$. Such a lattice acts on the unit ball ${\mathbb B}^n\subset{\mathbb C}^n$ seen as the $n$-dimensional complex hyperbolic space. We will assume that $\Gamma$ is torsion free and uniform, so that the quotient $M=\Gamma\backslash{\mathbb B}^n$ is a closed manifold.
Let $G$ be a classical Hermitian Lie group, and $\rho$ a group homomorphism from $\Gamma$ to $G$. There is a number, the Toledo invariant, which measures the complex size of $\rho$. One can show that this number is bounded by a quantity depending only on the volume of $M$ and the rank of the symmetric space $X$ associated to $G$. The representation $\rho$ is said to be maximal if this bound is attained.
We prove that if $\rho:\Gamma\rightarrow G$ is a maximal representation then necessarily $G={\rm SU(p,q)}$, with $p\geq qn$, and $\rho$ essentially extends to a homomorphism ${\rm SU}(n,1)\rightarrow G$. More precisely, there exists a special holomorphic totally geodesic $\rho$-equivariant embedding from complex hyperbolic space ${\mathbb B}^n$ to the symmetric space $X$. This is a joint work with Vincent Koziarz.
A complex structure is an almost complex structure which is integrable. A local description of such a structure reveals a lot of algebraic equations. Sergei Merkulov has studied the Nijenhuis integrability condition and he has proposed a simple interpretation of the equations characterizing Nijenhuis structures in terms of homotopy algebras. Following this attempt to define "homotopy geometry", we make use of the curved Koszul duality and of a hint of differential geometry to describe complex manifolds as homotopy algebras.
Abstract:
I will give an introduction to the theory of Spectral
Networks, developed by Gaiotto, Moore and Neitzke during their
research about supersymmetry.
These objects have an independent interest for mathematicians, mostly in the theory of surfaces. They can be described as combinatorical objects on the surface, or, equivalently as some orbits associated to a conformal structure on the surface and a collection of holomorphic differentials.
The main focus will be on how to use these objects to give coordinates on the moduli spaces of representations of surface groups. These coordinates generalise Fenchel-Nielsen coordinates and Fock-Goncharov coordinates in an especially intriguing way.
The moduli space M_g of Riemann surfaces of genus g is the quotient of a complex manifold by a finite group and so it makes sense to speak of its Dolbeault cohomological dimension, namely the greatest q=q_g such that H^{0,q}(M_g,E) is not zero for a suitable holomorphic vector bundle E on M_g. It is conjectured that q_g=g-2, which is verified for g=2,3,4,5.
In this talk I will discuss a result which is not optimal but valid for all g: the number q_g is not greater than 2g-2. Since the cohomological dimension behaves well under smooth proper fibrations, the idea is to stratify the projectivized Hodge bundle over M_g and prove that the Dolbeault cohomology of such locally closed strata (the so-called moduli spaces of translation surfaces with fixed singularities) vanishes in high degrees. Such vanishing follows from the existence of an exhaustion function whose complex Hessian has controlled index. The construction of such functions relies on certain elementary geometric properties of translation surfaces.
In this talk I consider surfaces with conical singularities and finite area convex angular sectors in the plane. I start with the eigenvalue problem for the Dirichlet Laplacian. I give the definition of the zeta regularized determinant of the Laplacian. I explain how to use conformal transformations to differentiate this determinant with respect to the opening angle of the sector or of the cone. This leads to a variational Polyakov formula, when the variation is taken in the direction of a conformal factor with a logarithmic singularity. The results presented are in collaboration with Julie Rowlett and Werner Mueller.
An anti-de Sitter (AdS) manifold is a manifold provided with a Lorentz metric of constant curvature -1. As a consequence of theorems of Kulkarni-Raymond and Kassel, closed AdS 3-manifolds are, up to a finite cover, non trivial circle bundles over a closed surface of genus greater than 2. Here I will describe the space of all AdS structures on such a circle bundle and explain how to compute their volume.
This talk will describe various examples of surface groups in SO(4,1) in terms of fundamental domain of their action on S^3. This includes the first examples by Gromov-Lawson-Thurston, and new examples. We will also look at the quotient 3-manifolds which are circle bundles over closed surfaces and give a proof of a soft bound on the Euler number of such circle bundles.
A bi-Lagrangian structure on a symplectic manifold is given by a pair of transverse Lagrangian foliations. First I will describe such structures and their properties, and study their possible relations with hyperkaehler structures -which are the quaternionic equivalent of Kaehler structures. I will then show that these structures are relevant in Teichmueller theory, notably in the description of the geometry of quasi-Fuchsian space. This is work in progress joint with Andy Sanders.
A Kaehler metric is a Riemannian metric admitting a parallel field J of endomorphisms of the tangent space such that JÂ²=-Id. For a Riemannian metric that is not a product, such a J is the only possible type of non trivial parallel endomorphism field. This is not true for a pseudo-Riemannian metric ; on the contrary, those may admit an algebra A of parallel endomorphism fields of arbitrary dimension. I give structure results for A, together with the sets of germs of metric corresponding to each possible case. In the talk, I will focus on the most specific ''non Riemannian'' phenomenon : when the metric admits a field of parallel nilpotent endomorphism. A natural description of such metrics follows from an analogy with complex geometry : analogues of power series developments, of the holomorphic derivation, of a Kaehler potential etc., appear.
I will discuss how topological surface defects can be implemented in the study of a three-dimensional topological field theory (TFT). There turns out to be an obstruction, which takes values in the Witt group of modular tensor categories. Certain functors associated with a subclass of surface defects give rise to symmetries of the TFT. In particular, in the case of a Dijkgraaf-Witten topological field theoriy based on an abelian gauge group G they furnish a bijection between the Brauer-Picard group of invertible bimodule categories over the fusion category Vect(G) of G-graded vector spaces and braided auto-equivalences of the Drinfeld center of Vect(G).
One can associate to a labeled graph an Artin group, as well as various configuration spaces. We will discuss the geometry and topology of such groups and spaces, with an emphasis on cohomogical aspects.
We start by shortly reviewing standard algebras up to homotopy; the standard example being associative algebras. Deformations also naturally lead to Gerstenhaber algebras. We simultaneously present a graphical language and a geometric realization of this language. There is a tower of these algebras which we mention. They are tied to configurations of little cubes. We show how this fits in neatly with surfaces, hyperbolic geometry and moduli spaces.