Organizers: Clara Aldana, Vincent Pecastaing.
Unless otherwise specified, the seminar will be on Monday, 3-4pm and will take place in Room S-6A (MNO).
In this talk I shall present some hints of the current knowledge on positive mass « theorems » as well as a few new results.
A complete hyperbolic manifold of finite volume is said to bound geometrically if it is isometric to the boundary of a complete finite-volume hyperbolic manifold with totally geodesic boundary. This is a non-trivial invariant for hyperbolic 3-manifolds.
We show that the number of non-compact geometrically bounding hyperbolic 3-manifolds with bounded volume grows asymptotically at least super-exponentially with the bound on the volume, both in the arithmetic and non-arithmetic case.
This is part of a work in progress joint with Alexander Kolpakov.
We define and derive several properties of a distance on the space of convex bodies in the n-dimensional Euclidean space, up to translations and homotheties, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient Lorentzian structure is an extension of the area form of convex bodies.
We deduce that the space of shapes of convex bodies (i.e. convex bodies up to similarities) has a proper distance with curvature bounded from below by -1. In dimension 3, this space naturally identifies with the space
of distances with non-negative curvature on the 2-sphere.
Joint work with Clément Debin
SOL is one of the classical eight Thurston's homogeneous geometries (perhaps the most exotic one).
A model of SOL is R^3 with Riemannian metric ds^2 = dz^2 + exp(2z)dx^2 + exp(-2z)dy^2.
Suppose one wants to "see" the shape of large spheres in SOl (in the coordinate xyz-space),
one should then be able to compute the distance between 2 points. But that is very complicated.
On the other hand if one replaces the Riemannian metric by a specific Finsler metric then one can explicitly
compute distances and draw spheres. The Finsler metric is not the Riemannian metric of the original problem,
but it is asymptotic in a precise sense and therefore the Finsler balls are very accurate models of the Riemannian balls.
The Finsler metric is inspired by cardboards models in architecture and will be defined and discussed in the talk.
The method can be generalized to other (Solvable groups) geometries.
The systole of a closed surface is the length of a shortest
non-contractible, closed curve on the surface.
A minimal surface $\Sigma$ in an closed ambient three manifold $M$ is a
submanifold, that is a critical point for the area functional or,
equivalently, the mean curvature of $\Sigma$ is identically zero.
There are a many results on the space of minimal surfaces of bounded
genus or index if the ambient manifold has some positivity condition on
the curvature, e.g. positive Ricci or scalar curvature.
In contrast, there are only few results describing asymptotic properties
of a sequence of minimal surfaces $\Sigma_j \subset M$ with
$\genus(\Sigma_j) \to \infty$.
We show that for a such a sequence the systole of $\Sigma_j$ has to tend
to zero, if $M$ has positive Ricci curvature.
This is joint work with Anna Siffert.
The flip-graph of a convex polygon π is the graph whose vertices are the triangulations of π and whose edges correspond to flips between them. The eccentricity of a triangulation T of π is the largest possible distance in this graph from T to any triangulation of π. Let n stand for the number of vertices of π. It is well known that, when all n-3 interior edges of T are incident to a given vertex, the eccentricity of T in the flip-graph of π is exactly n-3. The purpose of this talk is to generalize this statement to arbitrary triangulations of π: if n-3-k denotes the largest number of interior edges of T incident to a vertex, and if k≤n/2-2, the eccentricity of T in the flip-graph of π is exactly n-3+k. Surprisingly, the value of k turns out to characterize eccentricities if it is small enough. More precisely, when k≤n/8-5/2, T has eccentricity n-3+k if and only if exactly n-3-k of its interior edges are incident to a given vertex. A number of related questions will be mentioned and discussed.
The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve. Given a curve represented by closed walks of length at most l on a combinatorial surface of complexity n we describe a simple algorithm to compute the geometric intersection number of the curve in O(n + l^2 ) time. We also propose an algorithm of complexity O(n + l.log^2(l)) to decide if the geometric intersection number of a curve is zero, i.e. if the curve is homotopic to a simple curve. I will first explain how the problem can be handled from a mathematical point of view and then how we choose to discretize it to obtain efficient algorithms.
The deformation space AH(M) of a compact hyperbolic 3-manifold is the analogous of the Teichmüller space for a 2-manifold. The interior of AH(M) is a disjoint union of balls with the same dimension but the space itself may have unexpected topological features (non local connectivity for example).
I will describe these features et explain how they appear. Then I will give examples of points in the neighborhood of which the topology of AH(M) is controlled.
In his original proof of the uniformisation theorem of Haken manifolds, Thurston stated a theorem which is now the bounded image theorem. In all of books, surveys etc on Thurston’s uniformisation theorem, only a weaker version of this theorem was proved. In this talk, I shall present a proof of the original stronger version.
The n-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus g with n-punctures which becomes trivial on the closed surface. I am interested in the least dilatation of pseudo-Anosov pure surface braids. For the n=1 case, upper and lower bounds on the least dilatation were proved by Dowdall and Aougab—Taylor, respectively. In this talk, I will describe the upper and lower bounds I have proved as a function of g and n.
We shall describe and partially classify those Riemannian manifolds carrying a nonconstant function satisfying an Obata-like-equation.
Ongoing joint work with Ines Kath (Greifswald) and Georges Habib (Beirut).
Common work with Pierre Mounoud (Bordeaux). Its motivation is twofold.
Here M is a (complete & simply connected) Riemannian or
pseudo-Riemannian manifold.
On the one hand, if the tangent bundle of M splits in a direct
orthogonal sum which is stable by parallel transport, this distribution
"integrates" and M is a Riemannian product. This is a well-known result
by G. de Rham and H. Wu. But, in the pseudo-Riemannian case, if the
tangent bundle contains a parallel sub-bundle without parallel
orthogonal complement (hence isotropic), the situation is far more
complicated. Only a few local results are known, and still fewer global
ones. We study the simplest compact case: the Lorentzian 3-manifolds
with a parallel isotropic vector field. We classify them.
On the other hand, we were interested in studying the group Aff(M) of
affine morphisms of M, i.e. morphisms preserving its connection, which
is a rigid structure. It contains Isom(M). Now think that Conf(M) is
another isomorphism group of a rigid structure, containing Isom(M), and
that it has the following remarkable property in the compact riemannian
case: if it is "big", i.e. Conf(M)≠Isom(M) for all metric in the
conformal class, then M is very specific (round sphere). Does Aff(M)
satisfy a similar property? The riemannian case is treated (K. Yano, A.
Zeghib). Now the pseudo-riemannian case, in dimension 3, amounts to
study the Lorentzian manifolds with a parallel vector field: the problem
presented above.
This work is an occasion to look at a range of typical global
pseudo-riemannian phenomena, and at the dynamical questions they let arise.
The « Spherics » by Menelaus of Alexandria (1st-2nd c. A.D.) is probably the most important treatise ever written on spherical geometry.
It is a profound work, introducing new methods in geometry, intrinsic to the sphere, containing 91 propositions, some of which are very difficult to prove.
An edition, from Arabic texts (the Greek original does not survive), was just published by De Gruyter, in their series Scientia Graeco-Arabica, No. 21.
https://www.degruyter.com/view/product/496630
This publication contains the first English translation of Menelaus’ treatise together with an extensive commentary.
In this talk, I will present the content of the Spherics and explain some of the major theorems it contains.
On a "complicated" surface (homeomorphic to a sphere with a number of
handles attached), how to find the shortest closed curve that cannot be
shrunk to a point by letting it slide on the surface (the shortest
non-contractible closed curve)? How to cut a surface to make it planar
(homeomorphic to a disk)? How to shorten a curve as much as possible on
a surface by deformation (homotopy)?
Since the 2000s, the field of computational topology of graphs on
surfaces has emerged, which aims at studying such problems, revisiting
natural questions of surface topology with an algorithmic viewpoint. I
will survey some of the results in this field, by different authors,
introduce the main techniques, and conclude with a brief description of
some more recent results, which require more advanced tools from
systolic geometry and mapping class group theory.
It is well-known that a simply connected homogeneous Riemannian manifold satisfies the following "local-to-global rigidity" property:
any simply connected Riemannian manifold N whose balls of radius 1 are isometric to the ball of radius 1 of M must be isometric to M.
In this talk we shall study this property in the setting of vertex-transitive graphs.
In particular, we characterize local-to-global rigid building among those of SL(n,K) where K is a local field (not necessarily commutative).
This is a joint work with Mikael de la Salle.
A similarity structure on a manifold consists in giving a Riemannian metric on a neighborhood of each point such that the metrics on the intersection of two neighborhoods are homothetic, i.e. proportional by means of a locally constant function. The basic example is that of the cone over a Riemannian manifold which gives in the case of the standard sphere a (flat) Hopf manifold. The question is whether there exist more complicated examples?
W.P. Thurston has defined two assymetric norm on the Teichmüller space. The most famous is the one related
to the minimization problem of the Lipschitz constant of a map between two hyperbolic surfaces. In the same paper,
he also defined another one, dual to the first one in some meaning, which is defined as the length of measured geodesic
laminations, once identified in the correct way tangent vectors with measured geidesic laminations. I will show that this
construction can be generalized to some assymetric Finsler norm on $H^1(G, R^{1,n})$ where $G$ is a cocompact lattice of $SO(1,n)$.
I will also comment on an useful tool related to this: the co-Minkowski space, i.e. the space of spacelike hyperplanes in the
Minkowski space.
This is a work in collaboration with F. Fillastre.
We will answer the following question:
Question(Agol): Is there a 3-dimensional manifold M with no divisible subgroups in \pi_1(M) that is locally hyperbolic but not hyperbolic?
Specifically we construct an example of such a 3-manifold. Time permitting, we will state a characterisation of hyperbolizable 3-manifolds in a reasonable class and maybe give an example of a non residually finite Kleinian group.
Spacelike surfaces in 3-dimensional Minkowski space with constant mean curvature (CMC), respectively constant Gaussian curvature (CGC), give a nice way to think about harmonic maps (resp. minimal Lagrangian maps) between hyperbolic surfaces. Recently, Francesco Bonsante and Andrea Seppi characterized a large class of constant CGC surfaces in terms of their asymptotics. Together with Bonsante and Seppi, we extend this to a complete characterization of all properly embedded CGC surfaces in R^2,1 and do the same for CMC surfaces.
Symmetric spaces of non-compact type have analogs of infinite dimension with a pleasant geometry as soon they have finite rank. The first example of such a space is the infinite dimensional hyperbolic space. We will consider actions by isometries of lattices of semi-simple Lie groups and prove some rigidity results of higher rank lattices or complex hyperbolic lattices.
The systole of a Riemannian manifold M is defined as the length of a shortest non-contractible closed geodesic in M. Hyperbolic surfaces with arbitrary short systole can be constructed by using Teichmüller theory. In 1991 P. Buser and P. Sarnak showed that congruence coverings S_p of an arithmetic hyperbolic surface have systole growing at least as 4/3 log(area(S_p)). In this talk we will present recent results generalizing this to arithmetic hyperbolic manifolds and Hilbert modular varieties in any dimension.
In joint work with Yves Benoist, we study the action of the affine group G of R^d on the affine Grassmannian X_{k,d}, that is, the set of affine k-spaces in R^d. When G is endowed with a Zariski-dense probability measure, we give a criterion for the existence of an invariant probability measure. Such a measure, if it exists, is unique.
We consider problems where a Laplace-like operator is defined on a sequence of spaces that change either in dimension or in topology and that converge towards a limit space. We define a notion of convergence for the Laplace-like operators. Examples of such families are thin manifolds converging to a topological graph in the limit, discrete graphs converging to fractals or manifold with many holes of small radius removed. The convergence is formulated in an abstract level and can be applied in many situations.
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.
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.
References