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Post-edited  My favorite groups
Ghys, Etienne (Auteur de la Conférence) | CIRM (Editeur )

The world of groups is vast and meant for wandering! During this week, I will give seven short talks describing seven groups, or class of groups, that I find fascinating. These seven talks will be independent and I will have no intention of being exhaustive (this would be silly since there are uncountably many groups, even finitely generated!). In each talk, I will introduce the hero, state one or two results, and formulate one or two conjectures. The world of groups is vast and meant for wandering! During this week, I will give seven short talks describing seven groups, or class of groups, that I find fascinating. These seven talks will be independent and I will have no intention of being exhaustive (this would be silly since there are uncountably many groups, even finitely generated!). In each talk, I will introduce the hero, state one or two results, and formulate one or two c...

57S30 ; 58D05

There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what such a guide should look like.
It is quite a risky undertaking because it is all too easy to offend by omission, misrepresentation or other. I have not attempted a complete literature survey and inevitably these notes reflects my personal view, jaundiced as it may often be. My apologies for any offence caused.
I would like to express my warm thanks to Lukas Lewark, Alex Shumakovitch, Liam Watson and Ben Webster.
There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what ...

57M25 ; 57M27

A sub-Riemannian distance is obtained when minimizing lengths of paths which are tangent to a distribution of planes. Such distances differ substantially from Riemannian distances, even in the simplest example, the 3-dimensional Heisenberg group. This raises many questions in metric geometry: embeddability in Banach spaces, bi-Lipschitz or bi-Hölder comparison of various examples. Emphasis will be put on Gromov's results on the Hölder homeomorphism problem, and on a quasisymmetric version of it motivated by Riemannian geometry. A sub-Riemannian distance is obtained when minimizing lengths of paths which are tangent to a distribution of planes. Such distances differ substantially from Riemannian distances, even in the simplest example, the 3-dimensional Heisenberg group. This raises many questions in metric geometry: embeddability in Banach spaces, bi-Lipschitz or bi-Hölder comparison of various examples. Emphasis will be put on Gromov's results on the Hölder ...

53C20 ; 53C15

I will discuss recent applications of microlocal analysis to the study of hyperbolic flows, including geodesic flows on negatively curved manifolds. The key idea is to view the equation $(X + \lambda)u = f$ , where $X$ is the generator of the flow, as a scattering problem. The role of spatial infinity is taken by the infinity in the frequency space. We will concentrate on the case of noncompact manifolds, featuring a delicate interplay between shift to higher frequencies and escaping in the physical space. I will show meromorphic continuation of the resolvent of $X$; the poles, known as Pollicott-Ruelle resonances, describe exponential decay of correlations. As an application, I will prove that the Ruelle zeta function continues meromorphically for flows on non-compact manifolds (the compact case, known as Smale's conjecture, was recently settled by Giulietti-Liverani- Pollicott and a simple microlocal proof was given by Zworski and the speaker). Joint work with Colin Guillarmou. I will discuss recent applications of microlocal analysis to the study of hyperbolic flows, including geodesic flows on negatively curved manifolds. The key idea is to view the equation $(X + \lambda)u = f$ , where $X$ is the generator of the flow, as a scattering problem. The role of spatial infinity is taken by the infinity in the frequency space. We will concentrate on the case of noncompact manifolds, featuring a delicate interplay between ...

37D50 ; 53D25 ; 37D20 ; 35B34 ; 35P25

La géométrie stochastique est l'étude d'objets issus de la géométrie euclidienne dont le comportement relève du hasard. Si les premiers problèmes de probabilités géométriques ont été posés sous la forme de casse-têtes mathématiques, le domaine s'est considérablement développé depuis une cinquantaine d'années de part ses multiples applications, notamment en sciences expérimentales, et aussi ses liens avec l'analyse d'algorithmes géométriques. L'exposé sera centré sur la description des polytopes aléatoires qui sont construits comme enveloppes convexes d'un ensemble aléatoire de points. On s'intéressera plus particulièrement aux cas d'un nuage de points uniformes dans un corps convexe fixé ou d'un nuage de points gaussiens et on se focalisera sur l'étude asymptotique de grandeurs aléatoires associées, en particulier via des calculs de variances limites. Seront également évoqués d'autres modèles classiques de la géométrie aléatoire tels que la mosaïque de Poisson-Voronoi. La géométrie stochastique est l'étude d'objets issus de la géométrie euclidienne dont le comportement relève du hasard. Si les premiers problèmes de probabilités géométriques ont été posés sous la forme de casse-têtes mathématiques, le domaine s'est considérablement développé depuis une cinquantaine d'années de part ses multiples applications, notamment en sciences expérimentales, et aussi ses liens avec l'analyse d'algorithmes géométriques. ...

60D05 ; 60F05 ; 52A22 ; 60G55

I will discuss work in progress aimed towards defining contact homology using "virtual" holomorphic curve counting techniques.

37J10 ; 53D35 ; 53D40 ; 53D42 ; 53D45 ; 57R17

I will present results of three studies, performed in collaboration with M.Benli, L.Bowen, A.Dudko, R.Kravchenko and T.Nagnibeda, concerning the invariant and characteristic random subgroups in some groups of geometric origin, including hyperbolic groups, mapping class groups, groups of intermediate growth and branch groups. The role of totally non free actions will be emphasized. This will be used to explain why branch groups have infinitely many factor representations of type $II_1$. I will present results of three studies, performed in collaboration with M.Benli, L.Bowen, A.Dudko, R.Kravchenko and T.Nagnibeda, concerning the invariant and characteristic random subgroups in some groups of geometric origin, including hyperbolic groups, mapping class groups, groups of intermediate growth and branch groups. The role of totally non free actions will be emphasized. This will be used to explain why branch groups have infinitely ...

20E08 ; 20F65 ; 37B05

An endomorphism of a finitely generated free group naturally descends to an injective endomorphism on the stable quotient. We establish a geometric incarnation of this fact : an expanding irreducible train track map inducing an endomorphism of the fundamental group determines an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group G depends only on the component of the BNS invariant $\sum \left ( G \right )$ containing the associated homomorphism to the integers. In particular, it follows that if G is the mapping torus of an atoroidal fully irreducible automorphism of a free group and if the union of $\sum \left ( G \right ) $ and $\sum \left ( G \right )$ is connected then for every splitting of $G$ as a (f.g. free)-by-(infinite cyclic) group the monodromy is fully irreducible.
This talk is based on joint work with Spencer Dowdall and Christopher Leininger.
An endomorphism of a finitely generated free group naturally descends to an injective endomorphism on the stable quotient. We establish a geometric incarnation of this fact : an expanding irreducible train track map inducing an endomorphism of the fundamental group determines an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully ...

20F65 ; 57Mxx ; 37BXX ; 37Dxx

Theory of persistence modules is a rapidly developing field lying on the borderline between algebra, geometry and topology. It provides a very useful viewpoint at Morse theory, and at the same time is one of the cornerstones of topological data analysis. In the course I'll review foundations of this theory and focus on its applications to symplectic topology. In parts, the course is based on a recent work with Egor Shelukhin arXiv:1412.8277

37Cxx ; 37Jxx ; 53D25 ; 53D40 ; 53D42

We will give a survey of recent research progress on ancient or eternal solutions to geometric flows such as the Ricci flow, the Mean Curvature flow and the Yamabe flow.
We will address the classification of ancient solutions to parabolic equations as well as the construction of new ancient solutions from the gluing of two or more solitons.

53C44

Recently, Armstrong, Reiner and Rhoades associated with any (well generated) complex reflection group two parking spaces, and conjectured their isomorphism. This has to be seen as a generalisation of the bijection between non-crossing and non-nesting partitions, both counted by the Catalan numbers. In this talk, I will review the conjecture and discuss a new approach towards its proof, based on the geometry of the discriminant of a complex reflection group. This is an ongoing joint project with Iain Gordon. Recently, Armstrong, Reiner and Rhoades associated with any (well generated) complex reflection group two parking spaces, and conjectured their isomorphism. This has to be seen as a generalisation of the bijection between non-crossing and non-nesting partitions, both counted by the Catalan numbers. In this talk, I will review the conjecture and discuss a new approach towards its proof, based on the geometry of the discriminant of a complex ...

06B15 ; 05A19 ; 55R80

We discuss some new results for the Cheeger constant in dimension two, including:
- a polygonal version of Faber-Krahn inequality;
- a reverse isoperimetric inequality for convex bodies;
- a Mahler-type inequality in the axisymmetric setting;
- asymptotic behaviour of optimal partition problems.
Based on some recent joint works with D.Bucur,
and for the last part also with B.Velichkov and G.Verzini.

49Q10 ; 52B60 ; 35P15 ; 52A40 ; 52A10 ; 35A15

In this lecture I will describe a framework for the Fredholm analysis of non-elliptic problems both on manifolds without boundary and manifolds with boundary, with a view towards wave propagation on Kerr-de-Sitter spaces, which is the key analytic ingredient for showing the stability of black holes (see Peter Hintz' lecture). This lecture focuses on the general setup such as microlocal ellipticity, real principal type propagation, radial points and generalizations, as well as (potentially) normally hyperbolic trapping, as well as the role of resonances. In this lecture I will describe a framework for the Fredholm analysis of non-elliptic problems both on manifolds without boundary and manifolds with boundary, with a view towards wave propagation on Kerr-de-Sitter spaces, which is the key analytic ingredient for showing the stability of black holes (see Peter Hintz' lecture). This lecture focuses on the general setup such as microlocal ellipticity, real principal type propagation, radial points ...

35A21 ; 35A27 ; 35B34 ; 35B40 ; 58J40 ; 58J47 ; 83C35 ; 83C57

L'écologie est une discipline quantitative dans laquelle les mathématiques sont présentes sous différentes formes depuis très longtemps. En conséquence, l'arrivée massive d'ordinateurs de plus en plus puissants dans les laboratoires dans les dernières décennies, a conduit à une explosion de la modélisation dans ce domaine, sous forme de calculs numériques mais également par l'analyse mathématique de modèles relativement simples. Cette croissance importante de l'activité de modélisation mathématique a été accompagnée par une augmentation de la complexité des modèles d'écologie qui tentent d'intégrer la plus grosse quantité de processus connus possible. Parallèlement, les moyens d'expérimentations et d'observation du milieu naturel n'ont pas cessé de s'améliorer, produisant ainsi des bases de données de plus en plus complètes dans la description du fonctionnement des écosystèmes. Paradoxalement, la formulation de base des processus utilisée dans les modèles complexes est toujours la même et fondée sur des expérimentations réalisées dans des conditions homogènes de laboratoire au cours du XXème siècle. Nous posons la question de l'intérêt d'une description adéquate d'un écosystème pour comprendre ses réponses à différentes perturbations. Une approche consiste à utiliser des formulations mécanistes des processus, c'est à dire des formulations fondées sur des détails expliquant la cause de la réalisation des processus, plutôt que des formulations empiriques acquises dans des conditions différentes du milieu dans lequel on les applique. Cette prise en compte des mécanismes induit encore un surcroit de complexité. Les mathématiques fournissent un ensemble d'idées et de méthodes permettant tout d'abord de produire des formulations adaptées à la prise en compte des mécanismes et également d'aborder cette complexité des modèles écosystémiques, voire dans certains cas de la réduire. Nous illustrerons cette démarche à travers des exemples d'applications variés. L'écologie est une discipline quantitative dans laquelle les mathématiques sont présentes sous différentes formes depuis très longtemps. En conséquence, l'arrivée massive d'ordinateurs de plus en plus puissants dans les laboratoires dans les dernières décennies, a conduit à une explosion de la modélisation dans ce domaine, sous forme de calculs numériques mais également par l'analyse mathématique de modèles relativement simples. Cette croissance ...

34E13 ; 34E15 ; 34E20 ; 92D25 ; 92D40

Multi angle  Pseudo-Anosov braids are generic
Wiest, Bert (Auteur de la Conférence) | CIRM (Editeur )

We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with $n\geq 3$ strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius $l$ tends to $1$ exponentially quickly as $l$ tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic pairs of elements of the braid group, the conjugacy search problem can be solved in quadratic time. The idea behind both results is that generic braids can be conjugated ''easily'' into a rigid braid.
braid groups - Garside groups - Nielsen-Thurston classification - pseudo-Anosov - conjugacy problem
We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with $n\geq 3$ strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius $l$ tends to $1$ exponentially quickly as $l$ tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic pairs of elements of the braid group, the ...

20F36 ; 20F10 ; 20F65

How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart).
Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and vice-versa. This implies a conjecture by Przytycka and Przytycki from 1993, a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov's systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions.
Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length $O(g^{3/2}n^{1/2})$ for any triangulated combinatorial surface of genus g with n triangles, and describe an $O(gn)$-time algorithm to compute such a decomposition.
Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations.
systolic geometry - computational topology - topological graph theory - graphs on surfaces - triangulations - random graphs
How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart).
Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed ...

05C10 ; 68U05 ; 53C23 ; 57M15 ; 68R10

In this talk we shall present some results concerning global smooth solutions to the three-dimensional Navier-Stokes equations set in the whole space $(\mathbb{R}^3)$ :
$\partial_tu+u\cdot \nabla u-\Delta u = -\nabla p$, div $u=0$
We shall more particularly be interested in the geometry of the set $\mathcal{G}$ of initial data giving rise to a global smooth solution.
The question we shall address is the following: given an initial data $u_0$ in $\mathcal{G}$ and a sequence of divergence free vector fields converging towards $u_0$ in the sense of distributions, is the sequence itself in $\mathcal{G}$ ? The related question of strong stability was studied in [1] and [2] some years ago; the weak stability result is a recent work, joint with H. Bahouri and J.-Y. Chemin (see [3]-[4]). As we shall explain, it is necessary to restrict the study to sequences converging weakly up to rescaling (under the natural rescaling of the equation). Then weak stability can be proved, using profile decompositions in the spirit of P. Gerard's work [5], in an anisotropic context.
In this talk we shall present some results concerning global smooth solutions to the three-dimensional Navier-Stokes equations set in the whole space $(\mathbb{R}^3)$ :
$\partial_tu+u\cdot \nabla u-\Delta u = -\nabla p$, div $u=0$
We shall more particularly be interested in the geometry of the set $\mathcal{G}$ of initial data giving rise to a global smooth solution.
The question we shall address is the following: given an initial data $u_0$ in ...

35Q30 ; 76E09

There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what such a guide should look like. It is quite a risky undertaking because it is all too easy to offend by omission, misrepresentation or other. I have not attempted a complete literature survey and inevitably these notes reflects my personal view, jaundiced as it may often be. My apologies for any offence caused. I would like to express my warm thanks to Lukas Lewark, Alex Shumakovitch,Liam Watson and Ben Webster. There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what ...

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