Séminaire GT3

organisé par l'équipe Géométrie

• Alexey Sossinsky

Normal forms of closed planar polygonal lines: mathematical theorems and experiments, biological applications

8 janvier 2017 - 16:30Salle de conférences IRMA

Abstract.--- The subject matter of the talk originates in a simple biological model due to I.~Shapiro-Pyatetsky and studied by Yu.~Ilyashenko, A.~Leontovich and others. But the talk will begin with a purely mathematical theorem, which can be called the {\it Whitney theorem for polygonal curves}. It asserts that any regular closed polygonal line $\gamma : S^1 \to R^2$ is classified up to regular homotopy by its winding number $w(\gamma)$. (Here the word {\it regular} for polygonal lines means that successive edges of the line cannot have any common interior points and $F: \s^1 \times [0,1]\to R^2$ is a regular homotopy if $F(\cdot, t)$ is a regular polygonal line for all $t\in [0,1]$.) When the winding number is nonzero, Shapiro-Pyatetsky defined the {\it normal form} of a closed polygonal line as a line with constant edge length inscribed in a circle (a regular polygon if $w(\gamma)=1$, a closed polygonal line with equal edges inscribed in a circle and going around it twice if $w(\gamma)=2$, etc.). He posed the following problem: to construct an algorithm that takes each regular closed polygonal line to normal form by means if small moves defined by local conditions. One can also define the normal form of a closed polygonal line in the case $w(\gamma)=0$ as kind of figure eight curve inscribed in a lemniscate and pose the same problem for {\it all} regular closed polygonal lines, not only those for which $w(\gamma)>0$. \medskip I will describe results obtained in this direction by various authors and demonstrate videos that show how such algorithms experimentally solve this problem (joint work with S.~Avvakumov) and discuss the biological applications of these results, as well as the biological interpretation of some of my previous results obtained jointly with S.~Avvakumov and O.~Karpenkov.
• Thomas Mettler

Minimal Lagrangian connections

9 janvier 2017 - 14:00Salle de conférences IRMA

Abstract: A connection on the tangent bundle of a smooth manifold M can be understood as a map into an affine bundle over M whose total space carries a pseudo-Riemannian metric as well as a symplectic form, both of which can be constructed in a canonical fashion from the projective equivalence class of the connection. This viewpoint gives rise to the notion of a minimal Lagrangian connection. I will discuss the classification of minimal Lagrangian connections on compact oriented surfaces of non-vanishing Euler characteristic and talk about relations to convex projective geometry and non Ricci-flat Einstein metrics.
• Ivan Babenko

TBA

15 janvier 2017 - 14:00Salle de séminaires IRMA

• Andrea Seppi

Area-preserving diffeomorphisms of the hyperbolic plane and convex surfaces in Anti-de Sitter space.

16 janvier 2017 - 14:00Salle de séminaires IRMA

Teichmüller spaces of surfaces of infinite type, II

23 janvier 2017 - 14:00Salle de séminaires IRMA

I will explain the main problems and present recent results
• Stéphane Marseglia

Variétés projectives convexes de volume fini

6 février 2017 - 14:00Salle de séminaires IRMA

Une variété projective convexe est le quotient d’un ouvert proprement convexe de l’espace projectif par un groupe discret de transformations projectives. On s’intéresse aux holonomies des structures projectives strictement convexes. Lorsque les variétés projectives convexes sont compactes, Benoist (pour la fermeture) et Koszul (pour l’ouverture) ont montré que ces holonomies forment une union de composantes connexes de la variété des représentations. Afin d’étendre ce résultat au cas des variétés projectives convexes de volume fini, nous étudierons la preuve de la fermeture. L’ouverture a été prouvée récemment par Cooper, Long et Tillmann.
• Federica Fanoni

Basmajian-type inequalities for maximal representations

13 février 2017 - 14:00Salle de séminaires IRMA

Basmajian's celebrated identity gives a way to compute the length of the boundary of a hyperbolic surface in terms of the lengths of the so-called orthogeodesics (geodesics orthogonal to the boundary at both endpoints). This identity can be generalized to the context of maximal representations. This is a class of representations of the fundamental group of a surface that can be seen as a generalization of Teichmüller space. I will describe the classical identity, introduce maximal representations and discuss Basmajian's identity in this setup. Joint work with Beatrice Pozzetti.
• Andrew Yarmola

Hyperbolic 3-manifolds with low cusp volume

27 février 2017 - 14:00Salle de séminaires IRMA

For a cusped hyperbolic 3-manifold, one can 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. Our classification also gives a direct path to classify the first 3 smallest volume closed hyperbolic manifolds. As in some other results on hyperbolic 3-manifolds of low volume, our technique utilizes a rigorous computer assisted search. This 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.
• Florent Schaffhauser

Composantes de Hitchin pour les groupes fondamentaux d'orbi-surfaces compactes

6 mars 2017 - 14:00Salle de séminaires IRMA

Soit Y une orbi-surface compacte connexe de caractéristique d'Euler négative et soit \Pi son groupe fondamental orbifold. Soit R(\Pi, n) l'espace des représentations orbifold de \Pi dans PSL(n;R). Le but de l'exposé est de montrer que R(\Pi, n) possède des composantes connexes homéomorphes à une boule dont on sait calculer explicitement la dimension (pour n=2 et 3, on retrouve des formules connues, dues respectivement à Thurston et à Choi et Goldman). On donne ensuite des applications à l'étude des propriétés de rigidité des groupes de Coxeter hyperboliques. Travail en commun avec Daniele Alessandrini et Gye-Seon Lee (Heidelberg).
• Melkana Brakalova

Local and boundary behavior of quasiconformal maps

20 mars 2017 - 14:00Salle de séminaires IRMA

Abstract: "Quasiconformal maps are solutions to the Beltrami equation and posses many interesting properties. When certain conditions are imposed on the Beltrami coefficient (complex dilatation), the solutions exhibit additional nice geometric behavior that can be used to study of the universal Teichm\"uller space. We will discuss some such conditions and the corresponding behavior of the solutions."
• Neomie Combe

A’Campo’s geometric pictures for the space of complex polynomials

27 mars 2017 - 14:00Salle de séminaires IRMA

Abstract: A cellulation for the space of complex, polynomials $P$ of degree $d\geq 1$ is given. Each polynomial is characterized by A’Campo's geometric pictures’’, which are the inverse images of the union of the real and imaginary axis. These pictures provide a semi-algebraic stratification for the space. The strata are contractible by Riemann's theorem on the conformal structure of $S^{2}$.
Using \L{}ojasiewicz's triangulation, we provide a new cell decomposition. From this cell decomposition follow the cohomology groups for the space of polynomials.
This approach is reminiscent of the Grothendieck dessin d'enfants'', but is far from the construction of Shabat and Grothendieck, concerning only polynomials having two critical values.
• Dmitry Millionschikov

Lie algebras of slow growth and projective geometry

24 mai 2017 - 14:00Salle de séminaires IRMA

• Lizhen Ji

The Schottky problem from the metric geometric perspective

29 mai 2017 - 14:00Salle de séminaires IRMA

Abstract.--- The moduli space of compact Riemann surfaces of genus 1 can be identified with the quotient of the upper half plane by the modular group SL(2, Z). It admits two important generalizations: the moduli space M_g of compact Riemann surfaces of genus g greater than or equal to 1, and the moduli space A_g of principally polarized abelian varieties of dimension g. Besides various similarities between them, there is a period (or Jacobian) map from M_g to A_g. The classical Schottky problem is to understand the image of M_g in A_g. Besides being a quasi-projective variety, A_g is also a locally symmetric space of finite volume with respect to the invariant metric. We will discuss several results on the size, location and shape of the image of M_g with respect to this complete metric of A_g.
• Ken'ichi Ohshika

Le mouvement discontinu des applications de Cannon-Thurston

12 juin 2017 - 14:00Salle de séminaires IRMA

• Sonia Cannas

Algebraic structures on musical chords and their geometric visualizations

19 juin 2017 - 14:00Salle de séminaires IRMA

The subject is to show some applications of algebra and geometry in music theory. The main idea of transformational theory in music is to model musical transformations using algebraic structures. The most famous example is the neo-Riemannian group called PLR. (The terms neo-Ridmannian refers to the famous music theorist Hugo Riemann, and not to Bernhard Riemann.) Its transformations can be modeled by several geometric structures, of which the most important is the Tonnetz, a graph discovered by Euler in his musical investigations. I will present a generalization of the PLR group to seventh chords to describe the parsimonious voice leading.
• Hemangi Shah

Asymptotically harmonic manifolds with minimal horospheres.

3 juillet 2017 - 14:00Salle de séminaires IRMA

Abstract: Let (M,g) be an asymptotically harmonic manifold with minimal horospheres. Let {ei} be an orthonormal basis of TpM and let bei be the corresponding Busemann functions on M. Then we show that : (1) The vector space V ={bv | v ∈ TpM } is finite dimensional and dim V = dim M = n. (2) F : M → Rn defined by F(x) = (be1(x); be2(x); …; ben(x)) is an isometry and therefore, M is flat. Thus, the flatness of M is shown by using the strongest criterion.

On the work of Maryam Mirzakhani

4 septembre 2017 - 14:00Salle de séminaires IRMA

• Firat Yasar

Riemann surfaces and complex functions

13 septembre 2017 - 14:00Salle de conférences IRMA

• Elena Frenkel

Sur un problème de Schubert en géométrie non-euclidienne

22 septembre 2017 - 15:00Salle de séminaires IRMA

• Olivier Guichard

Groupes convexes cocompacts en rang supérieur

16 octobre 2017 - 14:00Salle de séminaires IRMA

[répétition du séminaire Bourbaki du 21/10]
• Jonas Beyrer

Cross ratios on Furstenberg boundaries

6 novembre 2017 - 14:00Salle de séminaires IRMA

A rank one symmetric space of non-compact type carries naturally a cross ratio on its visual boundary, which has many interesting applications. In particular the cross ratio characterizes the isometry group by its boundary action. We will use a similar geometric construction as for a rank one space to define cross ratios on Furstenberg boundaries of higher rank symmetric spaces of non-compact type. By showing several properties of those cross ratios, in particular that they characterize the isometry group of the symmetric space, we motivate that we get a reasonable generalization of the rank one case.
• Gye-Seon Lee

Exotic quasi-Fuchsian groups

13 novembre 2017 - 14:00Salle de séminaires IRMA

Let G(d) be the isometry group of the d-dimensional hyperbolic space. A subgroup Q of G(d+1) is quasi-Fuchsian if Q is a convex cocompact discrete subgroup of G(d+1) and the limit set of Q is homeomorphic to the (d-1)-dimensional sphere. In this talk, I will explain how to construct examples of quasi-Fuchsian groups which are not isomorphic to any uniform lattice of G(d) using the Tits-Vinberg representation of Coxeter groups. Joint work with Ludovic Marquis.