Séminaire de topologie, géométrie et algèbre (archives)

Weierstrass points on algebraic curves are special points of high importance in algebraic geometry and arithmetic geometry. In this talk, we study how those special points behave when the algebraic curve degenerates to a nodal curve. To this end, we first explain why tropical geometry is a relevant formalism for studying degeneration questions. We then define a tropical analogue on metric graphs (seen as tropical curves) for Weierstrass points, and explore the properties of the so-called “tropical Weierstrass locus”. We also associate intrinsic weights to the connected components of this locus, and show that their total sum for a given metric graph and divisor is a function of few combinatorial parameters (degree and rank of the divisor, genus of the metric graph).

In 2000, Eliashberg and Polterovich introduced the notion of orderability to investigate the structure of the group of contact diffeomorphisms and the structure of isotopy classes of Legendrian submanifolds. Roughly speaking, a group of contact diffeomorphisms is orderable if the relation induced by the partial order on contact hamiltonian maps induces a partial order on the associated time-one flows. In this talk, we will explain why orderability is equivalent to the existence of spectral selectors and how these selectors can be used to derive multiple geometric properties in the orderable situation. This is a joint work with Pierre-Alexandre Arlove.

In this talk, I will present recent results concerning special values of certain combinatorial zeta functions counting geodesic paths in triangulations. I will show that those values are related to some topological invariants. As such, we recover the first Betti number or L^2-Betti number of a compact manifold, as well as the linking number of knots in a 3-manifold. This is a joint work with Léo Bénard, Viet Dang and Thomas Schick.

(The talk will be in French.) Je décrirai la première construction, due à Hyde et Lodha, de groupes simples de présentation finie agissant fidèlement sur la droite. Il s'agit de groupes d'homéomorphismes affines par morceaux convenablement définis. L'exposé ne nécessite de prérequis ni en dynamique, ni en théorie des groupes. Il s'agit d'une variante d'une construction due à R. Thompson dans les années 60, qui donnait alors des groupes simples infinis de présentation finie agissant fidèlement sur le cercle - qui étaient alors les premiers exemples connus de groupes simples infinis de présentation finie.

Abstract: We will discuss smooth embeddings of 3-manifolds in 4-manifolds from both constructive and obstructive perspectives. We will present new results for the pair of Brieskorn spheres and sums of complex projective planes. Our results are based on Kirby calculus and Floer and gauge theoretic cobordism invariants.

An important application of tropical geometry is Mikhalkin's correspondence theorem. It states that counting algebraic curves on toric surfaces is the same as counting tropical curves with multiplicities. Several multiplicities can be chosen. In particular, the count with the Block-Göttsche multiplicities leads to the tropical refined invariant, which is a polynomial. In this talk we will investigate the polynomial behavior of the coefficients of this invariant.

(Joint with Erwan Brugallé and Lucía López de Medrano.) Combinatorial patchworking is a technique introduced by Viro to study the topology of real algebraic hypersurfaces. It's base ingredients are a triangulation of a lattice polytope and a sign distribution on its vertices. If the triangulation is "convex", the construction can be translated to the dual setting of tropical varieties. In recent years, this dual viewpoint has promoted new results using tropical homology theories. For example, Renaudineau and Shaw proved a conjecture by Itenberg bounding the Betti numbers of the real part of a (unimodular) patchwork hypersurface in terms of the Hodge numbers of the complex part.

Ce titre est un peu sibyllin. Il désigne un théorème prouvé par Jean Cerf à la fin des années 60, énonçant que tout difféomorphisme de la sphère de dimension trois se prolonge en difféomorphisme de la 4-boule. Par conséquent, aucune potentielle 4-sphère exotique ne pourra être obtenue par un recollement, aussi "exotique" soit-il, de deux hémisphères.

Le vrai théorème de Cerf (1968) énonce que tout difféomorphisme de la 3-sphère préservant l'orientation est isotope à l'identité. $\Gamma_4=0$ en est une conséquence immédiate.

I'll present new q-deformations of Lie algebras linked to the modular group and the q-rationnal numbers by Morier-Genoud and Ovsienko. In particular we describe deformations of sl2 and the Witt algebra. These deformations are realized as differential operators acting on the hyperbolic plane, giving new insights into q-rationnals.