Séminaire de Topologie et Géométrie — semestre d'automne 2019

Le séminaire de Topologie et Géométrie de la section de mathématiques de l'université de Genève a lieu le jeudi de 14:15 à 15:15. Il se déroule en salle 623 (6ème étage) du 2–4 rue du lièvre.

The Topology and Geometry seminar of the mathematics departement of Geneva university happens on Thursdays from 14:15 to 15:15. It takes place in room 623 (6th floor) at 2–4 rue du lièvre.

Exposés passés

  • Orateur: Roland van der Veen (Universiteit Leiden).
  • Titre: Drinfeld's double according to knot theory.
  • Résumé: The double construction is a way to produce a more interesting Hopf algebra by tensoring the original by its dual. All the commonly used quantum groups arise this way and this is usually explained in the context of Lie algebra theory. In this talk we show how Drinfeld's construction can derived naturally from simple knot theoretical considerations. No knowledge of quantum groups or even Hopf algebras is necessary to follow this talk because all these concepts have a natural topological interpretation that can be used instead. Time permitting we will also discuss how our point of view is especially useful in the case where the algebras are infinite dimensional. In some sense the simplest examples appear to be of this kind. For example, choosing the double of the enveloping algebra of the two-dimensional Lie algebra we will produce the Alexander polynomial of a knot. This is joint work with Dror Bar-Natan.

  • Orateur: Anthony Conway (Max Planck Institut für Mathematik).
  • Titre: Non-slice linear combinations of iterated torus knots.
  • Résumé: In 1976, L. Rudolph asked whether the set of algebraic knots is linearly independent in the knot concordance group. After introducing the problem and its history, we exhibit new infinite families of examples for which Rudolph's question is answered in the positive. The proof involves twisted Blanchfield forms. This is based on joint work with Min Hoon Kim and Wojciech Politarczyk.

  • Orateur: Jean Claude Hausmann (Université de Genève).
  • Titre: Sphères exotiques.
  • Résumé: En 1956, John Milnor surprend la communauté mathématique par sa découverte de variétés différentiables homéomorphes mais pas difféomorphes à la sphère 𝕊7. Moins de sept ans plus tard, ces "sphères exotiques" sont classifiées à difféomorphisme près en toute dimension > 4 par l'important travail de Kervaire–Milnor, qui a donné naissance à la théorie de la chirurgie. Dans cet exposé, on fera un survol de ces années d'effervescence de la topologie différentielle (talk in French, slides in English).

  • Orateur: Oliver Singh (Durham university).
  • Titre: Knotted Surfaces in 4-manifolds and Distances Between Them.
  • Résumé: I will discuss knotted surfaces, isotopy classes of embedded surfaces in a given 4-manifold, and will define two notions of distance between them. These distances are integer-valued and are defined topologically: one in terms of regular homotopy; another in terms of stabilisation, a form of embedded surgery. I will outline a proof of an inequality between these distances; the proof is constructive and draws upon ideas pioneered by Gabai in the proof of the 4-dimensional light bulb theorem.

  • Orateur: Livio Ferretti (Universität Bern).
  • Titre: On Kashaev’s Signature Conjecture.
  • Résumé: The Levine–Tristram signature of a link L is an invariant given by a function σL: 𝕊1 → ℤ. This invariant has many interesting properties and applications, and can be defined in a variety of ways. It is interesting to notice that all those definitions are strongly topological, and, to the best of our knowledge, combinatorial interpretations of the signature only exist for the value at some specific points of the unit circle. In 2018, Kashaev defined a new matrix associated to any link diagram and proved how to extract an invariant from it, conjecturally equal to the Levine–Tristram signature (then conjecturally providing a first combinatorial definition of σL). In this talk we will first recall the definition and some basic properties of the Levine–Tristram signature, then construct Kashaev’s matrix and finally present some partial results that provide strong evidence for the conjecture. Joint with David Cimasoni.

  • Orateur: Panagis Karazeris (University of Patras).
  • Titre: Towards a category-theoretic construction of Nori’s motives II.
  • Résumé: We present an account of Nori’s category of mixed motives, due to L. Barbieri-Viale, O. Caramello and L. Lafforgue (Selecta Mathematica, 24, 3619-3648, 2018). The category of mixed motives gives a universal way of realizing a directed graph, one that encodes the behaviour of a (co-)homological functor, into an abelian category. In this talk we will discuss the construction of a regular category out of data such as the representation of a directed graph in a category of finite-dimensional vector spaces and then recall the universal way of passing from a regular category to an effective one. We will show that the category so constructed has the required universal property and discuss to which extent it is independent from the specific representation we start with.

  • Orateur: François Guéritaud (Université de Lille 1).
  • Titre: Groupes de Coxeter et actions affines.
  • Résumé: On possède peu d'exemples de variétés affines (complètes), du moins à groupe fondamental non virtuellement résoluble. Je montrerai comment faire agir proprement discontinûment un groupe de Coxeter à angles droits, engendré par N involutions, sur l'espace affine à N(N-1)/2 dimensions. La construction généralise celle des "espaces-temps de Margulis" mais repose sur une nouvelle approche, de nature pseudo-métrique.
    Travail en collaboration avec Jeffrey Danciger et Fanny Kassel.

  • Orateur: Panagis Karazeris (University of Patras).
  • Titre: Towards a category-theoretic construction of Nori’s motives I.
  • Résumé: We set up the necessary category-theoretic machinery that will eventually lead to an account of Nori’s category of mixed motives and the proof of its universal property, as it appears in recent work by L. Barbieri-Viale, O. Caramello and L. Lafforgue (Selecta Mathematica, 24, 3619-3648, 2018). In this first talk we will review the non-additive analogue of abelian categories (Barr-exact or effective categories) and the weaker notion of regular categories. In the additive world the first are exemplified by the category of abelian groups and the later are exemplified by torsion-free abelian groups. In the non-additive world, while the former are exemplified by compact Hausdorff spaces, the later are exemplified by the category of compactly generated Hausdorff spaces. We will describe a universal way of passing from a regular category to an effective one that will be crucial for the construction of the category of motives in the second talk.

  • Orateur: Gregor Masbaum (Institut Mathématique de Jussieu – PRG et CNRS).
  • Titre: On the skein module of the product of a surface and a circle.
  • Résumé: We show how to use Witten–Reshetikhin–Turaev invariants and Topological Quantum Field Theory to study skein modules of 3-manifolds. As an application, we obtain a lower bound for the dimension, over the fraction field, of the Kauffman bracket skein module of the product of a closed oriented surface and a circle.