Séminaire de Topologie et Géométrie — 2023

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 1-15, Section de mathématiques, rue du Conseil-Général 7-9.

The Topology and Geometry seminar of the mathematics department of the University of Geneva takes place on Thursdays from 14:15 to 15:15. It takes place in room 1-15, Section de mathématiques, rue du Conseil-Général 7-9.

Exposés

01/12/2023
  • Orateur: Dror Bar-Natan (Toronto)
    • (changement d'heure 16h00, salle 6-13 )
  • Titre: Shifted Partial Quadratics, their Pushforwards, and Signature Invariants for Tangles.
  • Résumé: Following a general discussion of the computation of zombians of unfinished columbaria (with examples), I will tell you about my recent joint work with Jessica Liu on what we feel is the "textbook" extension of knot signatures to tangles, which for unknown reasons, is not in any of the textbooks that we know.


23/02/2023
  • Orateur: Evgeny Mukhin (Indiana University–Purdue University Indianapolis, IUPUI)
  • Titre: The quantum toroidal gl(1).
  • Résumé: The quantum toroidal gl(1) algebra appears in many areas of mathematics and physics. For example, it is isomorphic to the Hall algebra of the elliptic curve, it acts on equivariant K-theory of Hilbert scheme of points in C^2, it is a quantized version of W-algebra W_{1+\infty} in conformal field theory, it describes the action of Macdonald operators in the space of symmetric polynomials, etc. In this talk we will give an introduction to the quantum toroidal gl(1) algebra and a review of various related results. Our motivation originates in the integrable model defined by the transfer matrices related to the R-matrix of this algebra. In some limit, some instances of this model recover quantum models related to affine W-algebras of gl(m|n) type and are expected to be dual to affine Gaudin models. In particular, it includes the quantum KdV flows.


9/03/2023
  • Orateur: Martin Palmer (Mathematical Institute of the Romanian Academy)
  • Titre: The homology of big mapping class groups.
  • Résumé: "Big mapping class groups" -- mapping class groups of infinite-type surfaces -- have recently become the subject of intensive study, having connections for example with dynamical systems and geometric group theory. However, their homology (beyond degree 1) has so far been very little understood. I will describe two results, from joint work with Xiaolei Wu, that exhibit contrasting behaviour of the homology of big mapping class groups. First, using methods of homological stability, we find an uncountable family of big mapping class groups whose integral homology vanishes in all positive degrees. Second, by entirely different methods, we find another uncountable family of big mapping class groups whose integral homology is uncountable in each positive degree.


16/03/2023
  • Orateur: Pierre Dehornoy (Institut Fourier, Université Grenoble Alpes)
  • Titre: Sections de Birkhoff pour les flots de Reeb en dimension 3.
  • Résumé: Pour un flot dans une variété de dimension 3, une section de Birkhoff est une surface dont l’intérieur est plongé et transverse au flot, le bord constitué d’orbites périodiques, et la surface coupe toute orbite en temps fini. Une telle surface permet de réduire la dynamique, en particulier ses propriétés topologiques, à celle d'un difféomorphisme de surface. Dans un travail en commun avec V Colin, U Hryniewicz et A Rechtman, on montre que l’ensemble des flots de Reeb en dimension 3 qui admettent une section de Birkhoff contient un ouvert dense pour la topologie C-infini. La construction s’appuie sur les courbes pseudo-holomorphes fournies par l’homologie de contact plongée, des techniques de chirurgie dûes à Fried, et un résultat de densité des orbites périodiques d’Irie.


23/03/2023
  • Orateur: Peter Feller (ETH Zurich)
  • Titre: On the length of knots on a Heegaard surface of a 3-manifold.
  • Résumé: 3-manifold theory has expanded its tool box in recent decades: topological, (Floer and quantum) homological, and geometrical methods all have been employed with success. However, often the relation between these different approaches remains mysterious. In this talk we explore connections between the topology and the geometry of 3-manifolds by using Heegaard-splittings (topology) of a 3-manifold to describe hyperbolic structures (geometry) on it. There is no Ricci-flow machine running in the background. Instead, the motor behind what we do is an effective version of Thurston's hyperbolic Dehn surgery. Applications include a Ricci-flow free proof of Mather's result that random 3-manifolds (in the sense of Dunfield-Thurston) are hyperbolic, and bounds on the diameter and injectivity radius of a random 3-manifold. Based on work in progress with A. Sisto and G. Viaggi.


30/03/2023
  • Orateur: Christine Lescop (Institut Fourier, CNRS, Université Grenoble Alpes)
  • Titre: Un invariant sympathique des noeuds de genre un.
  • Résumé: Dans cet exposé élémentaire, on introduira un invariant topologique w facile à calculer pour les noeuds qui bordent des surfaces compactes orientables de genre un (dans l'espace ambiant ou dans une sphère d'homologie de dimension trois). La démonstration de l'invariance de w repose sur l'étude d'un invariant de telles surfaces de Seifert de genre un défini à partir du revêtement abélien libre maximal de leurs complémentaires.


27/04/2023
  • Orateur: Lukas Lewark (University of Regensburg)
  • Titre: Untwisting and surfaces in 4-space.
  • Résumé: Smooth surfaces in 4-dimensional space are generally easy to construct. In this talk, we will focus instead on surfaces that are topological, but not smooth submanifolds. We will see how they can nevertheless be constructed in a geometric fashion - namely by untwisting. In particular, this allows us to compute for some knots K the minimal genus of a compact oriented topological surface in the four-ball with boundary K.


4/05/2023
  • Orateur: Damien Calaque (Université de Montpellier)
    • (changement d'heure 13h15 - 14h15)
  • Titre: Comparing Calabi-Yau and quasi-bisymplectic structures for multiplicative preprojective algebras .
  • Résumé: Multiplicative preprojective algebras first appeared in the work of Crawley-Boevey and Shaw, in the course of their study of the Deligne-Simpson problem. Their representation varieties, known as multiplicative quiver varieties, naturally appear in various areas (character varieties, local systems on Riemann surfaces and perverse sheaves on nodal curves, integrable systems, etc...). Van den Bergh proved that these varieties can be obtained using a quasi-Hamiltonian reduction procedure (after Alekseev-Malkin-Meinreinken), and developed a noncommutative version of the quasi-Hamiltonian formalism so that all constructions actually hold directly at the level of the multiplicative preprojective algebras. In this talk I will explain that the moment map defining these multiplicative preprojective algebras carry a relative Calabi-Yau structure (a notion introduced by Brav and Dyckerhoff, following some earlier suggestion of Toën), and how the three notions (quasi-Hamiltonian structures, their non-commutative version, and relative Calabi-Yau structures) interact. The talk is based on joint works with Tristan Bozec and Sarah Scherotzke.


11/05/2023
  • Orateur: Roland van der Veen (Bernoulli Institute, University of Groningen)
  • Titre: Twisting quantum knot invariants.
  • Résumé: After giving an elementary introduction to quantum knot invariants we will show how one can twist this construction to allow a representation of the knot group. Some existing invariants such as the twisted Alexander polynomial and the ADO invariant are of this type and use Hopf algebraic techniques to show the degree of the invariant is related to the genus of the knot. This is joint work with Daniel Lopez Neumann, see https://arxiv.org/abs/2211.15010.


25/05/2023
  • Orateur: Gaetan Simian (Université de Genève)
  • Titre: Arf invariants of colored links.
  • Résumé: The Arf invariant is a modulo 2 integer associated to a non singular quadratic form over the field F_2. In 1965, Robertello applied this theory to Seifert forms, leading to an invariant of a certain class of oriented links. Colored links are a natural generalisation of oriented links, where the components are grouped into sublinks. Several invariants of oriented links, such as the Alexander module and the Levine-Tristram signature, admit natural multivariable extensions to colored links via so-called "generalised Seifert surfaces". In this talk, I will define these notions, and briefly explain how to use generalised Seifert surfaces to extend the Arf-Robertello invariant to colored links. Unfortunately, these extensions turn out to be determined by the linking numbers.


1/06/2023
  • Orateur: Emmanuel Wagner (Université de Paris)
  • Titre: Algebraic vs geometric categorification of the Alexander polynomial: a spectral sequel.
  • Résumé: We construct a spectral sequence from the gl0-homology, an algebraic categorification of the Alexander polynomial to the knot Floer homology. This spectral sequence is of Bockstein type and comes from a subtle manipulation of coefficients. The main tools are quantum traces of foams and of singular Soergel bimodules. This is a joint work with Anna Beliakova, Kris Putyra, and Louis-Hadrien Robert.