Séminaire de Topologie et Géométrie — 2024
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
22/02/2024
- Orateur: Ben-Michael Kohli (Yau Center, Tsinghua University)
- Titre: A lower bound for the genus of a knot using the Links-Gould invariant
- Résumé: For a long time, the Alexander polynomial was the main easily computable link invariant to be known. But in 1984, Jones discovered his well known polynomial link invariant, and that gave birth to the vast theory of quantum link invariants. However, unlike for the Alexander invariant, it is in general hard to deduce precise topological properties on a knot or link from the value quantum invariants take on that link. For instance, no genus bound is known for the Jones polynomial.
The Links-Gould invariants of oriented links \(LG^{m,n}(L,t_{0},t_{1})\) are two variable quantum invariants obtained by the Reshetikhin-Turaev construction applied to Hopf superalgebras \(U_{q}\mathfrak{gl}(m \vert n)\). These invariants are known to be generalizations of the Alexander invariant.
Using representation theory of \(U_{q}\mathfrak{gl}(2 \vert 1)\), we proved in recent work with Guillaume Tahar that the degree of the Links-Gould polynomial \(LG^{2,1}\) provides a lower bound on the Seifert genus of any knot, therefore improving the bound known as the Seifert inequality in the case of the Alexander invariant.
29/02/2024
- Orateur: Pavol Severa (Université de Genève)
- Titre: Braids and quantization (of Poisson-Hopf algebras and related things)
- Résumé: I will explain how the replacement of permutations by braids replaces also groups by Hopf algebras (by passing from "symmetric simplicial objects" to "braided simplicial objects"), and how it gives us a deformation quantization of Poisson brackets on commutative Hopf algebras. This story can be seen as a small corner of Chern-Simons theory with suitable boundary conditions. Based on a joint work with Jan Pulmann.
7/03/2024
- Orateur: Jean-Claude Hausmann (Université de Genève)
- Titre: Rigidité différentiable pour les plongements : les exemples d'Haefliger
- Résumé: En 1960, Haefliger a construit des plongements (par exemple de \(S^3\) dans \(\mathbb{R}^6\)) qui ne sont pas différentiablement isotopes, alors qu'il le sont topologiquement. Une telle "rigidité différentiable" (après celle des difféomorphismes découverte par Milnor) a étonné la communauté mathématique et apporté à Haefliger une indéniable célébrité.
Par rapport à mon exposé au colloque Haefliger de janvier, plusieurs points seront repris mais il y aura une certaine complémentarité : on sautera la preuve du théorème d'Haefliger sur les plongements métastables, pour se concentrer sur les idées de la démonstration de la rigidité différentiable.
Talk in French, slides in English.
14/03/2024
- Orateur: Ken'ichi Ohshika (Gakushuin University)
- Titre: La métrique de tremblement de terre sur l’espace de Teichmüller
- Résumé: La déformation de tremblement de terre introduite par Thurston est une déformation d’une structure hyperbolique d’une surface qui est une généralisation de la déformation de Fenchel-Nielsen. Thurston a remarqué dans sa prépublication célèbre que cette déformation induit une métrique finslerienne sur l’espace de Teichmüller, mais il n’y avait aucune étude sérieuse de cette métrique jusqu’à récemment. Nous montrons que cette métrique est asymétrique et incomplète avec la même complétion que la métrique de Weil-Petersson, et qu’elle a une rigidité par rapport à l’action du groupe modulaire.
Un travail en collaboration avec Yi Huang, Huping Pan et Athanase Papadopoulos.
21/03/2024
- Orateur: Mikami Hirasawa (Nagoya Institute of Technology)
- Titre: The equivariant genera of marked strongly invertible knots associated with \(2\)-bridge knots
- Résumé: A marked strongly invertible knot is a triple \((K,h,\delta)\) of a knot \(K\) in \(S^3\), a strong inversion \(h\) of \(K\), and a subarc \(\delta \subset \text{Fix}(h)\cong S^1\) bounded by \(\text{Fix}(h)\cap K\cong S^0\).
An invariant Seifert surface for \((K,h,\delta)\) is an \(h\)-invariant Seifert surface for \(K\) that intersects \(\text{Fix}(h)\) in the arc \(\delta\). In this paper, we completely determine the equivariant genus (the minimum of the genera of invariant Seifert surfaces for \((K,h,\delta)\)) of every marked strongly invertible knot \((K,h,\delta)\) with \(K\) a \(2\)-bridge knot. We concretely construct invariant Seifert surfaces of minimal equivariant genera.
This is a joint work with Makoto Sakuma and Ryota Hiura.
11/04/2024
- Orateur: François Costantino (Université Toulouse III Paul Sabatier)
- Titre: Stated skein modules of 3-manifolds and TQFTs
- Résumé: After reviewing the definition of stated skein modules for surfaces and 3-manifolds, I will detail how this recent notion allows to relate topological constructions (related to cut and paste techniques) to algebraic ones (braided tensor products of algebra objects in braided categories for instance).
I will explain how the stated skein algebra of some special surfaces provides a topological description for some notable algebras (e.g. the quantised functions ring \(O_q(\mathfrak{sl_2})\) or its "transmutation" \(BSL_2(q)\)). Then I will describe how stated skein moduli of 3-manifolds fit into a TQFT framework albeit a non completely standard one.
If time permits I will also discuss some unexpected non injectivity results in dimension 3.
(Joint work with Thang Le)
18/04/2024
- Orateur: Livio Ferretti (Université de Genève)
- Titre: The monodromy group of a positive braid, and invariant framings
- Résumé: The geometric monodromy group is a classical yet rather poorly understood topological invariant of isolated plane curve singularities. In this talk we will discuss a generalization of it to the setting of positive braids and see how working in this wider context can help understanding the original invariant of singularity theory. In particular we obtain that, for irreducible singularities not of type \(A_n\), up to finitely many exceptions the geometric monodromy group is determined by two simple knot invariants: the genus and the Arf invariant.
Our main technical tool are invariant framings on surfaces. If time permits, we will end with a discussion on the application of such techniques to the study of more general fibred links.
25/04/2024
- Orateur: Nicolas Orantin (Université de Genève)
- Titre: From Catalan numbers to knot invariants: a short introduction to
topological recursion
- Résumé: Catalan numbers are integers which appear in various counting
problems. In particular, they can be defined as the cardinal of a set
of graphs drawn on the sphere. From this perspective, they admit a
natural generalisation to higher genera as the cardinal of a specific
class of graphs embedded on a surface of fixed topology. In this talk, I
will use the computation of such numbers as an opportunity to present
how one can transform such a combinatorial problem into a very simple
problem of complex analysis on a Riemann surface through the
introduction of generating series. We will see that the solution of this
problem is given by an inductive procedure known as topological recursion.
In the second part of this talk, I will explain how this topological
recursion gives rise to a universal solution to many problems of
enumerative geometry and low dimensional topology. In particular, I will
explain its application to the computation of Gromov-Witten invariants
and volumes of moduli spaces of Riemann surfaces. Finally, I will
briefly explain how it is conjectured to give rise to knot invariants
and its relation to the AJ conjecture.
02/05/2024
- Orateur: Hugo Parlier (University of Luxembourg)
- Titre: Crossing the line: from graphs to curves
- Résumé: The crossing lemma for simple graphs gives a lower bound on the necessary number of crossings of any planar drawing of a graph in terms of its number of edges and vertices. Viewed through the lens of topology, this leads to other questions about arcs and curves on surfaces. Here is one: how many crossings do a collection of m homotopically distinct curves on a surface of genus g induce?
The talk will be about joint work with Alfredo Hubard where we explore some of these, using tools from the hyperbolic geometry of surfaces in the process.
16/05/2024
- Orateur: Diego Santoro (University of Vienna)
- Titre: Taut foliations from knot diagrams
- Résumé: Taut foliations have been a classical object of study in 3-manifolds theory. Recently, new interest in them has come from the investigation of the so-called "L-space conjecture", that predicts that Heegaard Floer L-spaces can be characterised as those 3-manifolds that do not admit coorientable taut foliations. A possible approach to the study of this conjecture is by analysing surgeries on knots and links. Most of the techniques employed for constructing taut foliations on Dehn surgeries usually make use of some special property of the exterior of the link (e.g. fiberedness). In this talk I will describe a procedure for constructing taut foliations that only makes use of diagrammatic properties of the knot.
23/05/2024
- Orateur: Alexis Virelizier (Université de Lille)
- Titre:
- Résumé:
30/05/2024
- Orateur: Françoise Michel (IMT Toulouse)
- Titre: Topologie des germes de surface complexe et topologie de la normalisation
- Résumé:
