26–30 August 2024
Nantes University Amphithéâtre A (Campus Lombarderie)

Schedule:

	Monday	Tuesday	Wednesday	Thursday	Friday
9:30-10:30		Starkston 2	Starkston 3	Marengon 2	Marengon 3
10:30-11:00	Registration	Coffee break	Coffee break	Coffee break	Coffee break
11:00-12:00	Starkston 1	Misev	Marengon 1	Genevois	Murakami
12:00-12:30			Short talk 7
12:30-14:00	Lunch	Lunch	Lunch	Lunch	Lunch
14:00-15:00	Queffelec	C. Anghel-Palmer		Casals
15:00-15:30	Coffee break	Coffee break		Coffee break
15:30-16:00	Short talk 1	Short talk 4		Short talk 8
16:00-16:30	Short talk 2	Short talk 5		Short talk 9
16:30-17:00	Short talk 3	Short talk 6

Mini-courses:

Marco Marengon (Rényi Institute)
Adjunction inequalities for embedded surfaces in 4-manifolds
I will survey the state of the art of adjunction inequalities in 4-manifolds, which provide lower bounds to the genus of smoothly embedded surfaces, and which are an invaluable tool to understand smooth 4-manifolds. A special emphasis will be placed on the relative case of properly embedded surfaces with boundary, where smooth genus bounds often incorporate a term from knot Floer homology or Khovanov homology.

Laura Starkston (UC Davis)
Four-genus and twist regions of random braids
We will talk about different ways to study symplectic surfaces in simple symplectic 4-manifolds. We will see how braids and their variations can be used to encode these surfaces and be used to compute invariants which might be able to distinguish them. We will also talk about ways to incorporate other methods from symplectic topology to study long standing questions in complex algebraic geometry and symplectic topology.

Plenary talks:

Cristina Anghel-Palmer (University of Leeds)
Universal coloured Alexander invariant via configurations of ovals in the disc
The coloured Jones and Alexander polynomials are quantum invariants that come from representation theory. There are important open problems in quantum topology regarding their geometric information. Our goal is to describe these invariants from a topological viewpoint, as intersections between submanifolds in configuration spaces. We show that the N^th coloured Jones and Alexander polynomials of a knot can be read off from Lagrangian intersections in a fixed configuration space. At the asymptotic level, we geometrically construct a universal ADO invariant for links, as a limit of invariants given by intersections in configuration spaces. The parallel question of providing an invariant unifying the colored Jones invariants is the subject of the universal Habiro invariant for knots. The universal ADO invariant that we construct recovers all of the coloured Alexander invariants (in particular, the Alexander polynomial in the first term).

Roger Casals (UC Davis)
Recent developments on braid varieties
First, I will explain how to associate an algebraic variety, and more generally a dg-category, to a positive braid. This is a geometric version, using contact topology, of an algebraic construction that goes back to P. Deligne. The construction is geometric in type A, in that it captures properties of (Legendrian) knots in 3-dimensional space, but also works for other Lie group types. Second, I will discuss recent results on the geometric properties of these varieties, known as braid varieties, and their categorical enhancements. These include having algebraic symplectic 2-forms, coming from relative CY-structures, and, more generally, their ring of regular functions being cluster algebras. A key ingredient for many of these results will be the diagrammatic calculus of weaves, which will be briefly discussed as well.

Anthony Genevois (CNRS and University of Montpellier)
A few words about asymptotically rigid mapping class groups
Since their introduction in the 1960s, the so-called Thompson groups have received a lot of attention in group theory, and many variations of their construction have been proposed. After a general introduction to Thompson-like groups, I will focus on the family of asymptotically rigid mapping class groups of some planar surfaces, which yield a way to bring together Thompson groups and braid groups. I will survey common works with Anne Lonjou and Christian Urech.

Filip Misev (Univesity of Regensburg)
On the ribbon number
The ribbon number of a ribbon knot is the smallest number of ribbon intersections among all ribbon disks bounding that knot. I would like to report on a project with Stefan Friedl and Alexander Zupan in which we study this relatively unexplored invariant, relate it to other invariants and obtain bounds, allowing to determine the ribbon numbers of knots up to crossing number 11 (with three exceptions). A key ingredient is that the set of all Alexander polynomials of ribbon knots with a given ribbon number is in fact finite, and computable.

Jun Murakami (Waseda University)
On complexified tetrahedron and volume conjecture for double twist knots
We first introduce the complexified tetrahedron from SL(2,C)-representations of the fundamental group of the complement of a double twist knot. The lengths and angles of a complexified tetrahedron are complex numbers corresponding to the eigenvalues of certain elements of the fundamental group. On the other hand, such complexified tetrahedron also corresponds to the volume potential function which is a certain continuous version of the colored Jones polynomial. By using this correspondence, the volume conjecture for double twist knots is proved.

Hoel Queffelec (CNRS and University of Montpellier)
Faithfulness questions for Burau representations
The Burau representation is one of the most classical ones for braid groups. Its (lack of) faithfulness has attracted quite a lot of attention, but the case of the 4-strand braid group still remains open. I'll review recent and ongoing work with Asilata Bapat and Thomas Haettel, in two directions: failed attempts to prove faithfulness by geometrical means, and more successful attempts to disprove faithfulness for other Artin–Tits classes.

Short talks:

Filippo Bianchi (University of Pisa)
Spin mapping class groups and 4-manifolds
We present a strategy to find a presentation of the even spin mapping class group. As an application, we explain a new proof of Rokhlin's theorem on the signature of spin 4-manifolds.

Jennifer Brown (University of Edinburgh)
Quantizing character varieties
Character varieties bring algebraic geometric techniques to study low-dimensional topology. Their quantizations reveal a beautiful connection to representation theory and have a satisfying diagrammatic description. We will introduce the basic constructions involved in this story and give a hint to its usefulness.

Subhankar Dey (Durham University)
Essential surfaces in link exteriors and link Floer homology
Knot/link Floer homology is a link invariant package, introduced independently by Ozsváth–Szabó and Rasmussen, has been shown to be quite useful to solve a number of questions in low dimensional topology in the last two decades. Although it is not a complete invariant of knots/links, a number of knots and links have been shown to be detected by this toolbox. The center of most of these results have been careful examination of certain essential surfaces in the knot/link exteriors and observing that operations on those surfaces can be kept track by the link/knot Floer homology of those knots/links. In this mostly self-contained talk, we will be talking about those results and some new ones. This is based on joint work with Fraser Binns, some of which is ongoing.

Alessio Di Prisa (University of Pisa)
Equivariant rational concordance of strongly invertible knots
In this talk, I will present a joint work with Oğuz Şavk on the study of equivariant rational concordance of strongly invertible knots. I will introduce the new theory with several explicit constructions (obtained from a refinement of Kawauchi's theorem) and obstructions (derived from the Fox–Milnor condition). I will also introduce the equivariant rational concordance group and discuss its relations with the other concordance groups.

Leonardo Ferrari (Institute of Mathematics of the Polish Academy of Sciences)
Cusps of hyperbolic 4-manifolds and rational homology spheres
Many interesting examples of knots are obtained with the aid of the satellite construction. Therefore it is crucial to understand how knot invariants behave under this construction. In this talk we will discuss a cabling formula for twisted Blanchfield forms and associated twisted signatures. As an example we will re-derive classical cabling formulas of Litherland for Casson–Gordon signatures. This is a joint work with Antony Conway and Maciej Borodzik.

Livio Ferretti (University of Geneva)
Diagrammatic computation of multivariable link invariants
The multivariable signature is a generalization of the classical Levine–Tristram signature to the setting of (colored) links. Its classical definition uses so-called C-complexes and generalized Seifert forms and is purely topological. In this talk, we will associate to every colored link diagram a symmetric matrix, constructed using only the combinatorial data of the diagram, and show how one can compute the multivariable signature from this matrix. The same matrix also allows to compute the multivariable Alexander polynomial. Joint with D. Cimasoni and J. Liu.

Benjamin Haïoun (University of Toulouse)
Skein algebras in non-semisimple settings
Skein algebras were first introduced as deformation quantization of character varieties. The representation theory of quantum groups has proven to be a key tool in their study, though it gets very tricky at roots of unity. In this talk, I will introduce a modified notion of skein algebra one can obtain from the representation-theoretic side at roots of unity, i.e. in the non-semisimple setting.

Martin Palmer-Anghel (Mathematical Institute of the Romanian Academy)
Compactly-supported homology classes for big mapping class groups
The Mumford conjecture—a consequence of the Madsen–Weiss theorem—describes the (rational) homology of the colimit of the mapping class groups Mod(Σ(g,1)) as g goes to infinity. One may alternatively take the colimit of the surfaces Σ(g,1) themselves, to obtain an infinite-type surface Σ(∞) and then consider the homology of its mapping class group Mod(Σ(∞)), which is uncountably generated in all positive degrees and whose precise structure is very mysterious. There is a natural homomorphism from the former to the latter, and it is a natural question to ask whether its image is non-zero. One may more generally ask, for any infinite-type surface S, whether Mod(S) admits non-zero homology classes supported on a compact subsurface of S. We will give a complete answer to this question when S has positive (possibly infinite) genus and a partial answer when S has genus zero. This represents joint work with Xiaolei Wu.

Acknowledgements: the conference is supported by the Réseau Thématique RTop Topologie algébrique et Géométrique, the ANR projects CoSy and SyTriQ, the Department of mathematics Jean Leray, the Centre Henri Lebesgue, the project ALL Ambition Lebesgue Loire of the Pays de la Loire region, and the regional project Étoile Montante PSyCo.