Algebra Days in Caen 2024: Algebraic aspects of configuration spaces and moduli spaces

March 18–20 2024

This issue of the series Algebra Days in Caen is devoted to configuration spaces and moduli spaces, which will be considered from various algebraic and topological perspectives. A particular focus will be made on dialogue between different communities working on the subject.

You will find the conference poster here.


  • Cristina Anghel (Leeds)
  • Andrea Bianchi (Bonn)
  • Rachael Boyd (Glasgow)
  • Tara Brendle (Glasgow)
  • John Guaschi (Caen)
  • Najib Idrissi (Paris)
  • Erik Lindell (Paris)

Practical information

To register, please follow this link before February 20.

If you have any questions, please contact the organisers, Arthur Soulié and Victoria Lebed.

We can provide financial support to junior participants. Please contact the organisers if you are interested.

Some hotels in Caen: Hôtel du Château, Hôtel des Quatrans, Hôtel la Fontaine.

Conference venue: Université de Caen Normandie, Campus 2, building Sciences 3, Lecture Hall S3 057 on Monday and Tuesday, and Amphi 500 on Wednesday. To get there, you may follow these instructions.

The list of participants is available here.

The conference dinner will take place on Tuesday, March 19 at 19:30 at the restaurant L'Aromate, 9 Rue Gemare, Caen.

Polaris, a sculpture in Caen. Image credit: Quentin Riel.


Monday, March 18
Location: S3-057

14:00-14:30 Welcome and registration
14:30-15:30 Najib Idrissi (mini-course)
15:30-16:00 Coffee break
16:00-17:00 Cristina Anghel

Tuesday, March 19
Location: S3-057

9:30-10:30 Najib Idrissi (mini-course)
10:30-11:00 Coffee break
11:00-12:00 Rachael Boyd
12:00-14:00 Lunch
14:00-15:00 Tara Brendle
15:00-15:30 Coffee break
15:30-16:30 Andrea Bianchi
16:45-17:45 Erik Lindell

Wednesday, March 20
Location: amphi 500

09:00-10:00 Najib Idrissi (mini-course)
10:00-11:00 John Guaschi
11:00-11:30 Coffee break

Titles and abstracts

Cristina Anghel: Universal coloured Alexander invariant via configurations on 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 Nth 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).

Andrea Bianchi: String topology and graph cobordisms

Let M be an oriented, closed manifold; the collection of homology groups H*(map(X,M)), for varying topological space X, carries additional operations; these come from the functoriality of map(-,M), and from Poincare duality of M. The most famous such operation is perhaps the Chas–Sullivan string product on H*(map(S1,M)). I will report on my current attempt to define quite general operations H*(map(X,M)) H*(map(Y,M)) using the homology of a suitable moduli space of graph cobordisms between X and Y.

Rachael Boyd: Diffeomorphisms of reducible 3-manifolds

I will talk about joint work with Corey Bregman and Jan Steinebrunner, in which we study the moduli space BDiff(M), for M a compact, connected, reducible 3-manifold. We prove that when M has non-empty boundary, BDiff(M rel ∂M) has the homotopy type of a finite CW-complex. This was conjectured by Kontsevich and previously proved in the case where M is irreducible by Hatcher and McCullough.

Tara Brendle: On the level 2 congruence subgroup of the mapping class group

We will survey work of Birman–Craggs, Johnson, and Sato on the abelianization of the level 2 congruence group of the mapping class group of a surface, and of the corresponding Torelli group. We will then describe recent work of Lewis providing a common framework for both abelianizations, with applications including a partial answer to a question of Johnson.

John Guaschi: Finite subgroups of surface braid groups

It is known since the 1960s that the 2-sphere S2 and the real projective plane RP2 are the only surfaces whose braid groups admit torsion. The isomorphism classes of the finite subgroups of the braid groups B n ( M ) , where M=S2 or RP2, and the maximal finite subgroups of B n ( S2 ) , were determined in previous work. In this talk, we discuss the use of fibrations and other geometric maps in this study, as well as recent work on the maximal finite subgroups of M=S2 or RP2, and the maximal finite subgroups of B n ( RP2 ) and their structure with respect to inclusion. In particular, we show that with respect to the case of S2, certain new phenomena appear for the finite subgroups of B n ( RP2 ) . This constitutes joint work with Daciberg Gonçalves (São Paulo).

Najib Idrissi: Operadic structures of configuration spaces (mini-course)

Configuration spaces of manifolds, that is, ordered finite collections of pairwise distinct points, are classical yet intriguing objects in algebraic topology. They admit a rich algebraic structure coming from the theory of operads. In this mini-course, I will explain how this structure is defined, and how one can show, using the extra algebraic structure, that the real homotopy type of the ambient manifold completely determines the real homotopy type of the configuration spaces under some hypotheses.
These talks are based on joint works with Campos, Ducoulombier, Lambrechts, and Willwacher.

Erik Lindell: The Torelli group and tautological cohomology classes with twisted coefficients

In the late 90's, Faber made a series of conjectures about the tautological subring of the Chow ring of Mg, the moduli space of algebraic curves, and certain variants of this moduli space. One of the conjectures states that the tautological ring should be a Gorenstein ring (i.e. satisfy Poincaré duality). This conjecture is no longer generally believed to hold and counterexamples have been found for some variants of Mg. In a previous (failed) project of mine I attempted to disprove the conjecture for a certain variant Mg, through an approach using homology of the Torelli group, an interesting subgroup of the mapping class group of a surface. In this talk, I will discuss the background of the problem in more detail, explain the intended approach and why it failed and if time permits, talk about the results this project led to instead.

Financial support:

Laboratoire de Mathématiques Nicolas Oresme, Université de Caen Normandie, CNRS