Journées thématiques
de Cergy-Pontoise

• Welcome coffee
• 11h-12h00: Francesco Costantino (joint with the Colloquium of the AGM) slides of the the talk
  Lunch
• 14h00-15h00: Francis Bonahon (1/3) slides of the the lectures
• 15h15-16h15: Renaud Detcherry (1/3)
  Break
• 16h45-17h45: Jennifer Brown

• 9h30-10h30: David Jordan (1/3)
• 11h00-12h00: Francis Bonahon (2/3)
  Lunch
• 14h00-15h00: Renaud Detcherry (2/3)
• 15h15-16h15: David Jordan (2/3)
  Break
• 16h45-17h45: Benjamin Haïoun

• 9h30-10h30: Francis Bonahon
  cofee break
• 11h00-12h00: Renaud Detcherry (3/3)
• 12h15-13h15: David Jordan (3/3)
  

• Francis Bonahon (University of Southern California and Michigan state University). Asymptotics of quantum invariants
  Abstract: There are now many volumes conjectures that relate to hyperbolic volumes the asymptotic behavior of quantum invariants when the quantum parameter is a root of unity whose order tends to infinity. I will discuss a particular one, for surface diffeomorphisms, where the asymptotic behavior is controlled by the (3-dimensional) mapping torus of the diffeomorphism considered. The emphasis will be on the analytic challenges, and on the tools used to address them.
  
  
• Renaud Detcherry (Université de Bourgogne Franche-Comté). The skein module, the character variety, and the incompressible surface
  Abstract: In this mini-course we will study the properties of Kauffman bracket skein modules S(M) of 3-manifolds M, which constitute the most famous kind of skein modules. They encode the combinatorics of links in a 3-manifold, up to the local relations satisfied by the Kauffman bracket of links in S^3. They also depend on a choice of parameter A in a ring. When A=-1, the skein module S(M) recovers the coordinate ring of the SL_2 character variety of M, by a fundamental theorem of Bullock, Przytycki and Sikora which we will review. When A is a root of unity, S(M) retains some connections with the character variety, by a beautiful theory initiated by Bonahon and Wong.
  When A is a generic parameter, S(M) is finite dimensional, by a striking result of Gunningham, Jordan and Safronov, and the computation of its dimension is a hard problem, on which we will present recent results, joint with Effie Kalfagianni and Adam Sikora. Finally, when A is the indeterminate in the ring Z[A,1/A] of Laurent polynomials, the existence of torsion in S(M) is conjectured to be connected to the existence of closed incompressible surfaces in M. We will present classical partial results of Przytycki on this question, as well as new results joint with Giulio Belletti.
  
  
• David Jordan (University of Edinburgh). Finiteness and computability in skein theory
  Abstract: In past work with Gunningham and Safronov we established the finite dimensionality of skein modules for closed 3-manifolds. Our proof however did not give much insight into what the promised dimensions are. In these lectures I hope to share:
  1. A conjecture we made with Gunningham and Vazirani about finite generation of skein categories of surfaces, and proofs in some cases.
  2. Resulting formulas for skein module dimensions of mapping tori, also obtained with Gunningham and Vazirani.
  3. A conjecture we made with Ben-Zvi, Gunningham and Safronov asserting Langlands duality for skein module dimensions.
  
  A common notion to all of these is a focus on G-skein theories for general reductive algebraic groups, not only for SL(2) as is commonly focused on. Another commonality is that the proofs are largely elementary and combinatorial and do not involve the algebraic geometry that went into our proof of the finiteness theorem.
  
  

• Jennifer Brown (University of Edinburgh). Non-semisimple skein categories
  Abstract: The skein category of a surface is a useful construction, from which algebras, modules, topological invariants, and a TQFT can all be defined. This process is well suited to semisimple inputs, and in that setting skein categories are known to be equivalent to the factorization homology of an appropriate E2-algebra.
  
  At the same time, skeins arise naturally in physics, where non-semisimple inputs frequently appear. Unfortunately, in this setting closed graphs (including links!) often vanish and must be forbidden by imposing admissibility conditions. We will explain what problems admissibility conditions cause for skein categories and factorization homology and what can be done to unify the two frameworks.
  
  This talk is based on arXiv:2406.08956, joint with Benjamin Haïoun.
  
  
• Benjamin Haïoun (University of Edinburgh). Non-semisimple skein TQFTs
  Abstract: Admissible skein modules extend to dimension 3 the notion of modified skein categories of surfaces introduced by Jennifer Brown on Monday. I will review joint work with F. Costantino, N. Geer and B. Patureau-Mirand on how to extend this construction further to 4-manifolds. In the well-studied modular case, the 4-dimensional content of this TQFT is very simple and I will explain how to recover the well-known non-semisimple (2+1)-TQFTs at its boundary. The general case however is still poorly studied, and I will discuss the new kind of algebraic data that our construction points at, and what it should satisfy to have a chance to detect exotic 4-manifolds!