Journée GRAAL (Géométrie-Représentations-Algèbre Amiens-Lille) à Lille

• 09h30 : accueil
• 10h00–10h45 : A geometric interpretation of the q-characters of quantum affine algebras (Luca Francone)
• 11h00–11h45 : The universal property of strict polynomial functors (Antoine Touzé)
• 12h00–13h30 : déjeuner 
• 13h30–14h15 : Brauer groups: applications to hyperkähler geometry (Dominique Mattei)
• 14h45–15h30 : The BPS decomposition theorem (Lucien Hennecart)

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A geometric interpretation of the q-characters of quantum affine algebras (Luca Francone)

The rough purpose of this talk is to show that some properties of the representation theory of affine quantum groups can be understood in terms of the geometry of complex reductive groups. Through the talk, we will focus on a geometric interpretation of the q-character homomorphism for quantum affine algebras of ADE type, which is described as the pullback along a morphism of pro-varieties called discrete Miura transformation. This result confirms a conjecture of Frenkel and Reshetikhin. The talk is based on a joint work with Bernard Leclerc.

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The universal property of strict polynomial functors (Antoine Touzé)

Strict polynomial functors are an avatar of polynomial representations of GLn(k). The aim of this talk is to explain that the category of strict polynomial functors has a universal property in the setting of tensor abelian categories (which explains why the representation theory of  GLn(k) may be interesting in unexpected settings). We will first explain this universal property when k is a field of characteristic zero (in which case, it is well-known), and then turn to the more subtle case of a field k of positive characteristic.

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Brauer groups: applications to hyperkähler geometry (Dominique Mattei)

I will present the construction of twisted relative Jacobians of curves, after introducing the notion of (special) Brauer group of a complex variety. I will explain how these spaces can be used to study hyperkähler manifolds. If time permits, I will tell a few words on an application to the so-called period-index problem.

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The BPS decomposition theorem (Lucien Hennecart)

Given a proper morphism between algebraic varieties, a fundamental question is to understand how the cohomology of its fibres varies. The foundational work of BBDG provides the formalism of perverse sheaves, which is particularly well suited to this question. When the source is smooth, the BBDG decomposition theorem expresses the derived pushforward of the constant sheaf as a direct sum of shifted simple perverse sheaves.
In this talk, I will broaden the scope to good moduli space morphisms of smooth algebraic stacks. I will explain how these morphisms satisfy an analogue of the BBDG decomposition theorem and give an explicit description of the summands appearing in the derived pushforward of the constant sheaf.
Rather than presenting the main theorem in its most general form, I will illustrate it through examples, showing the mathematical journey that led to these results.
This is based on various work, in particular with T. Kinjo.