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Mardi 10 mars 14:00-15:15 Syu Kato  (Université de Kyoto et IHP)
The formal model of semi-infinite flag manifolds and its application

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Lieu : salle 3L15 bâtiment 307

Résumé : Semi-infinite flag manifolds are variants of the affine flag manifolds of a
(simply connected) simple algebraic group G, that can be thought as enhancements
of the arc schemes of the corresponding flag manifolds. It was born approximately
the same time as the usual affine flag manifolds (by Lusztig and Drinfeld), but not
studied as much as them since its topology is a bit troublesome. In fact, a detailed
consideration suggests that they come up with several variants, including those we
call the ind-model and the formal model.
The ind-model of the semi-infinite flag manifolds is pursued by the works of
Braverman, Finkelberg, Mirkovic and their collaborators. They exhibit their moduli
interpretations, and connect them with the representation theory of Lie-theoretic objects
related to the Lusztig program.
In this talk, we first briefly recall some of the above results. After that, we explain our
explicit description of the semi-infinite flag manifolds as (universal) indschemes of
ind-infinite type. Also, we exhibit several basic material on them including the
Borel-Weil-Bott theorem and the description of its equivariant K-groups.
Then, we establish some connection between the formal models and the ind-models
of semi-infinite flag manifolds. Together with some interpretations provided by
Givental-Lee and Braverman-Finkelberg, this establishes an isomorphism between
(the suitable localizations of) the equivariant quantum K-group of a flag manifold G
and the equivariant K-group of the affine Grassmanian of G in such a way it resolves
a conjecture by Lam-Li-Mihalcea-Shimozono.

The formal model of semi-infinite flag manifolds and its application  Version PDF

Mardi 3 mars 14:00-15:15 Laurent Moret-Bailly  (Université de Rennes 1)
Points rationnels dans leur fibre : autour d’un théorème de Poonen

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Lieu : salle 3L15 bâtiment 307

Résumé : Soit f : X—>S un morphisme de schémas, de présentation finie et dominant
(on pourra supposer S et X affines intègres, et f surjectif). Lorsque S est une variété
de dimension >0 sur un corps (et aussi lorsque S=Spec(Z)), Poonen a montré l’existence
d’un point fermé x de X dont le corps résiduel est radiciel sur celui de f(x) ; en particulier,
si k est parfait, x est un point rationnel de sa fibre X_x vue comme schéma sur f(x).
L’objet de l’exposé est de préciser ce résultat et de le généraliser à d’autres schémas de
base S.

Points rationnels dans leur fibre : autour d’un théorème de Poonen  Version PDF

Mardi 25 février 14:00-15:15 Jonathan Brundan  (Université de l'Oregon et IHP)
Webs and tilting modules for the general linear group

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Lieu : salle 3L15 bâtiment 307

Résumé : I will talk about some strict monoidal categories which arise in the study of rational
representations of the general linear group. These include the Schur category, which may be
realized diagrammatically in terms of webs, and the closely related category Tilt GLn of tilting
modules for GLn. As an application, I will describe the structure of the semisimplification of
Tilt GLn, that is, the quotient of this category by the tensor ideal of all negligible morphisms.
In characteristic p > n, this is a well-known Verlinde category, but we also understand now the
situations when p <= n.

Webs and tilting modules for the general linear group  Version PDF