# Séminaire d'Algèbre

### salle 001, rez de chaussée, 11 rue Pierre et Marie Curie - 75005 Paris

Organisateurs : , , , , , et .

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Lundi 5 février 2018 à 14h00

Toshiki NAKASHIMA (Tokyo), Geometric crystals on cluster varietie.
[The notion of geometric crystal was initiated by A.Berenstein and D.Kazhdan to consider certain geometric analogue to the Kashiwara's crystal base theory. Their structures are described by rational maps and rational functions.
If all these rational maps are positive'', such geometric crystals are called positive'' and they can be tranfered to the Langlands dual crystal bases'' by tropicalization/ultra-discretization procedure.
V.Fock and A.Goncharov defined certain pair of varieties (A,X), called cluster ensemble'' which is obtained by glueing algebraic tori using the A-mutations and X-mutations'' respectively.
They gave the conjectures on tropical duality'' between cluster ensemble A-variety and X-variety (called Fock-Goncharov conjectures).
We shall define the positive geometric crystal structure on cluster varieties and then obtain the resulting tropicalized crystals, which will be a guide to understand the Fock-Goncharov conjectures in terms of crystal base theory.
Finally, we shall show some compatibility of geometric crystal structures on A-variety and X-variety in the classical type A case.]

Lundi 19 février 2018 à 14h00

Travis SCHEDLER (Imperial College London), Résolutions symplectiques des variétés de carquois et des caractères.
[Nous allons expliquer une classification des variétés de carquois (Nakajima) qui admettent une résolution symplectique, et similairement pour les variétés des caractères d’une surface de Riemann (espace moduli des systèmes locales). Nous expliquons également un résultat parallèle pour les espaces moduli de Higgs de A. Tirelli (doctorant). Ce travail est en collaboration avec G. Bellamy.]

Lundi 26 février 2018 à 14h00

Amnon YEKUTIELI (Be'er Sheva), The Derived Category of Sheaves of Commutative DG Rings.
[In modern algebraic geometry we encounter the notion of derived intersection of subschemes. This is a sophisticated way to encode what happens when two subschemes $Y_1$ and $Y_2$ of a given scheme X intersect non-transversely. The classical intersection multiplicity can be extracted from the derived intersection.
When the ambient scheme X is affine, it is not too hard to describe the derived intersection, by taking flat DG ring resolutions of the structure sheaves of the subschemes $Y_1$ or $Y_2$. This also works when the scheme X is quasi-projective. However, derived intersections in more general schemes X could only be treated using the very difficult homotopical methods of derived algebraic geometry.
Several months ago I discovered a cheap'' way to construct flat resolutions of sheaves of rings. The resolutions are by semi-pseudo-free sheaves of DG rings. The main advantage is that the geometry does not change: all the action takes place on the original topological space X.
Using semi-pseudo-free resolutions it is possible to produce derived intersections as above. It is also possible to get a direct presentation of the cotangent complex of a scheme (at least in characteristic 0). Presumably the derived adic completion of Shaul, so far studied only in the affine case, could be globalized using our our methods.
Lastly, the semi-pseudo-free resolutions give rise to a congruence on the category of sheaves of commutative DG rings on the space X, that we call relative quasi-homotopy. The functor from the homotopy category to the derived category turns out to be a faithful right Ore localization. This fact gives tight control on the derived category. It should be noted that in this situation there does not seem to exist a Quillen model structure, so the traditional approaches would fail.
In the talk I will explain the various ideas listed above. More details can be found in the eprint arxiv:1608.04265.]

Lundi 12 mars 2018 à 14h00

Julian KULSHAMMER (Stuttgart), Existence and uniqueness of exact Borel subalgebras.
[Quasi-hereditary algebras and their infinite analogues, highest weight categories, appear frequently in many areas of representation theory. In joint work with S. Koenig and S. Ovsienko, we showed that every quasi-hereditary algebra can up to Morita equivalence be obtained as the dual of a coring object in the tensor category of bimodules over a directed algebra, the exact Borel subalgebra.
Under an additional assumption, the exact Borel subalgebra as well as the coring object are in fact unique up to isomorphism. This is joint work with V. Miemietz.]

Lundi 26 mars 2018 à 14h00

Lara BOSSINGER (Cologne), Toric degenerations from representation theory, tropical geometry and cluster algebras.
[In this talk I will explain how toric degenerations arise from the tropicalization of a (projective) variety. In the context of varieties that are interesting from a representation theoretic point of view (e.g. Grassmannians or flag varieties) I will explain a construction of toric degenerations due to Fang, Fourier, and Littelmann called birational sequences and compare to degenerations obtained from the cluster structure on these varieties. I will present many examples and some results on how these constructions are related.
For example, I will present computational results on the tropicalization of the full flag variety for n=4 and 5 and compare the obtained toric degenerations to some classical degenerations from representation theory (string polytopes and the FFLV polytope) that arise in the context of birational sequences.]

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 Groupes, représentations et géométrie. Dernière modification : le 21/02/2018

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