Full exceptional collections on Isotropic Grassmannians

• Lyalya Guseva

Room: René Baire The bounded derived category of coherent sheaves D(X) is an important invariant of an algebraic variety X. While the structure of derived categories is generally quite intricate, in certain cases when D(X) admits a so-called full exceptional collection, D(X) can be described explicitly. Some of the earliest examples of full exceptional collections were constructed by Kapranov in 1983 for classical Grassmannians. Since then, a folklore conjecture says that full exceptional collections consisting of vector bundles exist in the derived categories of all rational homogeneous varieties. In my talk I will outline the proof of this conjecture for all rational homogeneous varieties associated with symplectic groups. This is joint work with Sasha Novikov.

La conjecture de Bloch-Kato et formes modulaires de poids 1

• Arthur Gérard

Room: René Baire Les formes modulaires de poids 1 et l'arithmétique des valeurs spéciales de fonction L semble assez mystérieuse. En effet, la fonction L de ces dernières ne possède pas de valeur critique au sens de Deligne, il semble alors nécessaire d'atteindre les formes modulaires de poids 1 par déformation p-adique, et le formalisme de la conjecture de Bloch-Kato se prête bien à cet exercice. Dans cet exposé, j'évoquerai certaines propriétés arithmétiques des formes modulaires afin de justifier l'intérêt porté aux valeurs spéciales de leur fonction L. Je présenterai ensuite les résultats et conjectures célèbres sur ces dernières, notamment la conjecture de Birch-Swinnerton-Dyer. Je présenterai ensuite le formalisme de Bloch-Kato et la reformulation motivique de BSD pour évoquer mon sujet de thèse ainsi qu'une méthode prometteuse pour sa résolution.

(Relative)A1-Contractibility of Smooth Affine Schemes over a Dedekind base

• Krishna Kumar Madhavan Vijayalakshmi

Room: René Baire In 1935 Whitehead published a purported proof on Poincaré conjecture unveiling an homotopical obstruction to unique characterization of R^n among open contractible n-manifolds. However, it is now a fact that for all n>2, R^n is the unique open contractible n-manifold that is simply connected at infinity. The analogous question in algebraic geometry is to characterize the affine n-space among smooth A1-contractible affine schemes. With much novelty, this is proven in affirmation in dimensions n < 3 over fields. This question is largely open for surfaces in positive characteristics and it utterly breaks down in higher dimensions. In this talk, we will upgrade this characterization to a Dedekind base by establishing a connection between the motivic homotopy theory and the classical affine algebraic geometry. To this end, we will see that over Dedekind scheme S, the affine n-space forms a Zariski torsor under an A^n-bundle for n < 3 and a vector bundle if S is affine. With further hypothesis, we can retrieve a similar characterization as that of fields.

Les F-isocristaux comme connexions de de Rham-Witt

• Rubén Muñoz–Bertrand

Room: René Baire Les F-isocristaux sont des objets importants en cohomologie p-adique des variétés algébriques en caractéristique strictement positive. Ceux-ci viennent avec une action du Frobenius qui induit un endomorphisme sur la cohomologie. Jusqu'à présent, la construction de ces objets et du Frobenius faisait intervenir soit des catégories de Grothendieck, soit des faisceaux sur lesquels on ne peut pas, en général, relever le Frobenius globalement. Nous expliquerons ici comment le complexe de de Rham-Witt permet de décrire les F-isocristaux d'une variété lisse sur un corps parfait de caractéristique strictement positive. Cette construction vient avec un Frobenius global et canonique, tout en conservant la topologie de Zariski.

Enumerative geometry using motivic theory

• Victor Chachay

Room: René Baire When computing enumerative invariants on algebraic surfaces, one often has to work over an algebraically closed field. We will use an example to highlight the problems on other fields and how the motivic theory is a way to go around it.

The method of Eisenstein congruences in Iwasawa theory.

• Ruichen Xu

Room: René Baire In Iwasawa theory, people study the mysterious relations between special values of L-functions and arithmetic objects (such as certain Galois cohomology groups called Selmer groups) as they vary in p-adic families. These relationships are formulated as the Iwasawa main conjectures. Among the successful approaches to these conjectures, one divisibility of them, namely “the lower bound of Selmer groups”, is often proved by the method of “Eisenstein congruences”. In this talk, I will discuss this method, beginning with the early days of Iwasawa theory, in the era of Iwasawa, Serre, and Ribet. If time allows, I will also introduce recent advances in this area.