Résume | 14h00--15h00 : **Oliver Braunling** (Universität Freiburg) *K-theory of locally compact modules*
We discuss the K-theory of categories of modules over a number field, but additionally equipped with a locally compact topology. Although this is not an abelian category, such a category has a natural exact structure, and Pontryagin duality makes it an exact category with duality. Generalizing results of Dustin Clausen, we explain how to compute the K-theory of such categories, and how it is connected to the adeles and other number-theoretical objects. Joint with Peter Arndt.
15h30--16h30 : **Kęstutis Česnavičius** (Université Paris-Sud, CNRS) *Grothendieck--Lefschetz for vector bundles*
According to the Grothendieck--Lefschetz theorem from SGA 2, there are no nontrivial line bundles on the punctured spectrum U_R of a local ring R that is a complete intersection of dimension ≥4. Hailong Dao isolated a condition on a vector bundle V of arbitrary rank on U_R, always satisfied by a line bundle, and conjectured that its validity is necessary and sufficient for V to be trivial. We use deformation theoretic techniques to settle Dao's conjecture and present examples showing that its assumptions are sharp.
17h00--18h00 : **Adam Topaz** (University of Oxford) *Recovering function fields from their ℓ-adic cohomology*
This talk will present some recent work in progress which shows that the function field of a higher-dimensional variety is determined, up-to isomorphism, from its ℓ-adic cohomology ring, when it is endowed with the Galois action of a "sufficiently global" base field. A comparison with Bogomolov's programme and the Bogomolov-Pop conjecture in birational anabelian geometry will also be discussed. |