| Résume | 14:00--15:00 Florian Ivorra (Freiburg, Rennes) On the six operations for perverse motivic sheaves
Let k be a subfield of the complex numbers. Though it is not known to satisfy all the expected properties (especially the relations with the Chow groups), Nori’s (unconditional) category of mixed motives has been so far one of the best candidates for Grothendieck's conjectural category of mixed motives over k.
After recalling Nori’s tannakian construction, I will explain how it is possible to use the triangulated categories of étale motives constructed by Morel-Voevodsky and studied by Ayoub to define an Abelian category of perverse motivic sheaves over any k-variety (so that a perverse motive over a point is simply a Nori motive). Then, I will discuss the construction of the six operations on (the derived categories of) perverse motives, focusing mainly on the issue of nearby cycles functors and inverse image functors.
This is joint work with Sophie Morel.
15:30--16:30 Hiroyasu Miyazaki (IMJ-PRG) On the nil higher Chow groups with modulus
The algebraic K-groups do not satisfy the homotopy invariance in general. The nil K-groups are defined as the obstruction to the homotopy invariance. C. Weibel proved that the nil K-groups admit module structures over the big Witt ring of the base field, and he deduced several structure theorems. Binda-Saito's higher Chow groups with modulus can be regarded as a cycle-theoretic analogue of the relative K-groups, which are not homotopy invariant in general.
In this talk, we will define the nil higher Chow groups with modulus as the obstruction to the homotopy invariance, and we will construct module structures over the big Witt ring of the base field. If time permits, I will also explain a similar result on Kahn-Saito-Yamazaki's reciprocity sheaves.
17:00--18:00 Matthew Morrow (IMJ-PRG) Some recent progress on motivic filtrations on algebraic K-theory
I will give an introduction to some aspects of the Lichtenbaum—Quillen conjecture and the existence of Atiyah—Hirzebruch motivic filtrations on algebraic K-theory, before presenting some new results in the case of p-adic K-theory in characteristic p via syntomic methods. This will include in particular a motivic cohomology theory with modulus, which describes the wild part of algebraic K-theory relative to a normal crossing divisor. Joint works with Bhargav Bhatt and Peter Scholze, and with Dustin Clausen and Akhil Mathew. |
