ANR Project HQDIAG - Workshop Lyon 2022

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Workshop UMPA Ens de Lyon - September 5-6 2022

Schedule

Monday 5th:
14h30-15h30: Niels Feld
16h30-17h30: Clémentine Lemarié-Rieusset
Tuesday 6th:
9h-10h: Ivan Rosas-Soto
10h30-11h30: Samuel Lerbet
12h-..: Discussion on the organization of the 2022-2023 working seminar

Program

Quadratic refinements in motivic homotopy theory and non-commutative geometry

Niels Feld
Abstract
In the nineties, Voevodsky proposed a radical unification of algebraic and topological methods. The amalgam of algebraic geometry and homotopy theory that he and Fabien Morel developed is known as motivic homotopy theory. Roughly speaking, motivic homotopy theory imports methods from simplicial homotopy theory and stable homotopy theory into algebraic geometry and uses the affine line to parameterize homotopies. Voevodsky developed this theory with a specific objective in mind: prove the Milnor conjecture. He succeeded in this goal and won the Fields Medal for his efforts in 2002. In this talk, we will start by recalling some facts in motivic homotopy theory, and we then present some results in motivic enumerative geometry (Euler characteristic, base change theorem, trace formula, ramifications). Finally, we will try to generalize these results by using ideas coming from non-commutative geometry.
The quadratic linking degree

Clémentine Lemarié-Rieusset
Abstract
In this talk I will present a new application of motivic homotopy theory: motivic knot theory. More specifically, I will present an analogue of the linking number of two oriented disjoint knots: the quadratic linking degree. To do this, I will recall some notions from knot theory as well as some notions about the Rost-Schmid complex (whose cohomology groups generalize Chow-Witt groups). There will be examples!
Motivic stable cohomotopy and unimodular rows

Samuel Lerbet
Abstract
Unimodular rows are fundamental objects in algebraic K-theory as they classify stably free projective modules. Geometrically, they correspond to morphisms to punctured affine space and may thus be thought of as an algebro-geometric analogue of maps to spheres in topology. The latter give rise to Borsuk's cohomotopy groups, a counterpart to which was constructed purely algebraically by van der Kallen in the 1980s using certain orbit sets of unimodular rows. Nowadays, using the homotopy theory of schemes, Borsuk's construction may be followed more directly to produce the motivic cohomotopy groups of Asok and Fasel. We prove that these two approaches of “algebraic” cohomotopy, namely van der Kallen's and Asok–Fasel's, coincide in that there is a natural map from orbit sets of unimodular rows to motivic cohomotopy groups which is a group isomorphism in the dimension range where both structures exist.
Étale motives and the Hodge conjecture

Ivan Rosas-Soto
Abstract
The Hodge conjecture states that for a smooth projective variety over $\mathbb{C}$ the image of the cycle class map from Chow groups to Betti cohomology is the group of Hodge classes (working with rational coefficients). Although the integral version of the conjecture is false, Rosenschon and Srinivas proved that the étale version of the integral Hodge conjecture, i.e. using Lichtenbaum cohomology groups instead of Chow groups, is equivalent to the Hodge conjecture with rational coefficients. The goal of this talk is to give an overview of étale motivic cohomology and the Hodge conjecture followed by a revisit to some of the counter-examples to the integral Hodge conjecture involving torsion classes, such as the ones of Atiyah-Hirzebruch and Benoist-Ottem. After that I will discuss the generalized Hodge conjecture from an étale motivic point of view.