Alexander Vishik

Motives of Quadrics with Applications to the Theory of Quadratic Forms

This text is the notes of my lectures at the mini-course “Methodes geometriques en theorie des formes quadratiques” at the Universite d’Artois, Lens, June 26-28, 2000. However, some extra material is added. I tried to make the material more accessible for the reader. So, complicated technical proofs are presented in a separate section. Applications are discussed in the last two sections. In particular, splitting patterns of quadratic forms of odd dimension ≤21 or of even dimension ≤12 are determined in the last section.

Fields of u-invariant 2 r + 1

In this article we provide a uniform construction of fields with all known u-invariants. We also obtain the new values for the u-invariant: 2 r + 1, for r > 3. The main tools here are the new discrete invariant of quadrics (so-called elementary discrete invariant), and the methods of [14] (which permit one to reduce the questions of rationality of elements of the Chow ring over the base field to that over bigger fields, the generic point of a quadric).

Koszul duality and Galois cohomology

It it shown that the Bloch-Kato conjecture on the norm residue homomorphism

Symmetric operations for all primes and Steenrod operations in algebraic cobordism

K^M(F)/l \to H^*(G_F,Z/l)

Motivic equivalence of affine quadrics

follows from its (partially known) low-degree part under the assumption that the Milnor K-theory algebra

Symmetric operations in algebraic cobordism

K^M(F)/l

On the Chow groups of quadratic Grassmannians.

modulo

Algebraic cobordisms of a Pfister quadric

l

Generic points of quadrics and Chow groups

is Koszul. This conclusion is a case of a general result on the cohomology of nilpotent (co-)algebras and Koszulity.

Quadratic forms with absolutely maximal splitting

In this article we construct symmetric operations for all primes (previously known only for