**LMS Workshop on Motives, Quadratic Forms and Algebraic Groups**

**LMS Workshop on Motives, Quadratic Forms and Algebraic Groups**

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*Program:*

Monday | Tuesday | Wednesday | Thursday | Friday | |

9:00 -9:30 | Registration | ||||

9:30 - 10:45 | Karpenko | Karpenko | Totaro | Weibel | Weibel |

11:15 - 12:30 | Weibel | Weibel | Totaro | Karpenko | Karpenko |

2:30 - 3:30 | Vishik | Karoubi | Excursion starting at 1:15 | Calmes | de Jeu |

4:00 - 5:00 | Neeman | Hoffmann | Hornbostel | ||

5:15 - 6:15 | Tignol | Saltman | Brosnan |

**Location:**

**Location:**

Room 005, Ground floor, (New) David Bates Building, School of Mathematics and Physics, 7 College Park.

Information sheet on what facilities (email, etc.) available.

*Courses:*

**Nikita Karpenko: ***Algebraic cycles on the square of a quadric*

*Four Lectures:* Let X be a smooth projective quadric over a field (of arbitrary characteristic). We summarize results, techniques, and open questions concerning the Chow group of algebraic cycles on X^2=XxX. Then we show how this is applied in the study of classical discrete invariants of quadratic forms.

Ref: *Algebraic and Geometric Theory of Quadratic Forms. *

Burt Totaro:* Quadrics and Birational Geometry*

*Lecture 1*** : **A longstanding theme in the theory of quadratic forms is the study of function fields of quadrics. Geometrically, this amounts to studying anisotropic projective quadrics up to birational equivalence. It is useful for applications to know when a quadric is ruled (that is, birational to the product of some variety with the projective

line over the base field). We discuss Karpenko's theorem that a quadric with Witt index 1 is non-ruled, and the conjecture that the converse also holds.

Ref: N. Karpenko, *On anisotropy of orthogonal involutions.*

B. Totaro, *The automorphism group of an affine quadric*

*Lecture 2:* The Sarkisov program is the standard method, coming from minimal model theory, to study birational equivalences between Fano varieties over any field. We look at what the Sarkisov program gives in the special case of quadrics. Only a few examples have been worked out so far.Chuck Weibel:

*Motivic Cohomology*

*Lecture 1:* We introduce the notion of sheaves and presheaves with transfers, define Motivic Cohomology, and explain the relation to Milnor K-theory. The "Tate motives" Z(i)[n] are chain complexes of sheaves with transfers.*Lecture 2:* Etale Sheaves with transfers are introduced; Suslin Rigidity says that they are often just Galois modules. Nisnevich sheaves with transfers are introduced, and this provides transfers on Motivic Cohomology.*Lecture 3:* The triangulated category DM of motives is defined. We show it contains the classical category of Chow Motives, due to Grothendieck, as well as the category of geometric motives, both of which have good duality theories. In this context, the Motivic Cohomology of X becomes the bigraded groups Hom(X,Z(i)[n]).

*Lecture 4:* Cohomology operations can be defined on Motivic Cohomology, and lead to a proof of the Milnor Conjecture. Similar ideas form the basis for the announced proof of the so-called Bloch-Kato Conjecture

by Voevodsky and Rost. A sketch of some key ideas involved in the proof will be given.

Ref: *Motivic Cohomology*

*Speakers:*

Patrick Brosnan: *Essential dimension and algebraic stacks*

I will discuss joint work with Zinovy Reichstein and Angelo Vistoli on
essential dimensions emphasizing lower bounds on the essential
dimension of the Spin groups and the applications of these lower bounds
to quadratic forms.

Baptiste Calmes: *Witt groups of Grassmann varieties*

The content of this lecture is joint work with Paul Balmer. We will give a computation of the Witt groups of Grassmannians by cellular decomposition. This will involve the usual combinatorics of Young diagrams, but also a new lemma giving a geometric description of the connecting homomorphism in a long exact sequence of localization in the context of a relative cellular space. This lemma shows that Witt groups behave quite differently from oriented cohomology theories

(such as Chow groups) when dealing with cellular spaces.

Detlev Hoffmann: *Becher's proof of the Pfister Factor Conjecture*

Karim Becher has recently found a surprisingly elementaryproof of the Pfister Factor Conjecture. This conjecture originallydue to Shapiro has been of great interest as it allows equivalentformulations in seemingly disparate contexts, such as composition ofquadratic forms or algebras with involution. We will mention some ofthese formulations and explain Becher's proof which makes clever use ofsome well known facts about function fields of conics and howquadratic forms and involutions behave when extending the base field to such a function field of a conic.

Jens Hornbostel: *Transfer morphisms for Witt groups*

We construct transfer morphisms for Witt groups with respect to proper morphisms. Then we establish some properties (e. g. base change) and discuss applications. This is a joint work with B. Calmes.

Rob de Jeu: *On the p-adic Beilinson conjecture for number fields*

We discuss a conjectural p-adic analogue of Borel's theorem that relates the regulators of the higher K-groups of a number field to the values of its zeta-function. We prove this in some cases and provide numerical evidence in

some other cases.

Max Karoubi: *Twisted K-theory *

Twisted K-theory has its origins in the author's PhD thesis and in the paper with P. Donovan.The objective of this lecture is to revisit the subject in the light of new developments inspired by Mathematical Physics. See for instance E. Witten (hep-th/9810188), J. Rosenberg, C. Laurent-Gentoux, J.-L. Tu, P. Xu (ArXiv math/0306138) and M.F. Atiyah, G. Segal (ArXiv

math/0407054), among many authors.

We also prove some new results in the subject: a Thom isomorphism in this setting, explicit computations in the equivariant case and new cohomology operations.

`Reference : M. Karoubi, Twisted K-theory, old and new`

Amnon Neeman:*An introduction to derived categories*

This is addressed primarily to students and non-specialists, meaning we should assume not much background.

David Saltman: *Division Algebras*

Division algebras have been studied for over 150 years and the basic question has not changed - what do they all look like? To be precise let me say that a ``division algebra'' is a noncommutative finite dimensional algebra over a field where every nonzero element has a multiplicative inverse. There have been various constructions of such objects - cyclic algebras, crossed products, etc., and each construction leads to the question of whether we now have them all. More generally, division algebras correspond to elements in a cohomology group, so the question then becomes to relate elements in cohomology to concrete algebras.

All these issues about division algebras come into clearer focus when we concentrate on those of so called prime degree, meaning those of dimension p^{2}over their center where p is prime. With this focus in mind, we will talk about the state of the art in describing all these algebras, and in particular about the question of the cyclicity of such algebras.

Jean-PierreTignol: *Central simple algebras with involution over Henselian fields*

Central simple algebras with anisotropic involution over a Henselian valued field with residue characteristic different from 2 are shown to carry a special kind of value function, which is an analogue of Schilling valuations on

division algebras. (Joint work with A. Wadsworth)

Alexander Vishik: *Symmetric and Steenrod operations in Algebraic Cobordism*

We will describe connection between Symmetric, Steenrod and Landweber-Novikov operations.