1. Annual Project Report Dec. 1, 1997  Nov. 30,
1998
Activities of the Network in 2000:
 Midterm Review Meeting and Postdoc
Conference, Bielefeld, Febr. 29  March 4, 2000,
organized by Ina Kersten and Ulf Rehmann.
 International Conference in
CIRM (MarseilleLuminy): Up to Homotopy, 1923 June 2000. Organized by JeanLouis Loday and Daniel
Guin.

CONFERENCE ON ALGEBRAIC KTHEORY AND HOMOTOPY
THEORY OF SCHEMES (Preliminary announcement)
Site : Université Paul Sabatier, Departement de Mathematiques
118, route de Narbonne, 31062 ToulouseCedex, FRANCE
Dates: Monday 3, July 2000  Friday 7, July 2000
This is the week before the
3rd European Congress of Mathematics
in Barcelona, 10 July  14 July (http://www.iec.es/3ecm/).
PROGRAM. Among those who have already agreed to speak are :
Spencer Bloch, Eric Friedlander, Marc Levine, Markus Rost,
Carlos Simpson, Masaki Hanamura, Annette Huber,
Emmanuel Peyre, Burt Totaro
Scientific Committee : Bruno Kahn, Max Karoubi, Georges Maltsiniotis,
Fabien Morel, Christophe Soule, Joseph Tapia
Further activities of the Network:
Further coordinated workshops on actual topics, among them:
 Thanksgiving Workshop on Controlled Topology,
Geometry and Algebra, organized by H. J. Munkholm.
 Workshop on various topics of the project, organized by
C. Pedrini.
Activities of the Network in the Past:
1999:
 Conférence "Algèbre
homotopique", Strasbourg, March 25  28, 1999,
organized by JeanLouis Loday (Strasbourg) and Teimuraz Pirashvili (Tbilisi).
 Workshop on Algebraic Cycles, Genova,
March 26  28, 1999, organized by
Alberto Albano, Luca Barbieri Viale, Gian Pietro Pirola..
 Oberwolfach Conference on Quadratic
and Hermitean Forms, May 16  22, 1999, organized by
Albrecht Pfister (Mainz), Winfried Scharlau (Münster), JeanPierre
Tignol (LouvainlaNeuve).
 Workshop on Quadratic Forms and
Related Topics, Besancon, May 25 28, 1999, organized by Eva
BayerFluckiger and Detlev Hoffmann.
 MiniSummerSchool "Involutions et groupes
algebriques". Le 7,8 et 9 juin a la faculte Jean Perrin, rue Jean
Souvraz a Lens. Organisateurs : J. Buresi, P. Mammone, JP. Tignol.
 Conference on "Quadratic Forms and their Applications" in
Dublin, July 410, 1999, organized by
E. Bayer (Besancon), D. Lewis (Dublin) and A. Ranicki (Edinburgh).
 Oberwolfach Conference on Algebraic
KTheory, Sept. 26  Oct. 2, 1999, organized by Daniel R. Grayson (Urbana),
Uwe Jannsen (Köln), Bruno Kahn (Paris)
1998:
 SUMMER SCHOOL on "KTheory and Algebraic Groups",
June 28July 3 1998, in Trento, organized by C. Pedrini
 Conference on Linear Algebraic Groups and Related
Structures,
Aug. 915, 1998 in Bielefeld, organized by I. Kersten, A. Merkurev, U. Rehmann.
 Conference on Topology and
Geometry,
Aug. 915, 1998 in Aarhus, organized by Karsten Grove, Ib Madsen and Erik
Kjær Pedersen.
The two above conferences are satellite conferences of the
International Congress of Mathematicians, Berlin, 1998
Organization of the Network:
 Network Coordinator:
Ulf Rehmann,
Fakultät für Mathematik, Universitätsstraße 25,
Universität Bielefeld,
D33501 Bielefeld, Germany
Tel +49 521 106 5039, Fax +49 521 106 4743
email: rehmann@mathematik.unibielefeld.de
 Local Coordinators at EC Countries:
 Claus
Scheiderer, Universität Duisburg, Germany
 Manfred Knebusch, Universität Regensburg, Germany
 Eva BayerFluckiger, Université de Franche Comté, Besancon, France
 Max Karoubi, Université Paris 7Denis Diderot, France
 JeanLouis
Loday, Université Louis Pasteur, Strasbourg, France
 JeanPierre Tignol, Université catholique de Louvain, Belgium
 Claudio Pedrini, Universitá di Genova, Italy
 David Lewis, University College Dublin, Ireland
 HansJ. Munkholm, Odense
Universitet, Denmark
 Local Coordinators at NonEC Countries:
 Viacheslav Yanchevskii, Institute of Mathematics, National Academy of Sciences, Belarus
 Teimuraz Pirashvili, A. Razmadze Mathematical Institute, Georgia
 Andrei Suslin, St. Petersburg Branch of Steklov Mathematical Institute,
Russian Academy of Sciences, Russia
Postdoc Positions to be offered by the Project
The project gets funding for 304 manmonths of postdoc
positions. Scientists from EC countries who are interested in one of
these positions should contact one of the above mentioned coordinators
from the EC countries (for closer information see the
work programme of the project).
Postdoc vacancies of the network
Research Objectives
Algebraic Ktheory is a young but well established and highly
interdisciplinary area in pure mathematics. Using methodical
approaches from algebra and topology, it has many beautiful
applications. It allows to study many different types of structures
like manifolds, schemes, quadratic forms and their automorphism
groups. Thereby it often not only gives new insight into the subject
but also provides a deeper understanding of "classical" results. The
theory of linear algebraic groups has a similarly universal nature, as
it allows to understand a wide area of related algebraic structures
under a unifying perspective, because the algebraic
groups occur as automorphism groups of these structures. Examples of
such structures are quadratic or Hermitian forms and
Azumaya algebras, which each have formed their own mathematical culture.
All these disciplines are closely related in various aspects and they
are vivid fields in the sense that they produce an abundance of
stimulating and fascinating problems.
It is the objective of this network project to perform advanced
research in the fields of Algebraic KTheory, Linear Algebraic Groups
and Related Structures, by using methodological approaches from
algebra and topology.
``Related Structures'' are for example manifolds, schemes, quadratic or
Hermitian forms, Azumaya algebras, Lie algebras, Brauer groups,
and their associated homology and cohomology theories.
It is the main objective of this proposal to bring together the
expertises of these distinct fields in order to promote their research
significantly by mutual benefit, and, in particular, to attract young
researchers at the postdoc level into this broad area by international
exchange and intensive training.
Main research areas will be: Stable Ktheory and Hochschild homology,
Ktheory of spaces, equivariant Ktheory, algebraic models of spaces,
Leibniz homology, controlled Ktheory, Ktheory of varieties, linear
algebraic groups over arbitrary fields, projective homogeneous
varieties, Hasse principles for principal homogeneous spaces and other
objects, splitting varieties, quadratic forms over arbitrary fields,
trace forms, Azumaya algebras, algebras with involution, Chow groups,
Brauer groups, Requivalence and birational questions, toric
varieties, cohomology ring of finite group schemes.
The detailed Work Programme: Postscript
Some historical remarks and
illustrations concerning certain
aspects of the research by this network can be found here.
Results and Achievements (Dec, 1, 1997  Nov. 30, 1998):
During the reporting period, several manuscripts and one monograph
(``The book of involutions'', 580 pages) have been published involving
two or three network partners each; several others of this type have
been produced as preprints or have been announced. These manuscripts
describe results on algebras with and without involutions, quadratic
froms, various algebraic groups, in particular on unitary and on
orthogonal as well as on exceptional groups, and also on cylic
homology. In particular, the publication of the monograph is a
milestone of the project:
This book is an exposition of the theory of central simple
algebras with involution,
in relation to linear algebraic groups. It provides the
algebratheoretic foundations for
much of the recent work on linear algebraic groups over arbitrary
fields.
The research of the network was certainly stronly enhanced and
influenced by the following three major networking activities:
 Summer School
on ``KTheory and algebraic groups'', June 28  July 3, 1998
(Levico Terme Italy, organized by C.~Pedrini with a scientific
committee from two other participants.
This had a particular training effect, since high level
curses for researchers on the postdoc level have been given.
There have been two sessions: a morning session where the following 4
series of lecture have been given:
1) An introduction to KTheory and its recent developments.
2) KTheory and Algebraic Groups.
3) KTheory of algebraic integers.
4) Ktheory and quadratic forms over fields.
Written notes oof these curses were distributed to all participants.
In the afternoon sessions there have been 11 research talks given by
young researchers on topics concerning
Ktheory, algebraic groups, number theory, algebraic geometry, quadratic
forms, Brauer groups etc.
g
 Conference on ``Linear Algebraic Groups and Related
Structures'', August 9  15, 1998 (University of Bielefeld, Germany),
organized by I. Kersten and U. Rehmann. 32 Talks were given (4560
min each), 67 participants from 17 countries attended the
conference. This conference was funded by SFB 343 ``Diskrete
Strukturen in der Mathematik'' at the University of Bielefeld, by this
TMR project, and by INTAS project INTAS932618EXT.
 Conference on ``Geometry and Topology'', August 9 
15, 1998 (University of Aarhus, Denmark), organized by K. Grove,
I. Madsen, E. K. Pedersen. 51 Talks were given (4560 min each), 139
participants attended the conference.
Last Updated: 21012000