# Jerusalem Mathematics Colloquium

Thursday, 25 March 1999, 4:00 pm

Mathematics Bldg., lecture hall 2

##
Professor Nikolai Vavilov

(Saint-Petersburg State University)

"Decomposition of transvections: a theme with variations"

** Abstract: **

The classical work of Bass and Milnor on "lower
algebraic K-theory" attracted major attention to the
following questions, among others:

- Normality of the elementary subgroup E(n,R)
in the general linear group GL(n,R);
- Classification of subgroups in GL(n,R),
normalised by E(n,R);
- Centrality of the extension St(n,R) --> E(n,R).

There are the so called STANDARD answers to these questions,
which mimic the behaviour of these groups over fields and at
the stable level. These standard answers are known as the
MAIN STRUCTURE THEOREMS and there are several methods to prove
these results, using geometry, K-theory or ring theory in various
proportions.

In the present talk (based on a joint paper with
Alexei Stepanov, to appear in `K-theory') we present new purely
geometric proofs of the main structure theorems and some related
results (like, say, description of certain classes of subgroups in
GL(n,R)) for the case of a commutative ring, based on the method
we call DECOMPOSITION OF UNIPOTENTS. Even for the case of fields
these proofs are much simpler than the proofs of results like
Jordan-Dickson theorem, usually reproduced in textbooks.

We discuss also some generalisations of these results
to non-commutative rings (a very challenging and not fully solved
problem is that of an exact class of rings when the standard answers
hold) and mention generalisation to other groups and modules (based
on our joint works with Eugene Plotkin, Anthony Bak, Elena Perelman
and Michail Gavrilovich).

Coffee, Cookies, Company at the faculty lounge at 3:30.

List of talks, 1998-99

List of talks, 1997-98