ATMW Classical Algebraic K - theory (2013)

Brief description of the workshop:-

1. Quillen-Suslin-Bak, localization and patching
   Quillen-Suslin-Bak, localization and patchingOne of the most powerful ideas in the study of groups of points of reductivegroups over rings is localization. It allows to reduce many important prob-lems over arbitrary commutative rings, to similar problems for semi-localrings. Localization comes in a number of versions. The two most famil-iar ones are localization and patching, proposed by Daniel Quillen [31] andAndrei Suslin [47], and localization–completion, proposed by Anthony Bak[3].Recently, these methods were further generalized to unitary groups, andisotropic reductive groups, and used to study nilpotent structure of K1 -groupof various classical-like groups by Bak, Basu, Hazrat, Petrov, Rao, Stepanov,Vavilov and others [4, 9, 5, ?, ?, ?, 10, ?, ?, ?, ?, 28, 29, 30, 35, ?]. One part ofthe workshop is talks to present various work done recently in this direction.Talks will be given by Rao, Stepanov and Vavilov. Rao will emphasize theconnection between some of these topics with the next topic.
2. Study of projective modules over an affine algebra
   Study of projective modules over an affine algebra[23, ?] This relates to recent progress in this subject by Fasel-Rao-Swan on aquestion of Suslin, and also reinterprets the earlier results of Suslin in light1of the modern approach proposed.If X = Spec(R) is a smooth affine threefold over a field k of characteristicdifferent from 2 then the elementary symplectic Witt group WE (R) has a nicecohomological description as H2 (X, K3 ). Using Bloch-Kato for K1 , K2 it is shown that this is a divisible group prime to the characteristic. The followingcancellation theorem is obtained from this result: If X is a non-singular affinevariety of dimension d over an algebraically closed field k, and if char(k) ≥ dthen any stably trivial vector bundle of rank (d − 1) over X is trivial. Thehypothesis that X is non-singular can be weakened to X is normal if d ≥ 4.The general case when X is singular is also discussed.The new idea behind these results is the use of the theory of Grothendieck-Witt groups (aka Hermitian K-theory). Talks will be given by Fasel, Schlicht-ing on this topic. Rao will talk on some applications.There will also be talks on the recent work of J.Fasel and Aravind Asokon classification of projective modules over affine threefolds, and some talksby Aravind Asok on A1 -homotopy.
   
   
3. The Bass-Suslin conjecture
   The Bass-Suslin conjectureRecent progress on this long standing conjecture by Rao-Swan in [25] willbe described by Rao. The connection with the theory of Suslin matrices willalso be be outlined. Talks will be given by Rao.There will be six to seven streams of 4-6 lectures each, one stream oftutorials, and a Conference for the last 2 or 3 days. The number of lecturesper day will not exceed 5.

Possible invited speakers

1. Localization techniques (N. Vavilov, A. Stepanov )
2. Stabilization and Prestabilization (N. Vavilov, R.A. Rao)
3. K-theory of Exceptional groups. (N. Vavilov,  N. Petrov)
4. Witt and Grothendieck groups (B. Calme`s)
5. Higher Grothendieck-Witt groups (M. Schlichting)
6. Recent developments on classification of projective modules over an affine algebra (J. Fasel, Aravind Asok , perhaps R.A. Rao)
7. A1 -homotopy and classification problems (Aravind Asok )
8. Suslin matrices, Bass-Suslin conjecture (R.A. Rao)

References

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