Andreas Rosenschon (SUNY at Buffalo):
Lecture series on Algebraic Cycles and K-theory

The Strategic University Program in Pure Mathematics (Suprema) at the Department of Mathematics, University of Oslo, organizes a series of lectures by Andreas Rosenschon in May 2006. The lectures are introductory and aimed at faculty members, postdocs, and students interested in learning about crossroads between algebraic geometry and algebraic topology.

K-theory is a branch of mathematics which brings together algebraic geometry, analysis, linear algebra, number theory, and topology in a big melting pot of ideas originating in homotopy theory. It used to be true that there are only two types of K-theories: algebraic and topological. Nowadays the subject has matured into a widely applicable theory with a flurry of activities. In algebraic geometry, K-theory can naively be thought of as an infinite sequence of invariants playing an important role in algebraic cycle theories, which are devilishly difficult to compute whenever they differ from their topological analogues. Rosenschon will detail basic connections between cohomology theories (based on algebraic cycles), motives, and algebraic K-theory (based on vector bundles). Motives are part of a large-scale abstract algebraic geometry program initiated by Alexander Grothendieck. The consistency of a theory of motives still requires long-standing conjectures to be proven.

1. Algebraic cycles and K_0

2. Motivic cohomology and higher K-theory

3. Introduction to Motives

References:

1. Bloch, S.: Lectures on algebraic cycles, Duke University Press, 1980.

2. Bloch, S.: Algebraic cycles and higher K-theory, Adv. Math. 61, 1986, 267-304.

3. Murre, J.: Lecture on Motives, in: Transcendental Aspects of algebraic cycles, London Math. Soc. Lect. Notes Ser. 313, 123-170, Cambridge University Press, 2004.

1. Torsion algebraic cycles and algebraic K-theory

2. Regulator maps and the Bloch-Beilinson conjectures

References:

1. Colliot-Thelene, J.: Cycles algebriques de torsion et K-theorie algebrique, in: Arithmetic Algebraic Geometry (Trento 1991), 1-49, Springer Lecture Notes 1553, 1993.

2. Ramakrishnan, D.: Regulators, algebraic cycles, and values of L-functions, in: Algebraic K-theory and algebraic number theory (Honululu 1987), Contemp. Math, 183-310, 1989.

Support for accommodation and travel expenses will be available only for a limited number of participants from out of town. Please contact paularne at math.uio.no if you wish to attend some of the lectures.

Paul Arne Østvær, February 9, 2006.