Abstract for a colloquium talk by Markus Rost

The Bloch-Kato conjecture states the bijectivity of the norm residue homomorphism. This homomorphism is a natural map from Milnor's K-ring to the Galois cohomology ring of a field. All approaches to this conjecture are based on the investigation of the so called "norm varieties". In some cases these varieties are very classical objects like Brauer-Severi varieties or quadrics. Brauer-Severi varieties were used in proof of the theorem of Merkurjev-Suslin (which settles the Bloch-Kato conjecture in the weight 2). Quadrics associated to Pfister forms play an important role for the Milnor conjecture (which is the special case of the Bloch-Kato conjecture at the prime 2). In general there are no obvious candidates for norm varieties, however in recent years there has been considerable progress based on Voevodsky's observation that certain characteristic numbers of norm varieties should be nontrivial. This way there appeared a fruitful link between Galois cohomology and cobordism theory.

In the talk we will explain what Milnor's K-theory is and recall basic facts about Galois cohomology. We will define the norm residue homomorphism and consider various examples of norm varieties.

This abstract has been written for talks at the Duke University Mathematics Department on Tuesday, March 21, 2000 (Video) and at the Topology seminar of CUNYMath at the The City University of New York on Wednesday, April 12, 2000.