Deformation theory and cup products

Algebra Seminar
Monday, September 14, 2015 - 15:00
1 hour (actually 50 minutes)
Skiles 006
Brandeis University
We will introduce, through examples, the philosophy of Delignethat "in characteristic zero, a deformation problem is controlled by adifferential graded (or "dg-") Lie algebra." Focusing on the deformationtheory of representations of a group, we will give an extension of thisphilosophy to positive characteristic. This will be justified by thepresence of a dg-algebra controlling the deformations, and the fact thatthe cohomology of the dg-algebra has an A-infinity algebra structureexplicitly presenting the deformation problem. This structure can bethought of as "higher cup products" on group cohomology, extending theusual cup product and often computable as Massey products. We will writedown concrete, representation-theoretic questions that are answered bythese higher cup products. To conclude, we will show that the cup productstructure on Galois cohomology, which is the subject of e.g. the motivicBloch-Kato conjecture and its proofs, is enriched by these higher cupproducts, and that this enrichment reflects properties of the Galois group.Familiarity with dg-algebras and infinity-algebras will not be presumed.