## MAT 541 Spring 2006 Announcement

In the fall semester (MAT 540), homology theory was treated as
part of Geometry in terms of cell decompositions, the Steenrod
axioms --homotopy functor, exactness, excision (based on
transversality), and dimension-- and the geometric picture of a
cycle or closed object representing a homology class, e.g. a closed
oriented manifold with singularities in codimension at least two.

In the spring semester (MAT 541) there will be an independent
treatment of cohomology as part of Homotopy Theory and
Algebra, primarily as obstructions to building certain objects by
induction: maps into target spaces, homotopies between maps, cross
sections to projections of fibrations. Characteristc classes and
k-invariants are direct applications of these examples.

Similarly cohomology classes arise in making inductive constructions
in an algebraic setting: deforming the product in a commutative,
Lie, or associative algebra; constructing the BRST differential of
gauge theory in Physics; understanding the algebraic models of
rational homotopy theory based on differential forms. These latter
models are free graded commutative algebras with differential and
they can be viewed as strong homotopy versions of Lie algebras called
Lie-infinity algebras. Surprisingly, it will be clearer and easier
to discuss the deformations and their obstructions in the more
natural setting of commutative infinity, Lie infinity, and
associative infinity algebras.

If time permits we will construct, as an application to Analysis, an
infinity algebra structure on distributions and currents using
smoothing and these ideas.

Dennis Sullivan

January 11, 2006