MAT540 and MAT541
 Intermediate algebraic topology

Intermediate Topology 540 and 541 is really a year long course which may be attended in consecutive years. It treats elementary phenomena in algebraic and geometric topology from an advanced standpoint. This year it will treat a common thread in Cartan-deRham homotopy theory and String topology, called Lie infinity algebras.    " The deRham theorem relating differential forms to the singular cohomology of a manifold is only the tip of the iceberg. From the  algebra of smooth forms with its exterior derivative operator one can define deRham homotopy groups, a graded Lie bracket and  other higher order tensors. When the manifold is simply connected the first two pieces are the usual homotopy groups tensor the reals with the induced Whitehead Lie bracket of homotopy  elements. The higher  degree tensors determine a real version of the postnikov system of the space which builds the homotopy type out of the atoms of homotopy theory. From this algebraic model one can model other features like the free loop space or the equivariant cohomology defined by the Borel construction.When the manifold has special structures like symplectic, complex or Kaehler geometries there are interesting questions and results about the form these higher order  deRham invariants take". The photograph shows George deRham and Armand Borel in Switzerland.