Abstract: Melrose introduced the formalism of b-geometry as a tool for studying partial differential operators on a smooth manifold M that suffer a first order degeneracy along a given hypersurface Z. The b-tangent bundle is the vector bundle whose sections are smooth vector fields defined on all of M and tangent along Z. Many of the classical geometries admit "b-analogues" in which the b-tangent bundle fills the role of the usual tangent bundle (so one has symplectic b-geometry, Riemannian b-geometry, etc). Complex b-geometry was introduced by Mendoza. In this talk, we will discuss the "b-Newlander-Nirenberg theorem": every complex b-manifold is locally isomorphic to some standard model. This allows one to define complex b-manifolds in the way one might hope, in terms of appropriately defined "b-holomorphic charts". This is joint work with Tatyana Barron.

Abstract: I will discuss a new product structure on the two-sided bar constructions of singular cochains. This bar construction is used in the Eilenberg-Moore theorem to compute the cohomology of pull-backs of fibrations. The product structure is based on so-called homotopy Gerstenhaber operations on singular cochains, transforming the two-sided bar construction into an $A_\infty$-algebra. This type of algebra is associative up to a strong form of homotopy. The new product includes those previously defined by Baues, Gerstenhaber-Voronov, Kadeishvili-Saneblidze, and Carlson-Franz as special cases. Consequently, the multiplicative cohomology isomorphism from the Eilenberg-Moore theorem is elevated to a quasi-isomorphism of $A_\infty$-algebras.

Also, please make sure the event does not appear twice on the calendar. Thanks!