Abstracts for Abstract Homotopy Theory Seminar

he classical Freudenthal Suspension Theorem is the statement that, given a space X, the connectivity of the unit map X -> ΩΣX of the loop-suspension adjunction is (roughly) twice the connectivity of X. In this talk, I will discuss the setting of oriented (or directed) homotopy theory, in which spaces are replaced by higher, i.e. (∞,ω)-, categories and the cartesian product by the Gray tensor product, and generalize the Freudenthal Suspension Theorem in this context. This gives, in particular, evidence for a yet-elusive "oriented topos theory".

The notion of a differentiable stack in geometry can be used to study objects such as orbifolds, moduli spaces or classifying spaces, which despite not being manifolds carry similar differential geometric information. Examples in particular include quotients of manifolds by a Lie group action. These objects are closely related to equivariant cohomology, a cohomology theory on such manifolds with group actions that has many applications in different areas of mathematics and physics. It can be seen as a special case of the cohomology of differentiable stacks.

It is well known that Hochschild cohomology and, in particular, the algebraic structure it carries, plays a central role in studying the infinitesimal noncommutative deformations of geometric spaces. We construct a canonical solution to Deligne’s conjecture for Hochschild cochains on a scheme, even for the singular case, by exhibiting the Hochschild complex as an ∞-operadic center.