In Haynes Miller's proof of the Sullivan conjecture on maps from classifying spaces, Quillen's derived functor notion of homology (in the case of commutative algebras) is a critical ingredient. This suggests that homology for the larger class of algebraic structures parametrized by an operad O will also provide interesting and useful invariants. Working in the context of symmetric spectra, we prove a Whitehead theorem for topological Quillen homology of algebras and modules over operads. This is part of a larger goal to attack the problem: how much of an O-algebra can be recovered from its topological Quillen homology? We also prove analogous results for algebras and modules over operads in unbounded chain complexes. This talk is an introduction to these results (joint with K. Hess) with an emphasis on several of the motivating ideas.

The Theory of Real Submanifolds in a Complex Space (which is sometimes called, in some more general settings, "CR-geometry") goes back to H.Poincare and was deeply developed in further works of E.Cartan, N.Tanaka, S.Chern and J.Moser. In the present series of lectures we consider the classical aspects of this theory, as well as some recent results, focusing mainly on the holomorphic equivalence problem, groups of holomorphic symmetries and the holomorphic extension problem for real submanifolds in a complex space.

We will define operads, algebras over operads, modules over operads, and give various examples.

I will discuss how stability conditions and well adapted autoequivalences can be used to understand geometric information in derived categories. Following this discussion, I will provide an example of the usefulness of these techniques. In particular, I will show how to classify all compactifications of stable bundles on a class of genus 0 singular reducible curves.