Quillen's derived functor notion of homology provides interesting and useful invariants in a variety of homotopical contexts, and includes as special cases (i) singular homology of spaces, (ii) homology of groups, and (iii) Andre-Quillen homology of commutative rings. Working in the topological context of symmetric spectra, we study topological Quillen homology of commutative ring spectra, E_n ring spectra, and more generally, algebras over any operad O in spectra. Using a QH-completion construction---analogous to the Bousfield-Kan R-completion of spaces---we prove under appropriate conditions (a) strong convergence of the associated homotopy spectral sequence, and (b) that connected O-algebras are QH-complete---thus recovering the O-algebra from its topological Quillen homology plus extra structure. A key problem in usefully describing this extra structure was solved recently using homotopical ideas in joint work with Kathryn Hess that describes a rigidification of the derived comonad that coacts on the object underlying topological Quillen homology, and plays the analogous role (in symmetric spectra) of the Koszul cooperad associated to a Koszul operad in chain complexes. This talk is an introduction to these results with an emphasis on proving (a) and (b) which is joint work with Michael Ching.

Real hypersurfaces in a complex space $\mathbb C^N, N \geq 2$, satisfying the Levi non-degeneracy condition, were very well studied in the famous works of Poincare, Cartan, Tanaka, Chern and Moser and in a large number of further papers. The Levi-degenerate case, which is trivial for $N=2$ (all Levi degenerate hypersurfaces in this case are essentially flat), turns out to be absolutely non-trivial for $N=3$. The reason is that a hypersurface in $\mathbb C^3$ can have Levi form of rank $1$ at a generic point, and, in this case, is neither Levi-flat nor Levi non-degenerate. If, in addition, it satisfies some non-degeneracy condition, guaranteeing that it can not be reduced to a product of a hypersurface in $\mathbb C^2$ and a complex line, the hypersurface is called 2-nondegenerate. 2-nondegenerate hypersurfaces in $\mathbb C^3$ were deeply studied in a series of papers by Ebenfelt, Beloshapka, Zaitsev, Merker, Fels and Kaup and many other authors, but a lot of essential questions, concerned with their holomorphic classification and symmetry groups, remained opened. In the present talk we demonstrate a new approach to the study of 2-nondegenerate hypersurfaces, based on the consideration of degenerate quadratic models. This new point of view enables us to give a complete solution for most of the above open questions.