Abstract. We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an Eilenberg-Mac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasiisomorphic dgas are topologically equivalent, but we produce explicit counterexamples of the converse. We also develop an associated notion of topological Morita equivalence using a homotopical version of tilting. Contents

Abstract We give a functorial construction of k-invariants for ring spectra, and use these to classify extensions in the Postnikov tower of a ring spectrum. AMS Classification 55P43; 55S45

There are many interesting situations in which algebraic structure can be described

Abstract. We establish model category structures on algebras and modules over operads in symmetric spectra, and study when a morphism of operads induces a Quillen equivalence between corresponding categories of algebras (resp. modules) over operads. 1.

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We define a plus-construction on connective augmented algebras over operads in symmetric spectra using Quillen homology. For associative and commutative algebras, we show that this plus-construction is related to both Bousfield localization and Carlsson’s derived completion. 1

Abstract. This is a historical talk about the recent confluence of two lines of research in equivariant elliptic cohomology, one concerned with connected Lie groups, the other with the finite case. These themes come together in (what seems to me remarkable) work of N. Ganter, relating replicability of McKay-Thompson series to the theory of exponential cohomology operations.

Given a diagram of Π–algebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an obstruction theory, of somewhat wider application, formulated in terms of generalized Π– algebras. This extends a program begun by Dwyer, Kan, Stover, Blanc and Goerss [21, 10] to study the realization of a single Π–algebra. In particular, we explicitly analyze the simple case of a single map, and provide a detailed example, illustrating the connections to higher homotopy operations. 18G55; 55Q05, 55P65 1

We give a functorial construction of k–invariants for ring spectra and use these to classify extensions in the Postnikov tower of a ring spectrum. 55P43; 55S45 1