University of Regina Topology Seminar

Starting from graded modules and chain complexes, we introduce the category of differential graded algebras (DGAs).

I will discuss the homology of a DGA as well as quasi-isomorphisms, which provide a notion of equivalence between two DGAs.

I will present a few examples of DGAs coming from algebra, topology, and geometry.

We discuss tensor algebras and symmetric algebras equipped with differentials, then we introduce Massey products and time permitting the endomorphism algebra of a chain complex.

This talk will present a classification theorem for a certain class of modules over A(1), a subalgebra of the mod-2 Steenrod algebra. In order to give the module classification, there will be some background on Margolis homology (an invariant of modules over the Steenrod algebra) and the stable module category. Applications of the classification theorem to lifting A(1)-modules to modules over the Steenrod algebra and applications to the computation of certain localized Adams spectral sequences will also be discussed.

We will look at the definitions of the bar and cobar constructions, some examples and their adjointness relation.

I will talk about the Tor functor and Eilenberg-Moore spectral sequences.

We describe a cofibrantly generated model structure on DG-modules which induces a model structure on DG-algebras as the category of monoids in DG-modules.

Quillen's work on rational homotopy theory illustrates the duality between differential graded Lie algebras and differential graded cocommutative coalgebras. In homotopy theory, there is an analogous duality phenomenon, called Koszul duality, where the role of differential graded algebras is played by structured ring spectra. In this talk, I will talk about Koszul duality as well as some of my related recent work on Topological Quillen (TQ) localization and the homological Whitehead theorem of structured ring spectra. This is joint with John E. Harper.

Two rings are called Morita equivalent if their categories of modules are equivalent. We will review Morita's classic theorem providing a criterion for when that happens. We will then consider the question: When do two rings have equivalent derived categories? Rickard's theorem provides an answer. Lastly, we will look at the derived category of a DG-algebra and tackle the question: When do two DG-algebras have equivalent derived categories?

We will develop background on symmetric spectra and the smash product of spectra. After introducing Eilenberg-MacLane (EM) spectra, the talk will culminate with a sketch of the proof of the stable Dold-kan correspondence between EM-module spectra and unbounded chain complexes.

We will see how to get a Quillen equivalence between simplicial algebras and connective DG-algebras from the functors in the Dold-Kan correspondence.