Michael Donovan

wdono@wath.wit.edu (invert the three w's...)

E17-301H

I am a final year graduate student studying algebraic topology at MIT. My CV is here. My advisor is Haynes Miller. My principal research goal is to understand the homotopy theory of simplicial commutative algebras. My thesis will focus on tools and techniques to analyse the Adams spectral sequence in this homotopy theory.

Preprints available:

*Connected simplicial algebras are André-Quillen complete*

We prove that connected simplicial commutative algebras over a field are complete with respect to André-Quillen homology.

*Operations in the Adams spectral sequence for a simplicial commutative algebra*

We construct products, Steenrod operations, and higher divided squares in the Adams spectral sequence for a simplicial commutative algebra over F_{2}, drawing on a shifted and dualized version of Dwyer's*Higher divider squares in second-quadrant spectral sequences*.

*The homology of unstable Lie coalgebras*

We construct some tools for the calculation of the E^{2}page of the Adams spectral sequence for a simplicial commutative algebra over F_{2}. Namely, we construct a spectral sequence which calculates unstable Lie coalgebra homology and whose E^{2}page is given by further unstable Lie coalgebra homology. This spectral sequence may be applied iteratively, yielding bounds on the magnitude of the Adams E^{2}.

*The Adams spectral sequence for a simplicial commutative algebra sphere*

Combining the methods above, we expect to be able to calculate all of the differentials in the Adams spectral sequence for a (wedge of) simplicial commutative algebra sphere(s).