Abstracts for Workshop on "Unstable Homotopy Theory"

The synthetic analogue of the bar comonad controls the universal differentials in the bar spectral sequence of algebras over spectral operads. This can be viewed as a deformation of Koszul duality for such algebras. I will explain ongoing work with Burklund and Senger on identifying the universal differentials in the bar spectral sequence for spectral Lie algebras over $F_p$. This will also shed light on the mod p homology of labeled configuration spaces and $E_n$-Dyer-Lashof operations for Lubin-Tate theories via a theorem of Knudsen.

I will discuss various localizations of unstable homotopy theory arising from the chromatic perspective, namely the v${}_n$-periodic localizations and the localizations with respect to T(n)-homology. There are many ways to approach these through "algebraic models". For example, the v${}_n$-periodic localizations can be studied via the theory of spectral Lie algebras. Quite orthogonally, p-finite spaces can be studied through their cochain algebras with values in Morava E-theory.

I will explain how to approach the question of $E_2$-formality of differential graded algebras over a prime field via obstruction theory. In particular, $E_2$-algebras whose cohomology ring is a polynomial algebra on even degree classes are intrinsically formal. As a consequence we can prove $E_2$-formality of the classifying space of some compact Lie group or of Davis-Januszkiewicz spaces.

Let Fr be the category of finitely generated free groups, and D a stable infinity category. We consider polynomial functors from Fr to D. We show that there are equivalences between the following infinity categories

[Polynomial functors from Fr to D] = [Excisive functors from Top${}_*$ to D] = [Truncated right modules in D over the the commutative operad] = [Right modules in D over the Lie operad, whose Koszul dual is truncated]