# TTT113: Online

Hosted by Sheffield

Talks were online, using Google Meet. We used Gather.town for discussions during the break and after the talks.

## Date

15 December 2020

## Speakers

### Jeffrey Carlson (Imperial)

Multiplicative collapse of the Eilenberg–Moore spectral sequence

Multiplicative collapse of the Eilenberg–Moore spectral sequence

In 1960s and 1970s there was a flurry of activity developing A-infinity-algebraic techniques with an aim toward computing the Eilenberg–Moore spectral sequence of a homotopy pullback (for example, a loop space or homogeneous space). Arguably the most powerful result this program produced was the theorem of Munkholm that the sequence collapses when the three input spaces have polynomial cohomology over the chosen coefficient ring, which gives the whole story on cohomology groups but sheds little light on the ring structure.

In this talk I will review this history and extend Munkholm's theorem to a ring isomorphism. The proof hinges on properties of the model categories of differential graded algebras and coalgebras and the commutativity of some large but hopefully photogenic diagrams.

### Ai Guan (Lancaster)

Nonconilpotent Koszul duality

Nonconilpotent Koszul duality

Koszul duality is a phenomenon appearing in many areas of mathematics, such as rational homotopy theory, deformation theory and representation theory. For differential graded (dg) algebras, it is often formulated as a Quillen equivalence between model categories of augmented dg algebras and conilpotent dg coalgebras, and their corresponding dg modules and comodules.

In this talk we consider what happens when the conilpotence condition is removed; the result is an exotic model structure on dg modules that is 'of second kind', ie the weak equivalences are subtler than quasi-isomorphisms. This is joint work with Andrey Lazarev.

### Luca Pol (Regensburg)

Torsion model for tensor-triangulated categories

Torsion model for tensor-triangulated categories

In this talk I will discuss how one can construct a model for (sufficiently well-behaved) tensor triangulated categories built from the data of local and torsion objects. The idea is to mirror constructions in commutative algebra such as torsion and localisation at prime ideals. I will then discuss two interesting examples arising from algebra and equivariant stable homotopy theory.

This is joint work with S Balchin, J P C Greenlees and J Williamson.