# Equivariant derived algebraic geometry

June 13 to June 17, 2016

at the

American Institute of Mathematics, San Jose, California

organized by

Andrew Blumberg, Teena Gerhardt, Michael Hill, and Kyle Ormsby

## Original Announcement

This workshop will explore computations and examples that will help guide the development of the fledgling field of ''equivariant derived algebraic geometry''. Although ideas that fit under this rubric have been around for a long time, recent work on the foundations of equivariant stable homotopy theory (starting with the Hill-Hopkins-Ravenel work on the Kervaire invariant one problem) and Lurie's development of the foundations of ''derived algebraic geometry'' now allows systematic exploration and organization. Motivating examples come from the work of Hopkins and his collaborators on algebraic geometry in algebraic topology. Since motivic homotopy theory has also grappled with understanding commutative ring spectra in algebraic geometry, sharing of examples and experience will be of great benefit.

A broad overarching goal is to explore when a moduli problem in algebraic geometry which has a solution in commutative ring spectra with a $G$-action has in fact a solution in genuine commutative ring $G$-spectra which have tractable slices. We hope to identify a number of such settings in which we will describe underlying computations, explore foundational consequences, and flesh out possible strategies for proof.

Concrete topics for the workshop include:

1. Studying examples arising from duality and line bundles for topological modular forms,
2. Computations with topological modular forms with level structure, such as the computation of the $RO(G)$-graded homotopy groups of $Tmf(\Gamma)$,
3. Determining the computational impact in the theory of topological modular forms of the extra structure that arises on equivariant commutative rings,
4. Determining and computing the obstruction groups for passage from naive $G$-equivariant commutative ring spectra to genuine commutative ring spectra,
5. Exploring the connections with computations in motivic homotopy theory.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

Primes and fields in stable motivic homotopy theory
by  Jeremiah Heller and Kyle Ormsby,  Geom. Topol. 22 (2018), no. 4, 2187–2218  MR3784519