# 2017 Workshop

## Dualities in Algebra, Geometry, and Topology with John Greenlees and Mike Hill

Geestland, Lower Saxony, Germany

June 11th - June 17th, 2017

Participants at the 2017 workshop

### About

The third European Talbot workshop took place in Germany from June 11 to June 17, 2017. The goal was to bring together a group of 30-35 graduate students and post-docs to work on a focused topic under the guidance of two senior mentors.

Most of the talks were given by the participants, with enough free time in the afternoon and evenings for further discussions and interaction. The character of the workshop was expository in nature, starting with the basic ideas and leading to a survey of the most recent developments in the field. Since all participants stayed together at a group house, jointly responsible for cooking and cleaning, we hope to have created an informal and inspiring atmosphere.

### Topic

Algebraists, geometers and topologists have been fascinated by duality since the start of their subjects. As the infrastructure has developed, the congruence of these dualities has become steadily more apparent, to the extent that it is now possible to consider them all in a common formal homotopy invariant framework. The principal goal of the workshop is to understand various duality phenomena in algebra, geometry and topology from a homotopical point of view.

Perhaps the most familiar example is Poincare duality and its incarnations in all three areas. Going beyond things which are visibly finite, we see a number of deep phenomena (Benson--Carlson duality in representation theory, Gross--Hopkins duality in stable homotopy theory, ...) that can be viewed as Gorenstein duality in a homotopy invariant context. Similarly Serre and Grothendieck duality in algebraic geometry lead to Gorenstein duality for connective spectra and Anderson duality for various spectral counterparts of objects with Serre duality.

The three main themes of this workshop are

(a) Gorenstein duality in algebra

(b) Serre duality in geometry and

(c) group actions,

brought together in a homotopy invariant context. Throughout the week, the general theory will be developed in parallel to the discussion of illustrating examples from algebra, chromatic homotopy theory and modular forms. We thus hope to gather people with diverse backgrounds to explore an exciting modern framework for such duality phenomena.

### Syllabus

Here is a preliminary version of the syllabus.

### Schedule

**Monday: Classical Duality**

1. Introduction

2. Serre duality in algebraic geometry

3. Local cohomology

4. Gorenstein duality

**Tuesday: Algebraic Geometry**

5. Formal group laws, formal groups

6. Stacks

7. Elliptic curves, moduli of elliptic curves

8. Modular forms, j-invariant

**Wednesday: A motivating example & Spectra**

9. Duality in group cohomology

10. Spectra

(excursion)

**Thursday: Tmf Dualities**

11. Commutative ring spectra

12. Tmf I - statements and basic properties

13. Tmf II - unpacking and extensions to larger sites

14. Stojanoska's work on duality around Tmf

**Friday: Equivariant extension**

15. Unstable equivariant homotopy

16. Stable equivariant homotopy

17. Equivariant applications: tmf_{1}(3)

18. Equivariant applications: BP_{R}<n>** **

### Application

The application dealine has now passed.

### Funding

The local expenses for all participants will be covered by the workshop, in particular the costs for accommodation and meals.

Additional funds towards travel expenses might become available, but we strongly encourage participants to seek independent financial support.

### Contact

For questions or suggestions, please send an email to the organizers organisers@europeantalbot.org.

This workshop is organized by Bertram Arnold, Tobias Barthel, Magdalena Kedziorek, and Sean Tilson, and its concept has been inspired by the US Talbot workshop. We gratefully acknowledge funding and support from SPP 1786.