## Derived structures in the Langlands programme

### London Number Theory Study Group, Spring 2019

This term's study group is intended to introduce the proposals of Akshay Venkatesh and his collaborators of certain derived structures appearing in the Langlands correspondence. This especially concerns the derived Hecke algebra set up in reference (1), the derived Galois deformation ring set up in reference (2), and how their relationship reflects a free action of a free exterior algebra of a certain motivic cohomology group on the cohomology of arithmetic groups that was conjectured in (1).

The group will consist of two somewhat parallel tracks, which we will call

- "Galois" track, focusing especially on reference (2). This is held 12:00-13:30.
- "Hecke" track, focusing especially on reference (1), with some exposition of (3), (4), and (5) as well. This is held 14:00-15:30.

The goal is to give a thorough introduction to these topics. A pair of initial lectures will be followed by extensive review, providing both motivation and background.
#### The Program

The program for the term is available here. [pdf]

Please contact one of the organisers if you are interested in giving a talk.

#### Primary References

The main references we will consult are the following works.

- Venkatesh: Derived Hecke algebra and cohomology of arithmetic groups
[arXiv]
- Galatius and Venkatesh: Derived Galois deformation rings
[arXiv]
- Prasanna and Venkatesh: Automorphic cohomology, motivic cohomology, and the adjoint L-function
[arXiv]
- Hansen and Thorne: On the GL
_{n}-eigenvariety and a conjecture of Venkatesh
[pdf]
- Harris and Venkatesh: Derived Hecke algebra for weight one forms
[arXiv]
- Venkatesh: Cohomology of arithmetic groups and periods of automorphic forms (expository account)
[pdf]

#### Notes

Some notes from the study group are being posted on Ashwin Iyengar's webpage, here.

#### Location and Time

The study group meets on Wednesday afternoon at 12:00 at UCL, usually in room 500 or 505 of the maths department. For updates about the room, stay tuned to the mailing list.

#### Organizers

The organizers are Pol van Hoften (KCL), Ashwin Iyengar (KCL), Alice Pozzi (UCL), and Carl Wang-Erickson (ICL). Pol and Ashwin are the main contacts for the Galois track, and Alice and Carl are the main contacts for the Hecke track.

Back to the study group landing page.

Back to Carl's main webpage.