## Derived Structures in the Langlands Program

In Winter 2019, I co-organized the London Number Theory Study Group with Pol van Hoften, Alice Pozzi, and Carl Wang-Erickson. The topic of the study group was the work of Akshay Venkatesh, in particular his work on derived structures in the Langlands program, specifically his joint work Derived Galois Deformation Rings with Soren Galatius and the paper Derived Hecke Algebras.

The study group followed two tracks: a Galois track, following the first paper, and a Hecke track, which followed the second. In the last few talks, the comparison between the derived Galois deformation ring and the spherical derived Hecke algebra was discussed.

Galois Track Hecke Track
Date Title Speaker Title Speaker
January 9 Introduction I James Newton Introduction II Carl Wang-Erickson
January 16 Deformations of representations of Galois groups Misja Steinmetz Introduction to the cohomology of arithmetic groups, I: comparison with automorphic forms Chris Williams
January 23 The Taylor-Wiles method Fred Diamond Introduction to the cohomology of arithmetic groups, II: Matsushima's formula Joaquin Rodrigues Jacinto
January 30 The obstructed Taylor–Wiles method of Calegari–Geraghty Toby Gee Derived Hecke algebra, I: Definition of Derived Hecke algebra Alice Pozzi
February 6 Homotopy background I: Simplicial objects Lorenzo la Porta Derived Hecke algebra, II: Satake isomorphism, Iwahori Hecke algebra Robert Kurinczuk
February 13 Homotopy background II: Model categories Chris Birkbeck Derived Hecke algebra in the Taylor-Wiles setting, I Andrew Graham
February 20 Homotopy background III: Simplicial commutative rings Raffael Singer Derived Hecke algebra in the Taylor-Wiles setting, II: Reciprocity law Dan Gulotta
February 27 Derived deformation theory in general Dougal Davis The conjecture and its complex realization Alex Torzewski
March 6 Derived Galois deformation problems Rebecca Bellovin The complex realization of the conjecture Eran Assaf
March 13 The derived deformation ring and patching Ashwin Iyengar The $p$-adic ($\ell \neq p$) realization of the conjecture Ana Caraiani
March 20 The derived deformation ring and the derived Hecke algebra Pol van Hoften The conjecture on weight one modular forms Alice Pozzi
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