Hecke algebras in the mod p Langlands program

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The Langlands program, initiated in the 1960s, is a set of conjectures predicting a unification of number theory and of representation theory of groups. The numerous developments of this program under its classical form in the last 30 years have had remarkable consequences such as the proof of Fermat's last theorem, and of Serre's modularity conjecture. Around 2000, the question of a p-adic/mod p version of these conjectures was raised, motivated by natural questions of p-adic arithmetic geometry. Because of unexpected and poorly understood phenomena involving groups other than GL_2(Q_p), statements of a general p-adic/mod p local Langlands conjecture remain elusive. For GL_2(Q_p) however, a correspondence has been established by P. Colmez and V. Paskunas (and based on the work of many others) with spectacular consequences such as the proof of the Fontaine-Mazur conjecture (by M. Kisin, M. Emerton). Progress remains to be made for more general groups. The general principle that governs the Langlands conjectures is that the correspondences should have a natural, geometric realization.  This proposal suggests an approach to the mod p Langlands program via the representation theory of (derived) Hecke algebras. This approach, which is amenable to geometrization, has already proved enlightening in previous work of the applicant. We will study (joint with P. Schneider) a certain "derived" pro-p Iwahori Hecke algebra and its connections to the derived category of smooth mod p representations of a given p-adic reductive group. We will investigate a derived version of the inverse mod p Satake isomorphism and the possible interpretations of our results in terms of coherent sheaves on the affine flag variety. Ultimately, the goal will be to relate this side of the mod p Langlands correspondence to object of Galois nature such as the ones studied in the work of Colmez, Grosse-Klönne, Schneider-Vignéras. The proposal has several components that are suitable for HQP who will be introduced to a subject with ramifications in several very dynamic areas of number theory and representation theory.