The local Langlands correspondence via endoscopy, geometry, type theory, and their interplay

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The aim of this project is to investigate the local Langlands correspondence for p-adic groups using the tools of type-theory, algebraic and rigid geometry, and spectral theory. The local Langands correspondence is a mysterious conjectural relationship between two kinds of symmetries -- a geometric one, offered by groups of matrices with coefficients in real, complex, or p-adic fields, and an arithmetic one, offered by the Galois group of the real or a p-adic field. The last decade has seen a tremendous progress in this area, but the theory is still largely conjectural. The PI and his collaborators will attempt to establish new cases of the conjectural correspondence through the use of type-theory, as well as to elucidate the connection between types and the cohomology of Rapoport-Zink spaces, in particular in the case of low-rank unitary groups. The already existing constructions will be investigated more deeply using the theory of endoscopy. Furthermore, the PI and his collaborators will study both the local and the global Langlands correspondences for unitary groups of arbitrary rank, using the trace-formula techniques developed by Arthur.The Langlands program brings together several areas of mathematics, including arithmetic, geometry, analysis, and symmetry. By establishing deep and surprising ties between them, it provides the means for answering very hard questions about the arithmetic of numbers, as exemplified by the recent proof of Fermat's Last Theorem. A sophisticated understanding of the arithmetic of numbers is in turn fundamental to many recent technological developments, including error-correcting codes, compression, encryption and secure communication. This project aims at deepening our understanding of the Langlands program by exploring connections between several approaches to it, as well as by exploiting techniques which have only recently become available.