2.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1...

If ρ is a selfdual representation of a group G on a vector space V over C, we will say that ρ is orthogonal, resp. symplectic, if G leaves a nondegenerate symmetric, resp. alternating, bilinear form B : V ×V → C invariant. If ρ is irreducible, exactly one of these possibilities will occur, and we may define a sign c(ρ) ∈ {±1}, taken to be +1, resp. −1, in the orthogonal, resp. symplectic, case....

The local Langlands correspondence for GL(n) of a non-Archimedean local field F parametrizes irreducible admissible representations of GL(n, F ) in terms of representations of the Weil-Deligne group WDF of F . The correspondence, whose existence for p-adic fields was proved in joint work of the author with R. Taylor, and then more simply by G. Henniart, is characterized by its preservation of s...

We present a gauge-theoretic interpretation of the “analytic” version geometric Langlands program, in which Hitchin Hamiltonians and Hecke operators are viewed as concrete acting on Hilbert space quantum states. The gauge theory ingredients required to understand this construction—such electric–magnetic duality between Wilson ’t Hooft line four-dimensional theory—are same ones that enter unders...

Hrushovski's suggestion, given in ["Groupoids, imaginaries and internal covers," Turkish Journal of Mathematics , 2012], to capture the structure 1-analysable covers a theory T using simplicial groupoids definable is realized here. The ideas Haykazyan Moosa, found ["Functoriality uniformity groupoid-cover correspondence," Annals Pure Applied Logic 2018] are used, extended, define an equivalence...