Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in seminal work of Riemann Euler–Gauss function and has blossomed into an active field connections to many areas mathematics. In this paper, we construct Hecke eigensheaves whose eigenvalues irreducible systems, thus confirming a central conjecture geo...

1. Algebraic Numbers. Early Greeks quadratic irrationals, their existence, their nature: √ 2, √ 3, √ 5, . . . appear in Theatetus (one of Plato’s Socretic dialogues) before Plato simple surds, after Plato more elaborate theory. Found in the later books of Euclid’s elements, the 10 and 12. Combinations of simple surds, such as a √ 2 + b √ 3, a and b rational. Proof of linear independence, thus f...

My research at UCSD has focused on advancing our understanding of Khovanov’s homology theory for links and link cobordisms, first introduced in [10]. His theory is a “categorification” of the Jones polynomial V (L), so described because it associates to any link L a complex Kh(L) of bigraded modules whose graded Euler characteristic is V (L). Not only is the homotopy type of Kh(L) a link invari...

In this paper we study the theta correspondence for Unitary groups of the same size over local and global fields. This correspondence has been studied in many cases by several authors. We are able to unify and generalise all these known results in terms of two conjectures, one local and the other global. These conjectures are in terms of the parametrisation of irreducible admissible representat...

The Local Langlands Conjecture posits the existence of a bijection between certain classes of irreducible representations of a reductive group G, such as GLnpKq, and certain ndimensional representations of an additive extension of the Weil group WK (closely related to the Galois group). Here we focus solely on the case G “ GLn and K is a p-adic field (such as Qp). The Local Langlands conjecture...

We initiate the study of affine Deligne–Lusztig varieties with arbitrarily deep level structure for general reductive groups over local fields. prove that $${{\,\mathrm{GL}\,}}_n$$ and its inner forms, Lusztig’s semi-infinite construction is isomorphic to an variety at infinite level. their homology give geometric realizations Langlands Jacquet–Langlands correspondences in setting Weil paramete...

We compute the universal deformations of cuspidal representations π of GL2(F ) over Fl, where F is a local field of residue characteristic p and l is an odd prime different from p. When π is supercuspidal there is an irreducible, two dimensional representation ρ of GF that corresponds to π via the mod l local Langlands correspondence of [Vi2]; we show that there is a natural isomorphism between...

The local Langlands conjectures imply that to every generic supercuspidal irreducible representation of G2 over a p-adic field, one can associate a generic supercuspidal irreducible representation of either PGSp6 orPGL3. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations of G2 and other representations of PGSp6 and PGL3. This corresponden...

The general problem we discuss in this paper is how to prove stability properties for a linear combination of characters of irreductible discrete series of p-adic groups. Here we give ideas on how to reduce the case where the Langlands parameter is trivial on the wild ramification group to the case where this Langlands parameter factorizes through the Frobenius; we handle only the case of an od...