The cotangent bundle T X to a complex manifold X is classically endowed with the sheaf of k-algebras WT∗X of deformation quantization, where k := W{pt} is a subfield of C[[~, ~]. Here, we construct a new sheaf of k-algebras W TX which contains WT∗X as a subalgebra and an extra central parameter t. We give the symbol calculus for this algebra and prove that quantized symplectic transformations o...

We show that intersection homology extends Poincaré duality to manifold homotopically stratified spaces (satisfying mild restrictions). These spaces were introduced by Quinn to provide “a setting for the study of purely topological stratified phenomena, particularly group actions on manifolds.” The main proof techniques involve blending the global algebraic machinery of sheaf theory with local ...

We construct a sheaf-theoretic representation of quantum probabilistic structures, in terms of covering systems of Boolean measure algebras. These systems coordinatize quantum states by means of Boolean coefficients, interpreted as Boolean localization measures. The representation is based on the existence of a pair of adjoint functors between the category of presheaves of Boolean measure algeb...

We prove that every homomorphism O ζ → O ζ , with E and F Banach spaces and ζ ∈ C, is induced by a Hom(E,F )-valued holomorphic germ, provided that 1 ≤ m < ∞. A similar structure theorem is obtained for the homomorphisms of type O ζ → Sζ , where Sζ is a stalk of a coherent sheaf of positive mζ -depth. We later extend these results to sheaf homomorphisms, obtaining a condition on coherent sheave...

We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard’s conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of S5, S6, and S7 for involutions.

0.1. Background. Let X be a smooth, complete, geometrically connected curve over the finite field Fq. Denote by F the field of rational functions on X and by A the ring of adèles of F . The Langlands conjecture, recently proved by L. Lafforgue [Laf], establishes a correspondence between cuspidal automorphic forms on the group GLn(A) and irreducible, almost everywhere unramified, n–dimensional `...

This talk presents some new notions of the theory of complex algebraic curves which have appeared as algebraic tools in string theory (see [4] for more details). In a sense, we have materialized non-existed complex powers of invertible sheaves on algebraic curves introduced at the level of the Atiyah algebras of invertible sheaves by Beilinson and Schechtman [1]. The Atiyah algebra AL in the ca...

Let κ = Fq be a finite field of characteristic p > 0, and k be a fixed algebraic closure of κ. We fix a prime ` 6= p, and an isomorphism τ : Q` → C. Whenever we want to denote something (e.g. scheme, sheaf, morphism, etc.) defined over κ, we will put a subscript 0 (e.g. X0 is a scheme over κ, F0 is a Weil sheaf defined over X0, etc.), and when the subscript is dropped, this means the correspond...