(This paper is an extended abstract for an invited talk at 29th Linz Seminar on Fuzzy Set Theory, February 2008.) Höhle has shown that fuzzy sets, valued in a frame (complete Heyting algebra) Ω, can be identified with certain sheaves over Ω: they are the subsheaves of constant sheaves, and more general sheaves can be got as quotients of the fuzzy sets. The technical development is complicated b...

‎We consider Albeverio's linear representations of the braid groups $B_3$ and $B_4$‎. ‎We specialize the indeterminates used in defining these representations to non zero complex numbers‎. ‎We then consider the tensor products of the representations of $B_3$ and the tensor products of those of $B_4$‎. ‎We then determine necessary and sufficient conditions that guarantee the irreducibility of th...

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 both permutations and involut...

Part I gives algebraic basic notions necessary to generating a graded sheaf of rings from a Galois extension, i.e. essentially a specialization, called emergent, from a ring of polynomials A[x1, ..., xm] giving rise to a set of compact connected algebraic subgroups which correspond to the different sections of the sheaf of rings θ. Part II refers to the introduction of the Eisenstein homology b...

We define the notions of micro-support and regularity for indsheaves, and prove their invariance by contact transformations. We apply the results to the ind-sheaves of temperate holomorphic solutions of D-modules. We prove that the micro-support of such an ind-sheaf is the characteristic variety of the corresponding D-module and that the ind-sheaf is regular if the D-module is regular holonomic.

Let X be a projective scheme and let M and N be two coherent OX -modules. Given an integer m, we present an algorithm for computing the global extension module ExtX (M,N ). As a consequence, one may calculate the sheaf cohomology Hm(X,N ) and construct the sheaf corresponding to an element of the module ExtX(M,N ). This algorithm depends only on the computation of Gröbner bases and syzygies and...

We reformulate Hecke’s open problem of 1923, regarding the Fourier-analytic proof of higher reciprocity laws, as a theorem about morphisms involving stratified topological spaces. We achieve this by placing Kubota’s formulations of n-Hilbert reciprocity in a new topological context, suited to the introduction of derived categories of sheaf complexes. Subsequently, we begin to investigate condit...

This is an introduction to the notion of local subgroupoid introduced by the author and R. Brown. It can also serve as an introduction to an application of sheaf theory, and so could be useful to beginners in that theory. The main results are the construction of the holonomy groupoid and the notion of s-sheaf for certain local subgroupoids s. 2000 Mathematics Subject Classification: 18F20, 18F0...

In this paper, we establish results of a basic nature for the the K-theory and G-theory of algebraic stacks, i.e. Artin stacks. At the same time, we enlarge the framework a bit more so that these results not only hold for stacks, but also for what are called dg-stacks, i.e. algebraic stacks where the usual structure sheaf is replaced by a sheaf of dgas.

We study the arithmetic self-intersection number of the dualizing sheaf on arithmetic surfaces with respect to morphisms of a particular kind. We obtain upper bounds for the arithmetic self-intersection number of the dualizing sheaf on minimal regular models of the modular curves associated with congruence subgroups Γ0(N) with square free level, as well as for the modular curves X(N) and the Fe...