The Bisognano-Wichmann property on the geometric behavior of the modular group of the von Neumann algebras of local observables associated to wedge regions in Quantum Field Theory is shown to provide an intrinsic suucient criterion for the existence of a covariant action of the (universal covering of) the Poincar e group. In particular this gives, together with our previous results, an intrinsi...

The Bisognano-Wichmann property on the geometric behavior of the modular group of the von Neumann algebras of local observables associated to wedge regions in Quantum Field Theory is shown to provide an intrinsic suucient criterion for the existence of a covariant action of the (universal covering of) the Poincar e group. In particular this gives, together with our previous results, an intrinsi...

Hecke operators play an important role in the theory of automorphic forms, and automorphic forms are closely linked to various cohomology groups. This paper is mostly a survey of Hecke operators acting on certain types of cohomology groups. The class of cohomology on which Hecke operators are introduced includes the group cohomology of discrete subgroups of a semisimple Lie group, the de Rham c...

The Schur polynomials s ? are essential in understanding the representation theory of general linear group. They also describe cohomology ring Grassmannians. For ?=(n,n-1,?,1) a staircase shape and ??? subpartition, Stembridge equality states that ?/? =s T . This provides information about symmetry ring. stable Grothendieck G , dual g developed by Buch, Lam, Pylyavskyy, variants K-theory Using ...

We develop homological techniques for finding explicit combinatorial formulas for finite-type cohomology classes of spaces of knots in R, n ≥ 3, generalizing the Polyak—Viro formulas [9] for invariants (i.e., 0-dimensional cohomology classes) of knots in R. As the first applications, we give such formulas for the (reduced mod 2) generalized Teiblum—Turchin cocycle of order 3 (which is the simpl...

The need to calculate local system cohomolgy of the complement of an hyperplane arrangement arises in various contexts. Nevertheless until now very few it’s known on the direct computation of such cohomology. In this paper the author describes all generators of the first homology group of braid arrangement giving also a complete description of its first characteristic variety, fulfilling result...

It is well known that the homotopy category of connected CW-complexes X whose homotopy groups n;(X) are trivial for i> 1 is equivalent to the category of groups. One of the objects of this paper is to prove a similar equivalence for the connected CW-complexes X whose homotopy groups are trivial for i > n + 1 (where n is a fixed non-negative integer). For n = 1 the notion of crossed module inven...

Lectures notes from a topics course on the definition of supersymmetric Euclidean field theories and their relation to generalized cohomology. Notre Dame. Spring 2009 Date: June 15, 2009.