There are two de Rham complexes in diffeology. The original one is due to Souriau and the other singular complex defined by a simplicial differential graded algebra. We compare first cohomology groups of within Čech–de spectral sequence making use factor map which connects complexes. As consequence, it follows that algebra irrational torus $T_\theta$ isomorphic tensor product exterior genera...

Introduction A locally compact, second countable group G is called Kazhdan if for any unitary representation of G, the first continuous cohomology group is trivial H 1 ct (G, ρ) = 0. There are several other equivalent definitions, the reader should consult [6], esp. 1.14 and 4.7. For some time now Kazhdan groups have attracted attention. One of the main challenges is to understand them geometri...

We study a positive characteristic analog of the nonabelian Hodge structure constructed by Katzarkov, Pantev, and Toen on the homotopy type of a complex algebraic variety. Given a proper smooth scheme X over a perfect field of characteristic p and a Tannakian category C of isocrystals on X, we construct an object XC in a suitable homotopy category of simplicial presheaves whose category of loca...

In this paper we review the definition of the monodromy of an angle valued map based on linear relations as proposed in [3]. This definition provides an alternative treatment of the Jordan cells, topological persistence invariants of a circle valued maps introduced in [2]. We give a new proof that homotopic angle valued maps have the same monodromy, hence the same Jordan cells, and we show that...

We introduce a cap product pairing for homology and cohomology of tropical cycles on integral affine manifolds with singularities. show the is perfect over $\mathbb{Q}$ in degree one when manifold has at worst symple By joint work Siebert, computes period integrals its perfectness implies versality canonical Calabi-Yau degenerations. also give an intersection theoretic application Strominger-Ya...

Currently, there is much interest in the development of geometric integrators, which retain analogues of geometric properties of an approximated system. This paper provides a means of ensuring that finite difference schemes accurately mirror global properties of approximated systems. To this end, we introduce a cohomology theory for lattice varieties, on which finite difference schemes and othe...

We establish a formula which decomposes the cohomologies of various sheaves on a simple normal crossings divisor (SNC) D in terms of the simplicial cohomologies of the dual complex ∆(D) with coefficients in a presheaf of vector spaces. This presheaf consists precisely of the corresponding cohomology data on the components of D and on their intersections. We use this formula to give a Hodge deco...