The present paper is a short survey on the mathematical basics of Classical Field Theory including the Serre-Swan’ theorem, Clifford algebra bundles and spinor bundles over smooth Riemannian manifolds, Spin-structures, Dirac operators, exterior algebra bundles and Connes’ differential algebras in the commutative case, among other elements. We avoid the introduction of principal bundles and put ...

We establish a procedure for constructing compatible mixed Hodge structures for the cohomology of various topological compactifications of locally symmetric varieties, notably ones that are not algebraic varieties. This is carried out in full for the case of the reductive Borel-Serre compactification, and conditionally for the excentric compactifications. Introduction. In [BS], Borel and Serre ...

Serre’s reduction aims at reducing the number of unknowns and equations of a linear functional system (e.g., system of partial differential equations, system of differential time-delay equations, system of difference equations). Finding an equivalent representation of a linear functional system containing fewer equations and fewer unknowns generally simplifies the study of its structural proper...

We prove in this paper a noncommutative version of Leray Theorem and then Leray-Serre Spectral Theorem for noncommutative Serre fibrations: for NC Serre fibration there are converging spectral sequences with E2 terms as E2 p,q = HPp(A;HPq(B,A)) =⇒ HPp+q(B) and E 2 p,q = HPp(A;Kq(B,A)) =⇒ Kp+q(B). Introduction The ideas of using spectral sequences to operator algebras was started in [C]. In that...

Let R be an associative ring with non-zero identity. For a Serre subcategory C of the category R-mod of left R-modules, we consider the class AC of all modules that do not belong to C, but all of their proper submodules belong to C. Alongside of basic properties of such associated classes of modules, we will prove that every uniform module of AC has a local endomorphism ring. Moreover, if R is ...

1. Local coefficient systems 2 1.1. The fundamental groupoid 2 1.2. Coefficient systems 3 1.3. Path connected based spaces 4 1.4. Singular chains with local coefficients 4 1.5. Homology with local coefficients 5 1.6. Universal covers 7 2. Exact couples and spectral sequences 9 2.1. Spectral sequences 9 2.2. Exact couples 9 2.3. The spectral sequence associated to an exact couple 10 2.4. Example...

My memoirs of Paul Erdős will be almost all entirely personal, but I would like to begin with two small incidents illustrating attitudes of the mathematical community and the wider world to Paul. Both took place on trains. After I received my copy of the two-volume work The Mathematics of Paul Erdős [13] (published just after Paul’s death), I was reading it on the train home from work. Sitting ...

We prove standard results of group cohomology – namely, existence a long exact sequence, classification torsors via the first group, Shapiro’s lemma, Hochschild-Serre spectral decomposition cochain complex in direct product case, and Jannsen’s result on recovery problem for theories such as continuous, analytic, bounded, pro-analytic cohomology. also these certain monoids, applications we have ...

We prove in this paper a noncommutative version of Leray Theorem and then Serre Spectral Theorem for noncommutative Serre fibrations. We prove that for NC Serre fibration there are converging spectral sequences with E2 terms as E2 p,q = HPp(B,A;HPq(A)) =⇒ HPp+q(B) and E 2 p,q = HPp(B,A;Kq(A)) =⇒ Kp+q(B). Introduction The ideas of using spectral sequences to operator algebras was started in [C]....

In this note we prove the following special case of Serre’s conjecture on Intersection Multiplicity: Let (R,m) be a regular local ring and let P , Q be two prime ideals such that (R/(P + Q)) < ∞, dimR/P + dimR/Q = dimR and dimension of Gm(R/P ) ⊗Gm(R) Gm(R/Q) < 2. Then χ(R/P, R/Q) ≥ em(R/P )em(R/Q); here em(T ) denotes the Hilbert-Samuel multiplicity for any finitely generated module T with res...