Related papers: Riemann-Roch for $\overline{\text{Spec}\mathbb Z}$
We elucidate the vector space (twisted relative cohomology) that is Poincar\'e dual to the vector space of Feynman integrals (twisted cohomology) in general spacetime dimension. The pairing between these spaces - an algebraic invariant…
We introduce a new example of unital commutative $n$-dimensional group algebra $\mathbb{R}_n$ for $n \geq 2$. The algebra $\mathbb{R}_n$ and the complex numbers $\mathbb{C}$ are astonishingly alike. The zero divisor set of the algebra has…
In this article, with a new approach, which is not discussed in the literature yet, the limit of the Riemann zeta function or Euler-Riemann zeta function is approximately explored by applying Dirichlet's rearrangement theorem for absolutely…
Let $S$ be a finite dimensional noetherian scheme. For any proper morphism between smooth $S$-schemes, we prove a Riemann-Roch formula relating higher algebraic $K$-theory and motivic cohomology, thus with no projective hypothesis neither…
A cohomology theory, associated to a $n$-Lie algebra and a representation space of it, is introduced. It is observed that this cohomology theory is qualified to encode the generalized derivation extensions, and that it coincides, for $n=3$,…
In these lectures we first review the important properties of the Riemann $\zeta$-function that are necessary to understand the nature and importance of the Riemann hypothesis (RH). In particular this first part describes the analytic…
Given a holomorphic vector bundle $E:EX X$ over a compact K\"ahler manifold, one introduces twisted GW-invariants of $X$ replacing virtual fundamental cycles of moduli spaces of stable maps $f: \Sigma \to X$ by their cap-product with a…
In this article, we first establish the main tool - an integral formula for Riemannian manifolds with multiple boundary components (or without boundary). This formula generalizes Reilly's original formula from \cite{Re2} and the recent…
This work establishes the geometric component of Deligne's longstanding program on refined Grothendieck-Riemann-Roch formulas expressed through determinants of cohomology. The approach relies on a newly developed universal category of Chern…
Consider a non-archimedean valuation ring V (K its fraction field, in mixed characteristic): inspired by some views presented by Scholze, we introduce a new point of view on the non-archimedean analytic setting in terms of derived analytic…
We introduce the notion of 3-Hom-Lie-Rinehart algebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we consider extensions of a 3-Hom-Lie-Rinehart algebra and characterize the first…
Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincar\'e--Hopf theorem. Generalized Euler characteristics (additive topological…
We introduce a cohomology theory of grading-restricted vertex algebras. To construct the {\it correct} cohomologies, we consider linear maps from tensor powers of a grading-restricted vertex algebra to "rational functions valued in the…
Deformations of compact Riemann surfaces are considered using a \v{C}ech cohomology sliding overlaps approach. Cocycles are calculated for conformal cutting and regluing deformations at zeros of Abelian differentials. A second order…
The Nyman-Beurling criterion, equivalent to the Riemann hypothesis (RH), is an approximation problem in the space of square integrable functions on $(0,\infty)$, involving dilations of the fractional part function by factors…
A general structure theorem on higher order invariants is proven. For an arithmetic group, the structure of the corresponding Hecke module is determined. It is shown that the module does not contain any irreducible submodule. This explains…
By adopting the standard definition of diffeomorphisms for a Regge surface we give an exact expression of the Liouville action both for the sphere and the torus topology in the discretized case. The results are obtained in a general way by…
We introduce a notion of duality for a Lie-Rinehart algebra giving certain bilinear pairings in its cohomology generalizing the usual notions of Poincar\'e duality in Lie algebra cohomology and de Rham cohomology. We show that the duality…
Given the unital C$^*$-algebra $A$, the unitary orbit of the projector $p_0=\begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix}$ in the C$^*$-algebra $M_2(A)$ of $2\times 2$ matrices with coefficients in $A$ is called in this paper, the Riemann…
In this paper we define the coarse (co)homology of the complement of a subspace in a metric space, generalizing the coarse (co)homology of Roe. We give a model space which encodes coarse geometric structure of the complement. We also…