Related papers: Quillen's Fundamental Spectral Sequences Revisited
This article establishes the algebraic covering theory of quandles. For every connected quandle we explicitly construct a universal covering, which in turn leads us to define the algebraic fundamental group as the automorphism group of the…
Generalizations and variations of the fundamental lemma by Willems et al. are an active topic of recent research. In this note, we explore and formalize the links between kernel regression and some known nonlinear extensions of the…
We discuss spectral properties of a charged quantum particle confined to a chain graph consisting of an infinite array of rings under influence of a magnetic field assuming a $\delta$-coupling at the points where the rings touch. We start…
It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has…
Barr--Beck cohomology, put into the framework of model categories by Quillen, provides a cohomology theory for any algebraic structure, for example Andr\'e--Quillen cohomology of commutative rings. Quillen cohomology has been studied…
We explain how a simple twisting of the notion of spectral triple allows to incorporate type III examples, such as those arising from the transverse geometry of codimension one foliations. Since the twisting of the commutators turns the…
We study the spectral properties of one-dimensional quantum wire with a single defect. We reveal the existence of the non-trivial topological structures in the spectral space of the system, which are behind the exotic quantum phenomena that…
Through the theory of Lie bi-algebroids and generalized complex structures, one could define a cohomology theory naturally associated to a holomorphic Poisson structure. It is known that it is the hypercohomology of a bi-complex such that…
We uncover the very rich graph topology of generic bounded non-Hermitian spectra, distinct from the topology of conventional band invariants and complex spectral winding. The graph configuration of complex spectra are characterized by the…
Fix an odd prime $p$. Let $X$ be a pointed space whose $p$-completed K-theory $\mathrm{KU}_p^*(X)$ is an exterior algebra on a finite number of odd generators; examples include odd spheres and many H-spaces. We give a…
Given a double complex $X$ there are spectral sequences with the $E_2$ terms being either H$_I$ (H$_{II}(X))$ or H$_{II}($H$_I (X))$. But if $H_I(X)=H_{II}(X)=0$ both spectral sequences have all their terms 0. This can happen even though…
The theory of abelian categories proved very useful, providing an axiomatic framework for homology and cohomology of modules over a ring and, in particular, of abelian groups. For many years, a similar categorical framework has been lacking…
The study of unconventional phases and elucidation of correspondences between topological invariants and their intriguing properties are pivotal in topological physics. Here, we investigate a complex exceptional ring (CER), composed of a…
We associate a Taylor tower supplied by calculus of the embedding functor to the space of long knots and study its cohomology spectral sequence. The combinatorics of the spectral sequence along the line of total degree zero leads to chord…
We describe the singular cohomology ring, the K-ring of complex vector bundles, the Chow ring, and the Grothendieck ring of coherent sheaves of the total space of the fibre bundle with base space an irreducible nonsingular complete…
We discuss the Adams Spectral Sequence for R-modules based on commutative localized regular quotient ring spectra over a commutative S-algebra R in the sense of Elmendorf, Kriz, Mandell, May and Strickland. The formulation of this spectral…
Kempe equivalence is a classical and fundamental notion in graph coloring theory. In the present paper we establish a connection between Kempe equivalence and quadratic stable set ring, which are toric rings associated to graphs. In fact,…
We apply our recently developed theory of frequency-filtered and time-resolved N-photon correlations to study the two-photon spectra of a variety of systems of increasing complexity: single mode emitters with two limiting statistics (one…
The aim of this article is to consider the spectral sequences induced by tensor-hom adjunction, and provide a number of new results. Let $R$ be a commutative Noetherian local ring of dimension $d$. In the 1st part, it is proved that $R$ is…
We develop some new aspects of cohomology in the context of semi-abelian categories: we establish a Hochschild-Serre 5-term exact sequence extending the classical one for groups and Lie algebras; we prove that an object is perfect if and…