Related papers: Computing Adem Cohomology Operations
We compute the dimensions of the Hochschild cohomology of symmetric groups over prime fields in low degrees. This involves us in studying some partition identities and generating functions of the dimensions in any fixed degree of the…
Given a mechanical system $(M, \mathcal{F}(M))$, where $M$ is a Poisson manifold and $\mathcal{F}(M)$ the algebra of regular functions on $M$, it is important to be able to quantize it, in order to obtain more precise results than through…
We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These…
We describe the geometrical ladder of equations for Abelian bundles and gerbes, as well as higher generalisations, in terms of the cohomology of an operator that combines de Rham and Cech cohomology.
We establish an explicit isomorphism between the associated graded of the filtered chiral operad and the classical operad, which is useful for computing the cohomology of vertex algebras.
Soft sets, as a mathematical tool for dealing with uncertainty, have recently gained considerable attention, including some successful applications in information processing, decision, demand analysis, and forecasting. To construct new soft…
We study the cohomology of the group of rational points of a linear algebraic group on adelic functional spaces, and we explore the relation with Emerton's completed cohomology.
We introduce all six operations for D-cap-modules on smooth rigid analytic spaces by considering the derived category of complete bornological D-cap-modules. We then focus on a full subcategory which should be thought of as consisting of…
We enumerate complex algebraic hypersurfaces in $P^n$, of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equi-singular strata in the…
In this note, working in the context of simplicial sets, we give a detailed study of the complexity for computing chain level Steenrod squares, in terms of the number of face operators required. This analysis is based on the combinatorial…
We extend the notion of simplicial set with effective homology to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X \colon \mathcal{I} \to \mathsf{sSet}$ such that each simplicial set $X(i)$ has…
Topological data analysis has emerged as a powerful tool for analyzing large-scale data. An abstract simplicial complex, in principle, can be built from data points, and by using tools from homology, topological features could be…
Quantum computations operate in the quantum world. For their results to be useful in any way, there is an intrinsic necessity of cooperation and communication controlled by the classical world. As a consequence, full formal descriptions of…
We present a fast algorithm for computing discrete cubical homology of graphs over finite fields with an appropriate characteristic. This algorithm improves on several computational steps compared to constructions in the existing…
We algorithmically compute the intersection cohomology Betti numbers of any complete normal algebraic variety with a torus action of complexity one.
The aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples.
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and…
Operations on the cohomology of spaces are important tools enhancing the descriptive power of this computable invariant. For cohomology with mod 2 coefficients, Steenrod squares are the most significant of these operations. Their effective…
The central problem in computational algebraic topology is the computation of the homotopy groups of a given space, represented as a simplicial set. Algorithms have been found which achieve this, but the running times depend on the size of…