Related papers: Homological Methods in Equations of Mathematical P…
The methods of abstract simplicial homology and cohomology are reviewed and applied to the topology of electrical networks. Kirchhoffs laws of electrical circuits are shown to be manifestly homological in their origins. Since they are based…
These are lecture notes from the IMPANGA 2010 Summer School. The lectures survey some of the main features of equivariant cohomology at an introductory level. The first part is an overview, including basic definitions and examples. In the…
A rather simple natural outer derivation of the graded Lie algebra of all vector valued differential forms with the Fr\"olicher-Nijenhuis bracket turns out to be a differential and gives rise to a cohomology of the manifold, which is…
This set of lecture notes gives an introduction to holomorphic function spaces as used in mathematical physics. The emphasis is on the Segal-Bargmann space and the canonical commutation relations. Later sections describe more advanced…
In this paper, we introduce a representation theory of Hom-Lie conformal superalgebras and discuss the cases of adjoint representations. Furthermore, we develop cohomology theory of Hom-Lie conformal superalgebras and discuss some…
In this review paper, we consider three kinds of systems of differential equations, which are relevant in physics, control theory and other applications in engineering and applied mathematics; namely: Hamilton equations, singular…
One of the aims of this paper is to better explain the philosophy behind the computations in [E.Bifet, C.De Concini,C.Procesi Cohomology of Regular Embeddings ] and to place them in a wider conceptual setting. Another aim of the paper is to…
The purpose of this paper is to study global deformations of Hom-Leibniz algebras. We introduce a cohomology for Hom-Leibniz algebras with values in a Hom-module, characterize versal deformations and provide examples.
We study some classes of semi-linear differential equations including both well-posed and ill-posed cases that can generate cocycles (or cocycle correspondences with generating cocycles). Under exponential dichotomy condition with other…
We use differential cohomology to systematically construct a large class of topological actions in physics, including Chern-Simons terms, Wess-Zumino-Novikov-Witten terms, and theta terms (continuous or discrete). We introduce a notion of…
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…
The paper is devoted to several questions related to the notion of Cohomological Hall algebra (COHA for short) introduced few years ago by Maxim Kontsevich and the author. In particular we discuss a class of representations of COHA in the…
These notes partly touch the topic of the talk given by the author at the XXXVIII Workshop on Geometric Methods in Physics, hold in June-July 2019 in Bia\l{}owie\.{z}a, Poland. They consist of a short and self-contained introduction to the…
The dimensions of the graded quotients of the cohomology of a plane curve complement with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed…
We introduce a notion of oriented dialgebra and develop a cohomology theory for oriented dialgebras based on the possibility to mix the standard chain complexes computing group cohomology and associative dialgebra cohomology. We also…
These lecture notes contain an exposition of basic ideas of K-theory and cyclic cohomology. I begin with a list of examples of various situations in which the K-functor of Grothendieck appears naturally, including the rudiments of the…
These expository notes are dedicated to the study of the topology of configuration spaces of manifolds. We give detailed computations of many invariants, including the fundamental group of the configuration spaces of $\mathbb{R}^2$, the…
This is an exposition of some recent developments related to the object in the title, particularly the combinatorial computation of the (genus 0) Gromov-Witten invariants of the flag manifold and the quadratic algebra approach. The notes…
A survey of some results and open questions related to the following algebraic invariants of compact complex manifolds, that can be obtained from differential forms: cohomology groups, Chern classes, rational homotopy groups, and higher…
In this text we expose basic cases of some fundamental ideas and methods of topology. Namely, of homotopy, degree, fundamental group, covering, Whitehead invariant, etc. This is done by considering the elementary example: closed polygonal…