相关论文: A Geometric Approach to Differential Forms

We present two geometric interpretations for complex multivectors and determinants: a little known one in terms of square roots of volumes, and a new one which uses fractions of volumes and allows graphical representations. The fraction…

The Separation of Variables theory for the Hamilton-Jacobi equation is 'by definition' related to the use of special kinds of coordinates, for example Jacobi coordinates on the ellipsoid or St\"ackel systems in the Euclidean space. However,…

This paper aims to provide a careful and self-contained introduction to the theory of topological degree in Euclidean spaces. It is intended for people mostly interested in analysis and, in general, a heavy background in algebraic or…

The goal of this note is to describe a class of formal deformations of a symplectic manifold $M$ in the case when the base ring of the deformation problem involves parameters of non-positive degrees. The interesting feature of such…

These are expanded lecture notes of a mini-course whose objectives were to introduce the basic concepts, constructions and techniques of noncommutative geometry, as well as their uses as a framework for modelling quantum spacetime. Key…

The authors study the method of scaling in the context of the study of automorphism groups of complex domains in multiple dimensions. Various types of scaling techniques are compared and contrasted. Applications are given in a number of…

The purpose of this paper is to propose the implementation of some methods from algebraic geometry in the theory of gravitation, and more especially in the variational formalism. It has been assumed that the metric tensor depends on two…

These notes provide an introduction to the algebra and geometry of differential operators and jet bundles. Their point of view is guided by the leitmotiv that higher-spin gravity theories call for higher-order generalisations of Lie…

This is a collection of articles, written as sections, on arithmetic properties of differential equations, holomorphic foliations, Gauss-Manin connections and Hodge loci. Each section is independent from the others and it has its own…

This is the preliminary manuscript of a book on symplectic field theory based on a lecture course for PhD students given in 2015-16. It covers the essentials of the analytical theory of punctured pseudoholomorphic curves, taking the…

This article provides an exposition to the topic of formal moduli problems, emphasizing its connections with differential graded Lie algebras, and mainly following from Jacob Lurie's DAG X: Formal Moduli Problems. As such, this paper should…

In this paper we study differential forms and vector fields on the orbit space of a proper action of a Lie group on a smooth manifold, defining them as multilinear maps on the generators of infinitesimal diffeomorphisms, respectively. This…

The theorems of vector analysis (divergence theorem, etc.) are typically first applied in the undergraduate physics curriculum in the context of the electromagnetic field and the differential forms of Maxwell's equations. However, these…

Partial differential equations are fundamental tools in mathematics,sciences and engineering. This book is mainly an exposition of the various algebraic techniques of solving partial differential equations for exact solutions developed by…

This book offers an introduction to Hofer's metric on the group of Hamiltonian diffeomorphisms. It presents results on the diameter, geodesics, and the growth of one-parameter subgroups, along with applications to dynamics and ergodic…

We explicate a procedure to solve general linear differential equations, which connects the desired solutions to monomials x^m of an appropriate degree m. In the process the underlying symmetry of the equations under study, as well as that…

A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it…

These lecture notes provide an introduction to the theory and application of symmetry methods for ordinary differential equations, building on minimal prerequisites. Their primary purpose is to enable a quick and self-contained approach for…

These are lecture notes for the course "MATS4120 Geometry of geodesics" given at the University of Jyv\"askyl\"a in Spring 2020. Basic differential geometry or Riemannian geometry is useful background but is not strictly necessary. Exercise…

This is a written-up version of eight introductory lectures to the Hodge theory of projective manifolds. The table of contents should be self-explanatory. The only exception is section 8 where I discuss, in a simple example, a technique for…