Valentin Lychagin

Geometry of differential equations

We review geometric and algebraic methods of investigations of systems of partial differential equations. Classical and modern approaches are reported.

Invariants of pseudogroup actions: Homological methods and Finiteness theorem

We study the equivalence problem of submanifolds with respect to a transitive pseudogroup action. The corresponding differential invariants are determined via formal theory and lead to the notions of l-variants and l-covariants, even in the case of non-integrable pseudogroup. Their calculation is based on the cohomological machinery: we introduce a complex for covariants, define their cohomology and prove the finiteness theorem. This implies the well-known Lie–Tresse theorem about differential invariants. We also generalize this theorem to the case of pseudogroup action on differential equations.

Mayer brackets and solvability of PDEs -- II

A solvability of a system of partial differential equations with two variables is investigated. The smooth solvability conditions are formulated in terms of (generalized) Mayer brackets.  2002 Elsevier Science B.V. All rights reserved.

On the Blaschke Conjecture for 3-Webs

We find relative differential invariants of orders eight and nine for a planar nonparallelizable 3-web such that their vanishing is necessary and sufficient for a 3-web to be linearizable. This resolves the Blaschke conjecture for 3-webs. We also give the algorithm for determining whether a given 3-web is linearizable, find the linearity condition for 3-webs and establish its relationships to the condition that a plane curve consists of flexes and to the Euler equation in gas-dynamics.

Mayer brackets and solvability of PDEs—I

Abstract A solvability of a system of partial differential equations with two variables is investigated. The smooth solvability conditions are formulated in terms of (generalized) Mayer brackets.

Colour calculus and colour quantizations

A colour calculus linked with an any discrete groupG is developed. Colour differential operators and colour jets are introduced. Algebras colour differential forms and de Rham complexes are constructed. For colour differential equations, Spencer complexes are constructed. Relations between colour commutative algebras and quantizations of usual algebras are considered.

Symmetries of Distributions and Quadrature of Ordinary Differential Equations

We present a geometric exposition of S. Lies and E. Cartans theory of explicit integration of finite-type (in particular, ordinary) differential equations. Numerous examples of how this theory works are given. In one of these, we propose a method of hunting for particular solutions of partial differential equations via symmetry preserving overdetermination.

Calculus and Quantizations Over Hopf Algebras

In this paper we outline an approach to calculus over quasitriangular Hopf algebras. We construct braided differential operators and introduce a general notion of quantizations in monoidal categories. We discuss some applications to quantizations of differential operators.

Differential Equations - Geometry, Symmetries and Integrability

This article reviews some recent theoretical results about the structure of Darboux integrable differential systems and their relationship with symmetry reduction of exterior differential systems. The symmetry reduction representation of Darboux integrable equations is then used to derive some new and unusual transformations.

Invariants in relativity theory

In the relativistic theory the principle of naturality of physical laws leads to differential equations admitting diffeomorphisms of space-time as symmetries.In this paper we investigate differential invariants of actions of diffeomorphisms on (pseudo)- Riemann metrics, on solutions of the Einstein equation, and on solutions of the Einstein—Maxwell equation. We find the corresponding fields of rational differential invariants and the corresponding factor-equations.