The Diverse World of PDEs : Algebraic and Cohomological Aspects / edited by I.S. Krasil'shchik, A.B. Sossinsky, A.M. Verbovetsky.

Format:

Book

Contributor:

Krasilʹshchik, I. S. (Iosif Semenovich), editor.

Sosinskiĭ, A. B. (Alekseĭ Bronislavovich), editor.

Verbovetsky, Alexander, editor.

Series:

Contemporary mathematics (American Mathematical Society) ; Volume 789.

Contemporary Mathematics Series ; Volume 789

Language:

English

Subjects (All):

Differential equations, Nonlinear--Congresses.

Differential equations, Nonlinear.

Differential equations, Partial--Congresses.

Differential equations, Partial.

Geometry, Differential--Congresses.

Geometry, Differential.

Homology theory--Congresses.

Homology theory.

Physical Description:

1 online resource (236 pages)

Edition:

First edition.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2023]

Summary:

This volume contains the proceedings of the Alexandre Vinogradov Memorial Conference on Diffieties, Cohomological Physics, and Other Animals, held from December 13-17, 2021, at Independent University of Moscow and Moscow State University, Moscow, Russia.The papers reflect the modern interplay between partial differential equations and various aspects of algebra and computer science. The topics discussed are: relations between integrability and differential rings, supermanifolds, differential calculus over graded algebras, noncommutative generalizations of PDEs, quantum vector fields, generalized Nijenhuis torsion, cohomological approach to the geometry of differential equations, the argument shift method, Frölicher structures in the formal Kadomtsev-Petviashvili hierarchy, and computer-based determination of optimal systems of Lie subalgebras.The companion volume (Contemporary Mathematics, Volume 788) is devoted to Geometry and Mathematical Physics.

Contents:

Cover

Title page

Contents

The Editors Preface

Automatic determination of optimal systems of Lie subalgebras: The package SymboLie

1. Introduction

2. Theoretical preliminaries and notation

3. The algorithm for optimal systems of Lie subalgebras

4. Case studies

5. Conclusions

Acknowledgments

References

New Dubrovin-type integrability theory applications of differential rings

2. Differential-algebraic problem setting

3. The Dubrovin differential rings and the integrability criterion

4. Invariant ideals, derivations and the related linear finite -dimensional endomorphic representations

5. Conclusion

6. Acknowledgments

Non-abelian Painlevé systems with generalized Okamoto integral

2. Painlevé-6 systems

3. Systems of \PV-\PII type

4. Tree of degenerations

Appendices

Appendix A. Lists of non-abelian systems of Painlevé type that have Okamoto integral

Appendix B. List of Hamiltonian non-abelian systems of Painlevé type

Appendix C. Special cases of \PIIIpr type systems

Mathematical etudes on quantum computation

1. Motivation and layout

2. Circuit model of an ideal quantum computer. Quantum universality.

3. Ideal quantum advantage.

4. Given the miraculous properties of ideal quantum computers, do these computers exist in reality?

5. Quantum error-correcting codes and anyons

6. Non-Abelian anyons

7. Concluding remarks

Quantum vector fields via quantum doubles and their applications

2. Quantum doubles and partial derivatives

3. Quantum left vector fields and the invariant operators

4. Different forms of the Capelli identity

5. Quantum adjoint vector fields and quantum orbits.

6. Quantum partial derivatives on ( _{ }) background

Non-linear homomorphisms of algebras of functions are induced by thick morphisms

2. Thick morphisms and non-linear functionals

3. Proof of the Theorems

4. Proofs of lemmas

Appendix A. Thick morphisms and _{∞} maps

Appendix B. Polarisation of functionals

On the Buchstaber-Rees theory of "Frobenius -homomorphisms" and its generalization

2. Main tool: characteristic function for a linear map of algebras

3. Application to the theory of Buchstaber-Rees

4. Our generalization of the Buchstaber-Rees theory

Differential calculus over graded commutative algebras and vector bundles with inner structures

Introduction

1. Conventions, Notation and Terminologies

2. Inner Structures in Projective Modules

3. An 'Ecosystem' of Graded Algebras and their Differential Calculus

4. Basic Functors of Differential Calculus in -olic Algebras

Vinogradov's cohomological geometry of partial differential equations

1. Geometry of PDEs

2. Cohomology of PDEs

3. Homotopy of PDEs

Frölicher structures, diffieties, and a formal KP hierarchy

2. Preliminaries on Frölicher spaces

3. Diffieties and their Frölicher structures

4. KP hierarchy on a diffiety

5. Outlook

Quasi-derivations on _{ } and the argument shift method

2. The universal enveloping algebra _{ }

3. Quasi-derivations of _{ }

4. The map Θ

5. Quasi-derivtions and the map Θ

6. Relations to the symmetric group

7. Relation with differential operators

References.

Polarization of generalized Nijenhuis torsions

2. Generalized Nijenhuis torsions

3. Polarization of generalized Nijenhuis torsions

Acknowledgment

Back Cover.

Notes:

Description based on print version record.

Includes bibliographical references.

Other Format:

Print version: Krasil′shchik, I. S. The Diverse World of PDEs

ISBN:

1-4704-7409-3