Overview
- Presents Gröbner bases and quiver theories as providers of computing models for differential equations and systems
- Offers a historical background for a better understanding of how theories developed
- Appeals to a wide readership, from graduate students to researchers and scholars
Part of the book series: Algorithms and Computation in Mathematics (AACIM, volume 28)
Access this book
Tax calculation will be finalised at checkout
Other ways to access
About this book
This edited volume presents a fascinating collection of lecture notes focusing on differential equations from two viewpoints: formal calculus (through the theory of Gröbner bases) and geometry (via quiver theory). Gröbner bases serve as effective models for computation in algebras of various types. Although the theory of Gröbner bases was developed in the second half of the 20th century, many works on computational methods in algebra were published well before the introduction of the modern algebraic language. Since then, new algorithms have been developed and the theory itself has greatly expanded. In comparison, diagrammatic methods in representation theory are relatively new, with the quiver varieties only being introduced – with big impact – in the 1990s.
Divided into two parts, the book first discusses the theory of Gröbner bases in their commutative and noncommutative contexts, with a focus on algorithmic aspects and applications of Gröbner bases to analysison systems of partial differential equations, effective analysis on rings of differential operators, and homological algebra. It then introduces representations of quivers, quiver varieties and their applications to the moduli spaces of meromorphic connections on the complex projective line.
While no particular reader background is assumed, the book is intended for graduate students in mathematics, engineering and related fields, as well as researchers and scholars.
Similar content being viewed by others
Keywords
Table of contents (9 chapters)
-
Front Matter
-
First Algebraic Byway: Gröbner Bases
-
Front Matter
-
Editors and Affiliations
About the editors
Kenji Iohara is a Professor at Université Claude Bernard Lyon 1, France. His research focuses mainly on the Lie theory, singularity, and special functions. He co-authored “Representation Theory of the Virasoro Algebra” (978-0-85729-159-2), published with Springer.
Philippe Malbos is a Professor at Université Claude Bernard Lyon 1, France. His fields of research include algebraic rewriting, Gröbner bases, and homological algebra.
Masa-Hiko Saito is a Professor and Director of the Center for Mathematical and Data Sciences at Kobe University, Japan. His interests include algebraic geometry and its applications to integrable systems.
Nobuki Takayama is a Professor at Kobe University, Japan. His research fields comprise computer algebra, hypergeometric functions, D-modules, and algebraic statistics. He co-authored “Gröbner Deformations of Hypergeometric Differential Equations” (978-3-540-66065-1), published by Springer.
Bibliographic Information
Book Title: Two Algebraic Byways from Differential Equations: Gröbner Bases and Quivers
Editors: Kenji Iohara, Philippe Malbos, Masa-Hiko Saito, Nobuki Takayama
Series Title: Algorithms and Computation in Mathematics
DOI: https://doi.org/10.1007/978-3-030-26454-3
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2020
Hardcover ISBN: 978-3-030-26453-6Published: 21 February 2020
Softcover ISBN: 978-3-030-26456-7Published: 15 July 2021
eBook ISBN: 978-3-030-26454-3Published: 20 February 2020
Series ISSN: 1431-1550
Series E-ISSN: 2512-3254
Edition Number: 1
Number of Pages: XI, 371
Number of Illustrations: 55 b/w illustrations, 1 illustrations in colour
Topics: Field Theory and Polynomials, Algebraic Geometry, Associative Rings and Algebras, Category Theory, Homological Algebra, Ordinary Differential Equations, Partial Differential Equations