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Two Algebraic Byways from Differential Equations: Groebner Bases and Quivers

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Two Algebraic Byways from Differential Equations: Groebner Bases and Quivers

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This edited volume presents a fascinating collection of lecture notes focusing on differential equations from two viewpoints: formal calculus (through the theory of Groebner bases) and geometry (via quiver theory). Groebner bases serve as effective models for computation in algebras of various types. Although the theory of Groebner 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 Groebner bases in their commutative and noncommutative contexts, with a focus on algorithmic aspects and applications of Groebner bases to analysis on 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.

Author Biography

Kenji Iohara is a Professor at Universite 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 Universite Claude Bernard Lyon 1, France. His fields of research include algebraic rewriting, Groebner 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 "Groebner Deformations of Hypergeometric Differential Equations" (978-3-540-66065-1), published by Springer.
Release date NZ
January 14th, 2020
Edited by Kenji Iohara Edited by Masa-Hiko Saito Edited by Nobuki Takayama Edited by Philippe Malbos
Country of Publication
1st ed. 2019
12 Illustrations, black and white; X, 286 p. 12 illus.
Springer Nature Switzerland AG
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