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Two Algebraic Byways from Differential Equations: Gröbner Bases and Quivers

  • Kenji Iohara
  • Philippe Malbos
  • Masa-Hiko Saito
  • Nobuki Takayama
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Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 28)

Table of contents

  1. Front Matter
    Pages i-xi
  2. First Algebraic Byway: Gröbner Bases

  3. Second Algebraic Byway: Quivers

    1. Front Matter
      Pages 213-213
    2. Kenji Iohara
      Pages 215-229
    3. Yoshiyuki Kimura
      Pages 231-270
    4. Kazuki Hiroe
      Pages 271-323

About this book

Introduction

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 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.

Keywords

commutative and noncommutative Gröbner bases D-modules B-functions quiver varieties moduli spaces meromorphic connections

Editors and affiliations

  • Kenji Iohara
    • 1
  • Philippe Malbos
    • 2
  • Masa-Hiko Saito
    • 3
  • Nobuki Takayama
    • 4
  1. 1.Université Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille JordanVilleurbanneFrance
  2. 2.Université Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille JordanVilleurbanneFrance
  3. 3.Center for Mathematical and Data Sciences, Department of Mathematics, Graduate School of ScienceKobe UniversityKobeJapan
  4. 4.Department of Mathematics, Graduate School of ScienceKobe UniversityKobeJapan

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-030-26454-3
  • Copyright Information Springer Nature Switzerland AG 2020
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-26453-6
  • Online ISBN 978-3-030-26454-3
  • Series Print ISSN 1431-1550
  • Buy this book on publisher's site