© 2020

Two Algebraic Byways from Differential Equations: Gröbner Bases and Quivers

  • Kenji Iohara
  • Philippe Malbos
  • Masa-Hiko Saito
  • Nobuki Takayama

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


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.


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

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