Two algebraic byways from differential equations: Gröbner bases and quivers.

*(English)*Zbl 1444.14002
Algorithms and Computation in Mathematics 28. Cham: Springer (ISBN 978-3-030-26453-6/hbk; 978-3-030-26454-3/ebook). xi, 371 p. (2020).

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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.

The articles of this volume will be reviewed individually.

Indexed articles:

Iohara, Kenji; Malbos, Philippe, From analytical mechanics problems to rewriting theory through M. Janet’s work, 3-74 [Zbl 1446.13022]

Bahloul, Rouchdi, Gröbner bases in \(D\)-modules: application to Bernstein-Sato polynomials, 75-93 [Zbl 1442.14065]

Nakayama, Hiromasa; Takayama, Nobuki, Introduction to algorithms for \(D\)-modules with quiver \(D\)-modules, 95-114 [Zbl 1453.14057]

Malbos, Philippe, Noncommutative Gröbner bases: applications and generalizations, 115-183 [Zbl 1457.16046]

Aoki, Satoshi, Introduction to computational algebraic statistics, 185-212 [Zbl 1442.62765]

Iohara, Kenji, Introduction to representations of quivers, 215-229 [Zbl 1448.16019]

Kimura, Yoshiyuki, Introduction to quiver varieties, 231-270 [Zbl 1455.16012]

Hiroe, Kazuki, On additive Deligne-Simpson problems, 271-323 [Zbl 1442.14038]

Yamakawa, Daisuke, Applications of quiver varieties to moduli spaces of connections on \(\mathbb{P}^1\), 325-371 [Zbl 1442.14041]

##### MSC:

14-06 | Proceedings, conferences, collections, etc. pertaining to algebraic geometry |

16-06 | Proceedings, conferences, collections, etc. pertaining to associative rings and algebras |

62-06 | Proceedings, conferences, collections, etc. pertaining to statistics |

00B15 | Collections of articles of miscellaneous specific interest |

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

14H60 | Vector bundles on curves and their moduli |

14L24 | Geometric invariant theory |

14L30 | Group actions on varieties or schemes (quotients) |

16D40 | Free, projective, and flat modules and ideals in associative algebras |

16G20 | Representations of quivers and partially ordered sets |

16S37 | Quadratic and Koszul algebras |

18C10 | Theories (e.g., algebraic theories), structure, and semantics |

18N10 | 2-categories, bicategories, double categories |

18G10 | Resolutions; derived functors (category-theoretic aspects) |

34M35 | Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms |

34M40 | Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain |

35A25 | Other special methods applied to PDEs |

53D20 | Momentum maps; symplectic reduction |

58A15 | Exterior differential systems (Cartan theory) |

68Q42 | Grammars and rewriting systems |