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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).
Publisher’s description: 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 20-th 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.
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
##### Software:
4ti2; Risa/Asir; CafeOBJ; Macaulay2; SINGULAR
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