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2020 | Buch

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Two Algebraic Byways from Differential Equations: Gröbner Bases and Quivers

herausgegeben von: Prof. Kenji Iohara, Prof. Philippe Malbos, Prof. Masa-Hiko Saito, Nobuki Takayama

Verlag: Springer International Publishing

Buchreihe : Algorithms and Computation in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

First Algebraic Byway: Gröbner Bases

Frontmatter

Chapter 1. From Analytical Mechanics Problems to Rewriting Theory Through M. Janet’s Work

Abstract

This chapter is devoted to a survey of the historical background of Gröbner bases for D-modules and linear rewriting theory largely developed in algebra throughout the twentieth century and to present deep relationships between them. Completion methods are the main streams for these computational theories. In the theory of Gröbner bases, they were motivated by algorithmic problems in elimination theory such as computations in quotient polynomial rings modulo an ideal, manipulating algebraic equations, and computing Hilbert series. In rewriting theory, they were motivated by computation of normal forms and linear bases for algebras and computational problems in homological algebra.

Kenji Iohara, Philippe Malbos

Chapter 2. Gröbner Bases in D-Modules: Application to Bernstein-Sato Polynomials

Abstract

In this chapter, we introduce Gröbner bases in a particular non-commutative ring and we show how they can be applied in a geometric context. In Sect. 2, we introduce the Weyl algebra and we present Gröbner bases in this ring.

Rouchdi Bahloul

Chapter 3. Introduction to Algorithms for D-Modules with Quiver D-Modules

Abstract

The goal of this expository chapter is to illustrate how to use algorithmic methods for D-modules to make mathematical experiments for D-modules and cohomology groups with examples of quiver D-modules. The first section is based on a lecture by the second author given in the Kobe-Lyon summer school 2015 On Quivers: s Computational Aspects and Geometric Applications. The second author could attend several interesting lectures of the school and the Sects. 2 and 3 are written by an inspiration from these lectures and the interesting paper by Khoroshkin and Varchenko (IMRP Int Math Res Pap, Art ID 69590, 116, 2006, [5]).

Hiromasa Nakayama, Nobuki Takayama

Chapter 4. Noncommutative Gröbner Bases: Applications and Generalizations

Abstract

The aim of this chapter is to provide a summary of the theory of linear rewriting and the application of this theory to the construction of free resolutions for associative algebras. In Sect. 2, we present linear polygraphs as an algebraic setting for linear rewriting without a monomial order, and we review the fundamental notion of linear polygraphs. In Sect. 3, we recall several historical constructions on linear rewriting systems for associative algebras, and we show how the confluence properties are studied in these different approaches. We relate the notion of convergent linear polygraph with the notion of noncommutative Gröbner basis. In Sect. 4, we describe an algorithmic way to compute free resolutions for algebras using a method introduced by Anick. Section 5 deals with extension of linear polygraphs, seen as higher dimensional linear rewriting systems, into polygraphic resolutions for algebras. We show how to construct such a resolution starting from a convergent presentation. In the last section, we show how to relate Koszulness for algebras with the property of confluence.

Philippe Malbos

Chapter 5. Introduction to Computational Algebraic Statistics

Abstract

In this paper, we introduce the fundamental notion of a Markov basis, which is one of the first connections between commutative algebra and statistics. The notion of a Markov basis is first introduced by Diaconis and Sturmfels (Algebraic algorithms for sampling from conditional distributions. Ann Statist 26(1):363–397, 1998, [8]) for conditional testing problems on contingency tables by Markov chain Monte Carlo methods.

Satoshi Aoki

Second Algebraic Byway: Quivers

Frontmatter

Chapter 6. Introduction to Representations of Quivers

Abstract

The main purpose of this lecture note is to provide a quick introduction to quivers and their representations. In particular, as there already exists several introductory and complete texts on quivers, the author tries motivating the reader to develop the theory by showing several concrete examples.

Kenji Iohara

Chapter 7. Introduction to Quiver Varieties

Abstract

Quiver representations and Kac–Moody Lie algebras. The interaction between quiver representations and Kac–Moody Lie algebras has an origin in Gabriel’s theorem. Gabriel [15] classified the quivers which are finite representation types and showed the existence of the bijection between the set of isomorphism classes of indecomposable representations of Dynkin quivers Q and the set of positive roots of the corresponding simply laced Lie algebra \({\mathfrak {g}}_{Q}\) via dimension vectors. Bernšteĭn–Gel’fand–Ponomarev [2] gave a proof of Gabriel’s theorem using reflection functors and Coxeter functors. Using the theory of species introduced by Gabriel [16], Dlab–Ringel [11] extended Gabriel’s theorem to finite dimensional hereditary algebras over arbitrary fields. The classification of the indecomposable representations of affine quivers was studied by Weierstrass, Kronecker, Gel’fand–Ponomarev, Donovan–Freislich, Nazarova and Dlab–Ringel [12].

Yoshiyuki Kimura

Chapter 8. On Additive Deligne–Simpson Problems

Abstract

In this note, we explain the additive Deligne–Simpson problem and its generalization for differential equations with unramified irregular singularities. A correspondence between spaces of solutions of these additive Deligne–Simpson problems and quiver varieties is given. As an application, the geometry of moduli spaces of meromorphic connections with unramified irregular singularities is discussed, for example, the non-emptiness of the smooth parts of moduli spaces and their connectedness. The detail of this note can also be found in [17].

Kazuki Hiroe

Chapter 9. Applications of Quiver Varieties to Moduli Spaces of Connections on

Abstract

The aim of this lecture is to explain the main result of Hiroe and Yamakawa (Adv Math 266:120–151, (2014), [13]), which affirmatively solves Boalch’s conjecture (proposed in Boalch, Irregular connections and Kac-Moody root systems, 2008, [3]) on some relationship between meromorphic connections on the Riemann sphere \({\mathbb {P}}^1\) and the quiver varieties introduced by Nakajima (Duke Math J 76(2):365–416, (1994), [18]).

Daisuke Yamakawa

Titel: Two Algebraic Byways from Differential Equations: Gröbner Bases and Quivers
herausgegeben von: Prof. Kenji Iohara
Prof. Philippe Malbos
Prof. Masa-Hiko Saito
Nobuki Takayama
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-26454-3
Print ISBN: 978-3-030-26453-6
DOI: https://doi.org/10.1007/978-3-030-26454-3