Two Algebraic Byways from Differential Equations: Gröbner Bases and Quivers (Algorithms and Computation in Mathematics, 28)

Preface
Contents
Contributors
Part I First Algebraic Byway: Gröbner Bases
1 From Analytical Mechanics Problems  to Rewriting Theory Through M. Janet\'s Work
	1 Introduction
		1.1 From Analytical Mechanics Problems to Involutive Division
		1.2 Constructive Methods and Rewriting in Algebra Through the Twentieth Century
		1.3 Conventions and Notations
	2 Exterior Differential Systems
		2.1 Pfaff\'s Problem
		2.2 The Cartan–Kähler Theory
	3 Monomial PDE Systems
		3.1 Ring of Partial Differential Operators and Multiplicative Variables
		3.2 Completion Procedure
		3.3 Inversion of Differentiation
	4 Monomial Involutive Bases
		4.1 Involutive Division
		4.2 Involutive Completion Procedure
		4.3 Others Involutive Approaches
	5 Polynomial Partial Differential Equations Systems
		5.1 Parametric and Principal Derivatives
		5.2 First-Order PDE Systems
		5.3 Higher Order Finite Linear PDE Systems
		5.4 Completely Integrable Higher Order Linear PDE Systems
		5.5 Canonical Forms of Linear PDE Systems
		5.6 Reduction of a PDE System to a Canonical Form
		5.7 Algebra, Geometry, and PDEs
		5.8 Involutive Systems
		5.9 Concluding Remarks
	6 Polynomial Involutive Bases
		6.1 Involutive Reduction on Polynomials
		6.2 Involutive Bases
		6.3 Involutive Bases and Gröbner Bases
	References
2 Gröbner Bases in D-Modules: Application to Bernstein-Sato Polynomials
	1 Introduction
	2 Gröbner Bases and Rings of Differential Operators
		2.1 Rings of Differential Operators
		2.2 Orders
		2.3 Gröbner Bases in An(mathbbK)
	3 Bernstein-Sato Polynomials and Ideals
		3.1 Introduction of the Bernstein-Sato Polynomial of One Function
		3.2 Generalization to Several Functions
		3.3 Stratification Results
	4 Computation of Bernstein-Sato Ideals by Using Gröbner Bases
		4.1 Malgrange Point of View
		4.2 Algorithmic Point of View
	5 b-Function and V-Filtration
	References
3 Introduction to Algorithms for D-Modules with Quiver D-Modules
	1 Computation of Integration Functors in One Dimensional Case
	2 Quiver D-Modules
	3 Examples—D-Modules on Computer Algebra Systems
	References
4 Noncommutative Gröbner Bases: Applications and Generalizations
	1 Introduction
		1.1 Rewriting and Linear Rewriting
		1.2 Noncommutative Gröbner Bases: Applications and Generalizations
	2 Linear Rewriting
		2.1 Linear 2-Polygraphs
		2.2 Linear Rewriting Steps
		2.3 Termination of Linear 2-Polygraphs
		2.4 Monomial Orders
	3 Convergence in Linear Rewriting Systems
		3.1 Ideal of a Linear 1-Polygraph
		3.2 Confluence and Convergence
		3.3 Critical branching lemma
		3.4 Composition Lemma
		3.5 Reduction Operators
		3.6 Noncommutative Gröbner bases
	4 Anick\'s Resolution
		4.1 Homology of an Algebra
		4.2 Anick\'s Chains
		4.3 Anick\'s Resolution
		4.4 Anick\'s Resolution for a Monomial Algebra
		4.5 Computing Homology with Anick\'s Resolution
		4.6 Minimality of Anick\'s Resolution
	5 Higher Dimensional Linear Rewriting
		5.1 Coherent Presentations of Algebras
		5.2 Polygraphic Resolutions of Algebras
	6 Confluence and Koszulness
		6.1 Koszulness of Associative Algebras
		6.2 Confluence and Koszulness
	References
5 Introduction to Computational Algebraic Statistics
	1 Conditional Tests for Contingency Tables
	2 Markov Bases and Ideals
	References
Part II Second Algebraic Byway: Quivers
6 Introduction to Representations  of Quivers
	1 Quivers and Their Representations
		1.1 Quivers and Their Representations
		1.2 Path Algebras
		1.3 Examples and Exercises
	2 Classification of Indecomposable Representations
		2.1 Type of Representations
		2.2 Tits Form
		2.3 Main Theorem
	References
7 Introduction to Quiver Varieties
	1 Introduction
	2 Moduli Spaces of Quiver Representations
		2.1 Quivers and Their Representations
		2.2 Stability Conditions for Quiver Representations
		2.3 Harder–Narasimhan Filtration
		2.4 Jordan–Hölder Filtration
		2.5 Moduli Space of Quiver Representations
		2.6 Ice Quivers and Framed Moduli
	3 The Preprojective Algebra of a Quiver
		3.1 Reflection Functor for Quiver Representation
		3.2 Preprojective Algebra
		3.3 2-Calabi–Yau Property and the Crawley–Boevey Formula
		3.4 McKay Correspodence
		3.5 Moment Map and Hamiltonian Reduction
		3.6 Quiver Varieties
		3.7 Nilpotent Orbit and Springer Resolution
	References
8 On Additive Deligne–Simpson Problems
	1 Deligne–Simpson Problem and Riemann–Hilbert Problem
	2 Additive Deligne–Simpson Problem
		2.1 A Generalization of the Additive Deligne–Simpson Problem
		2.2 Moduli Spaces of Meromorphic Connections and Additive Deligne–Simpson Problem
	3 A Review of Representations of Quivers
		3.1 Representations of Quivers and Quiver Varieties
		3.2 Crawley-Boevey\'s Theorems for the Geometry of Quiver Varieties
	4 A Review of Fuchsian Cases
	5 Moduli Spaces of Meromorphic Connections and Quiver Varieties
		5.1 A Preliminary Example: Differential Equations with Poles of Order 2 and Representations of Quivers
		5.2 Truncated Orbits and Representations of Quivers
		5.3 Quivers Associated with Differential Equations
	6 Geometry of Moduli Spaces of Meromorphic Connections
	7 Outline of the Proofs of the Main Theorems
		7.1 A Review of Middle Convolutions
		7.2 Irreducibility and mathcalL-Irreducibility
		7.3 mathcalL-Fundamental Set
	References
9 Applications of Quiver Varieties to Moduli Spaces of Connections on mathbbP1
	1 Introduction
	2 Hamiltonian Geometry
		2.1 Hamiltonian Spaces
		2.2 Hamiltonian Reduction
	3 Quiver Varieties
		3.1 Open/Closed Quiver Varieties
		3.2 Closing, Gluing and Blowing Up
	4 Polar-Parts Manifolds
		4.1 Definition
		4.2 Relation to Stable Meromorphic Connections on mathbbP1
	5 Residue Manifolds and Star-Shaped Quiver Varieties
		5.1 Coadjoint Orbits of Type A and Quiver Varieties
		5.2 Residue Manifolds and Star-Shaped Quiver Varieties
	6 Higher Order Pole Case
		6.1 Normal Forms and G[[z]] 0-Coadjoint Orbits
		6.2 Triangular Decomposition of G[[z]] 0-Coadjoint Orbits
		6.3 Polar-Parts Manifolds and Quiver Varieties
		6.4 Further Directions
	References