Alphabeta MathHall Integration, Stability, DT/BPS Invariants, and Wall Crossing
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Fields and Cyclotomic Extensions
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules over a Principal Ideal Domain and the Canonical Forms
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Group Algebra and Representations of Finite Groups
Algebraic Geometry
- Affine Algebraic Sets and Coordinate Rings
- Algebraic Differentials Separability and Smooth Local Presentations
- Analytic Constructible Sheaf Foundations
- Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps
- Complex Algebraic Stratified Topology
- Constructible Incidence And Cochain Models
- Dimension Constructible Images and Dimensions of Fibres
- Morphisms Local Rings and Rational Maps of Affine Varieties
- Nonaffine Algebraic Groups, Barsotti-Chevalley, and Abelian Varieties
- Products Segre and Veronese Embeddings and Grassmannians
- Projective Algebraic Sets Projective Morphisms and Cones
- Unipotent and Solvable Groups and Borel Fixed Points
- Zariski Tangent Spaces, Regular Points, Smoothness, and Bertini
Algebraic Topology
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- Fibrations Fiber Bundles and Homotopy Exact Sequences
- Higher Homotopy Groups and Cofiber Sequences
- Homology Axioms Degree and Classical Applications
- Hurewicz Whitehead Freudenthal and Cw Approximation
- Leray–Hirsch, the Thom Isomorphism, and Gysin Sequences
- Local Coefficients, Twisted Homology, and Duality
- Obstruction Theory, Postnikov Towers, and Classifying Spaces
- Relative Homology Excision and Mayer Vietoris
- Simple Homotopy, Whitehead Groups, and Torsion
- Simplicial Complexes and Simplicial Homology
- Simplicial Subdivision and Simplicial Approximation
- Singular Chains and Singular Homology
- Singular Cohomology and Coefficient Theorems
- Stiefel Whitney and Euler Classes by Universal Constructions
- The Serre Spectral Sequence and Applications
- Topological Vector Bundles and Grassmannian Classification
Categorification and Geometric Representation Theory
- Cohomological Hall Algebras and Quiver Correspondences
- Derived Hall Algebras from dg-Enhanced Categories
- Equivariant Borel–Moore Correspondence Convolution and Constructible Ext
- Finitary Ringel–Hall Algebras
- Hall Algebra Modules, Fock Spaces, and Geometric Actions
- Motivic Hall Algebras and Integration Maps
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Subobject Lattices Generators and the Grothendieck Axioms
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Commutative Algebra
- Artinian Rings and Length
- Associated Primes and Primary Decomposition
- Dedekind Domains and Ideal Classes
- Depth and Cohen Macaulay Modules
- Flatness and Faithful Flatness
- Homogeneous Resultants and Projective Intersection Length
- Integral Extensions and Going Up
- Koszul Complexes and Regular Sequences
- Krull Dimension and Height Theorems
- Localisation of Modules and Support
- Noether Normalisation and Nullstellensatz
- Noetherian Rings and Hilbert Basis
- Normalization Finiteness for Affine Domains
- Prime Spectra and Radicals
- Rees Modules Artin Rees and Hilbert Samuel Theory
- Regular Local Rings and Homological Dimension
- Valuation Rings and Discrete Valuation Rings
- Zariski Topology on Prime Spectra
Complex Analysis
- Analyticity of Holomorphic Functions; Liouville and Morera
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Contour Integration
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Holomorphic Functions of Several Complex Variables
- Isolated Singularities and Laurent Series
- Riemann Surfaces, Branched Maps, and Differentials
- The Argument Principle and Rouché's Theorem
- The Holomorphic Inverse Function Theorem and Weierstrass Preparation
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Residue Theorem and the Evaluation of Real Integrals
- The Winding Number and the Global Cauchy Theorem
Differential Geometry
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Rank Theorems and Embedded Submanifolds
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Tangent Cotangent and the Differential
- Whitney Embedding Tubular Neighbourhoods and Approximation
Differential Topology
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Delta Functors and Universality
- Derived Categories
- Derived Functors
- Differential Graded Algebras and Homological Foundations
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Spectral Sequences
- Tor Flatness and Global Dimension
- Triangulated Categories
- Universal Coefficients and Kunneth Theorems
- Yoneda Extensions and Homological Dimension
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Exterior Powers, Orientation and Hodge Duality
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Bounded Variation and the Riemann–Stieltjes Integral
- Constant Rank, Submersions, Immersions and Regular Level Sets
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- pi: the Equivalent Characterizations
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Representation Theory of Groups
Scheme Theory
- Affine Group Schemes, Hopf Algebras, and Rational Representations
- Affine Schemes and the Structure Sheaf
- Algebraic Group Actions, Orbits, Stabilizers, and Controlled Quotients
- Algebraic Spaces, Stacks, and Derived Algebraic Geometry Foundations
- Analytic Derived Six Operations And Duality
- Blowups, Exceptional Divisors, and Strict Transforms
- Cohomology of Quasi Coherent Sheaves on Affine and Projective Schemes
- Compact Support Sheaf Calculus
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Finite Proper and Projective Morphisms
- Flat Smooth and Etale Morphisms
- Group Schemes of Finite Type over a Field
- Kahler Differentials Conormal Sequences and Infinitesimal Lifting
- Presheaves Sheaves Stalks and Sheafification
- Proj Projective Schemes Twisting Sheaves and Ampleness
- Quasi Coherent and Coherent Sheaves and Vector Bundles
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Cohomology Cech Cohomology and Comparison
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Covering Spaces and Lifting
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
Finite-field stability data for an acyclic quiver
Lemma
Finite acyclic quiver representations: heredity, Euler form, and finite-field finitarity
Lemma
The phase of a sum lies between upper-half-plane phases
Lemma
Semistable loci of finite-quiver representations are open
Lemma
Quiver Harder–Narasimhan filtrations exist, are unique, and descend
Lemma
Finite-field points of the semistable quiver stack give the Hall groupoid
Definition
Subobject-first Hall completion and quantum tori for finite quivers
Lemma
Finite-quiver Hall multiplication counts extensions by automorphism orbits
Theorem
Euler and skew integration are multiplicative for finite acyclic quivers
Theorem
Completed Hall wall crossing for finite acyclic quivers
Remark
Finite-quiver Hall counts and the limits of DT/BPS interpretation
Sources
- Tom Bridgeland, Hall algebras and Donaldson–Thomas invariants, §§4.4–4.6
- Markus Reineke, The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli, §§2–3
- Olivier Schiffmann, Lectures on Hall Algebras, §§1.1, 1.3 and 3.3
- Tom Bridgeland, Hall algebras and Donaldson–Thomas invariants, §§2.1 and 3.2–3.4
- Tom Bridgeland, Hall algebras and Donaldson–Thomas invariants, §4.4
- Markus Reineke, The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli, Lemma 2.2
- Markus Reineke, The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli, §3
- Markus Reineke, The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli, §2
- Tom Bridgeland, Hall algebras and Donaldson–Thomas invariants, §§2.1 and 4.4
- Tom Bridgeland, An Introduction to Motivic Hall Algebras, §§3.1–3.2
- Tom Bridgeland, Hall algebras and Donaldson–Thomas invariants, §§2.1, 3.3–3.4
- Tom Bridgeland, Hall algebras and Donaldson–Thomas invariants, Lemma 2.2
- Olivier Schiffmann, Lectures on Hall Algebras, §§1.1 and 1.3
- Tom Bridgeland, Hall algebras and Donaldson–Thomas invariants, §§3.3–3.4
- Markus Reineke, The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli, §6
- Markus Reineke, The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli, §§2 and 4
- Tom Bridgeland, Hall algebras and Donaldson–Thomas invariants, §§4.3–4.6
- Bertrand Toën, Derived Hall algebras, §§3–4
- Tom Bridgeland, An Introduction to Motivic Hall Algebras, §§3–5
- Maxim Kontsevich and Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, §§1–2
- Maxim Kontsevich and Yan Soibelman, Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, §§1–2 and 7