Alphabeta MathADHM Linear Data, Positive-FI Quotients and Framed Monads
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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 Groups and Presentations
- 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
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
Algebraic Geometry
- Affine Algebraic Sets and Coordinate Rings
- Algebraic Differentials Separability and Smooth Local Presentations
- Birational Morphisms, Contractions, and Surface Singularities
- Chow Groups, Intersection Products, and Grothendieck-Riemann-Roch
- Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps
- Classical Complex Algebraic Actions and Affine Embeddings
- Dimension Constructible Images and Dimensions of Fibres
- Morphisms Local Rings and Rational Maps of Affine Varieties
- Products Segre and Veronese Embeddings and Grassmannians
- Projective Algebraic Sets Projective Morphisms and Cones
- Reductive Affine Invariant Theory and Geometric Quotients
- Smooth-Projective Serre Duality and Flag-Variety Line Bundles
Algebraic Topology
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
- Algebraic Zariski Main for Quasi-Finite Morphisms
- 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
- Inverse Limits and Noetherian Completion
- 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
- Analytic Continuation, Monodromy, and Riemann Surfaces
- 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
- Divisors, Riemann--Roch, and Duality
- Domains of Holomorphy, Plurisubharmonicity and Pseudoconvexity
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Harmonic Functions and the Poisson Integral
- Holomorphic Functions of Several Complex Variables
- Hörmander Estimates and the Levi Problem
- Isolated Singularities and Laurent Series
- Riemann Surfaces, Branched Maps, and Differentials
- Subharmonic Functions and the Dirichlet Problem
- The Argument Principle and Rouché's Theorem
- The Dolbeault Complex and Integral Solutions
- 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 Riemann Sphere and Möbius Transformations
- The Winding Number and the Global Cauchy Theorem
Differential Geometry
- Chern–Weil Theory and Characteristic Forms
- Compact Laplace-type operators and harmonic representatives
- Connections Levi Civita and Parallel Transport
- Integration of Forms and the General Stokes Theorem
- Manifolds with Boundary Collars and Orientations
- Rank Theorems and Embedded Submanifolds
- Riemann Curvature and Riemannian Submanifolds
- Riemannian Metrics Length Distance and Volume
- Sard Theorem and Transversality
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- The Exterior Derivative and Cartan Calculus
- Vector Fields Flows and Lie Derivatives
Functional Analysis
- Banach Alaoglu Goldstine and Krein Milman
- Banach Valued Integration and the Radon Nikodym Property
- Bounded Linear Operators and Quotient Spaces
- Complex Norms and Realification
- Complex Subspace Realification
- Distributions Test Functions and Differentiation
- Dual Spaces Adjoint Operators and Annihilators
- Hilbert Space Geometry and Riesz Representation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Reflexivity and Eberlein Smulian
- Schwartz Space and the Plancherel Theorem
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
- Unbounded Self Adjoint Operators and Stones Theorem
- Weak and Weak Star Topologies
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Delta Functors and Universality
- Derived Categories
- Derived Functors
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Grothendieck Groups and Graded Cartan Pairings
- Grothendieck Spectral Sequences and Computations
- 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
Lie Theory
Linear Algebra
- Complexification, Realification and Real Structures
- 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
- Matrix Norms, Condition Numbers and Numerical Stability
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Radon Measures and the Riesz Markov Kakutani Theorem
- Sigma Algebras and Borel Sets
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
Number Theory
Numerical Differential Equations and Stochastic Dynamics
Partial Differential Equations
Quantum Field Theory Mathematics
- Faithful flatness of finite-type algebraic-to-analytic local rings
- Finite-dimensional coherent cohomology on compact projective space
- Local coherent acyclicity on sufficiently small polydiscs
- Oka coherence and local coherent free resolutions
- Projective Dolbeault comparison and scalar structure-sheaf cohomology
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Arc Length and Rectifiable Curves
- Areas of Elementary Plane Figures
- 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
- Convex and Semicontinuous Functions on Rⁿ
- Convexity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- 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
- 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
Scheme Theory
- Affine Schemes and the Structure Sheaf
- Blowups, Exceptional Divisors, and Strict Transforms
- Cartier and Weil Divisors Line Bundles and Picard Groups
- Cohomology of Quasi Coherent Sheaves on Affine and Projective Schemes
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Finite Proper and Projective Morphisms
- Flat Smooth and Etale Morphisms
- 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
- Complete Metrizability, Čech-Completeness, and Baire Category
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Convergence: Nets and Filters
- Countability Axioms and Cardinal Functions
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Ascoli–Arzelà Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
ADHM linear data, stability, costability and FI levels
Lemma
Stable ADHM quotient charts, properness and torus lifts
Theorem
Real and complex ADHM quotients at positive and regular zero FI
Theorem
Projective GAGA for framed coherent and monad data
Lemma
The projective-plane Beilinson monad for a framed torsion-free sheaf
Theorem
Framed P² monads and stable ADHM data
Sources
- Nikita A. Nekrasov, Seiberg–Witten Prepotential from Instanton Counting
- Hiraku Nakajima and Kota Yoshioka, Lectures on Instanton Counting
- John M. Lee, Introduction to Smooth Manifolds, second edition
- Jean-Pierre Serre, Géométrie algébrique et géométrie analytique
- F. Henni, Monads for framed torsion-free sheaves on multi-blow-ups of the projective plane
- Andrew J. Donaldson, The ADHM construction of Yang–Mills instantons