Alphabeta MathAnalytic Compactness, Transversality, Floer and Fukaya Operations
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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 Averaging and Character-Theory Prerequisites
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Free Products and Amalgamation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Maschke's Theorem, Complete Reducibility and the Structure of k[G]
- 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 Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Group Algebra and Representations of Finite Groups
Algebraic Geometry
Algebraic Topology
- Bocksteins Steenrod Squares and Cohomology Operations
- Chern and Pontryagin Classes by Splitting and Complexification
- 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
- Orientations Poincare Lefschetz and Alexander Duality
- Relative Homology Excision and Mayer Vietoris
- 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
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
Combinatorics
Commutative Algebra
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
- Divisors, Riemann--Roch, and Duality
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Hodge Theory on Compact Riemann Surfaces
- Holomorphic Functions of Several Complex Variables
- Isolated Singularities and Laurent Series
- Mittag-Leffler and Runge's Theorem
- Periods, Jacobians, and Abel--Jacobi Theory
- Riemann Surfaces, Branched Maps, and Differentials
- 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 Riemann Sphere and Möbius Transformations
- The Winding Number and the Global Cauchy Theorem
Differential Geometry
- Chern–Weil Theory and Characteristic Forms
- Clifford Bundles, Dirac Operators, and Index Interfaces
- Compact Laplace-type operators and harmonic representatives
- Connections Levi Civita and Parallel Transport
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Geodesics, the Exponential Map, Completeness, and Hopf–Rinow
- Integration of Forms and the General Stokes Theorem
- Manifolds with Boundary Collars and Orientations
- Measurable Densities and Radon Volume on Manifolds
- Rank Theorems and Embedded Submanifolds
- Riemann Curvature and Riemannian Submanifolds
- Riemannian Metrics Length Distance and Volume
- Sard Theorem and Transversality
- Singular Cochains Mayer Vietoris and Smooth Singular Comparison
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Symplectic Manifolds, Moser Stability, and Darboux–Weinstein Theory
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- The De Rham Complex Homotopy and Mayer Vietoris
- The de Rham Theorem and Degree
- The Exterior Derivative and Cartan Calculus
- Vector Fields Flows and Lie Derivatives
- Whitney Embedding Tubular Neighbourhoods and Approximation
Differential Topology
- Gradient Like Vector Fields and Morse Trajectories
- Handle Decompositions Duality and Rearrangement
- Intersection Pairings Self Intersection and Euler Classes
- Morse Critical Points Hessians and Indices
- Morse Functions Critical Values and Genericity
- Morse Homology Continuation and Comparison
- Morse Inequalities and the Handle Chain Complex
- Morse Trajectory Moduli Spaces and the Morse Differential
- Oriented and Mod Two Intersection Numbers
- Relative smooth triangulation and bundle fitting
- Stable Unstable Manifolds and Morse Smale Transversality
- Sublevel Deformation and the Handle Attachment Theorem
- Vector Field Index Euler Characteristic and Poincare Hopf
Fourier Analysis
Functional Analysis
- Banach Alaoglu Goldstine and Krein Milman
- Banach Valued Integration and the Radon Nikodym Property
- Banach-Space Differential Calculus and Banach Manifolds
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- Compact Operators and Riesz Schauder Theory
- Compact Self Adjoint Hilbert Schmidt and Trace Class Operators
- Complex Norms and Realification
- Complex Subspace Realification
- Distributions Test Functions and Differentiation
- Dual Spaces Adjoint Operators and Annihilators
- Finite Dimensional Normed Spaces and Riesz Lemma
- Fourier Transform Convolution and Approximate Identities
- Geometric Hahn Banach and Convex Separation
- Hilbert Space Geometry and Riesz Representation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Orthonormal Bases, Parseval and Fourier Series
- Reflexivity and Eberlein Smulian
- Schwartz Space and the Plancherel Theorem
- Tempered Distributions and the Fourier Transform
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
- Unbounded Self Adjoint Operators and Stones Theorem
- Weak and Weak Star Topologies
Group Theory
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Derived Categories
- Derived Functors
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Kunneth Exactness and Splittings over Principal Ideal Domains
- Long Exact Sequences in Homology
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- Projective and Injective Resolutions
- Spectral Sequences
- Triangulated Categories
- Universal Coefficients and Kunneth Theorems
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
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- Matrices, the Matrix of a Linear Map, and Change of Basis
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- 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
- Absolute Continuity and the Sharp Fundamental Theorem of Calculus
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Differentiation of Monotone Functions and the Vitali Covering Theorem
- 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
- Signed and Complex Measures Hahn and Jordan
- The Duality of Lᵖ and L^q
- 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
- The Radon Nikodym Theorem and Lebesgue Decomposition
Number Theory
Partial Differential Equations
- Analytic Majorants and the Cauchy–Kovalevskaya Theorem
- Euclidean Surface Measure, Divergence, and Green Identities
- Fredholm Elliptic Problems and the Elliptic Spectrum
- Harmonic Functions and Mean Values in Rn
- Interior and Boundary Sobolev Elliptic Regularity
- Lax--Milgram and Weak Elliptic Solutions
- Maximum Principles Harnack and Liouville in Rn
- Partial Differential Equations and Characteristics
- Poisson Problems and Interior Harmonic Estimates
- Rellich Kondrachov and Sobolev Compactness
- Smooth Approximation and Sobolev Extension
- Sobolev Poincare and Morrey Inequalities
- Sobolev Traces and Zero Boundary Values
- Weak Derivatives and Sobolev Spaces
Quantum Algebra and Representation Theory
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
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- 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
- Regular Surfaces and Surface Integrals
- 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 Divergence Theorem and Classical Stokes
- 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
- Volumes of Elementary Solids and Solids of Revolution
Representation Theory of Groups
Scheme Theory
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
String Theory
Topology
- Classification of Compact Connected Surfaces
- Classification of Covering Spaces
- 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
- 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
- Metrization: Urysohn, Nagata–Smirnov, Bing, Smirnov
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Stone–Weierstrass in General
- Subspaces, Products, and Quotients
- The Ascoli–Arzelà Theorem
- The Fundamental Group
- The Fundamental Group of the Circle
- The Seifert–van Kampen Theorem
- 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
Lemma
Weak Lp regularity for a small Cauchy–Riemann principal perturbation
Lemma
Planar Sobolev continuity estimates under Countable Choice
Lemma
A half-disk small-energy estimate with a nonlinear normal derivative bound
Lemma
A Heinz mean value estimate from small scalar energy
Lemma
Local higher derivative bounds for almost complex maps with bounded first derivative
Lemma
Uniform area quantum and finite homology classes for closed tame holomorphic curves
Lemma
Ordered real disk domains, actual plumbing and associahedral face orientations
Definition
Stable almost-complex maps and Gromov convergence
Definition
Capped Lagrangian Hamiltonian chords and action
Lemma
Smoothness of continuous weak almost complex strip solutions
Lemma
Interior small-energy gradient control for almost complex maps
Lemma
Cauchy–Riemann similarity and unique continuation under countable choice
Lemma
Tame transverse-strip coercivity and actual weighted end inverse
Lemma
Boundary small-energy gradient control along a compact totally real submanifold
Lemma
Exponential energy decay on small-gradient almost complex cylinder necks
Lemma
Exponential energy decay on totally real boundary necks
Lemma
Positive energy gap for nonconstant curves on a fixed closed domain
Lemma
Closed-surface real Cauchy–Riemann index and smooth adjoint obstructions
Lemma
Rational filtered curved transfer to all continuous singular chains
Definition
Rational capped Hamiltonian Floer modules and their graded submodules
Lemma
Finite-energy interior and totally real boundary puncture removability
Lemma
Small local energy forces cylinder neck collapse
Lemma
Smooth strip convergence outside finitely many energy concentration points
Lemma
Transverse strip action inequality, exponential ends and neck collapse
Lemma
Bordered CR Fredholm index, boundary regularity and evaluation obstructions
Definition
Spectral-cut cappings and weighted label-changing corners
Lemma
General totally real corner cappings and weighted Fredholm configurations
Lemma
Actual weighted nodal CR gluing and continuous linear kernels
Theorem
Regular aspherical Hamiltonian Floer differential and continuation contract
Lemma
Finite Floer matrices extend from the graded to the full energy completion
Lemma
Derivative concentration produces a finite-energy plane or half-plane bubble
Lemma
Soft rescaling captures an isolated interior energy atom
Theorem
Interior Gromov bubble-tree compactness under Countable Choice
Lemma
Soft rescaling captures an isolated totally real boundary energy atom
Theorem
Zinger fixed-domain interior Gromov compactness
Theorem
Totally real boundary bubble-tree compactness under Countable Choice
Lemma
Local domain slices and a countable compact exhaustion for simple closed curves
Lemma
Geometrically tame disk compactness and actual real-nodal atlas
Theorem
Wendl generic regularity for somewhere-injective closed curves
Theorem
Fixed-boundary disk bubble compactness contract
Theorem
Lagrangian Floer strip breaking and bubbling contract
Definition
Pardon implicit-atlas specialization to Floer cylinder moduli
Lemma
Relative-spin disk determinants and the actual suspended boundary sign
Theorem
Wendl generic one-parameter regularity for somewhere-injective curves
Theorem
General virtual Hamiltonian Floer differential and continuation
Theorem
Rational collared disk virtual correspondences and curved Morse operations
Definition
FOOO single-object filtered curved A-infinity algebra
Theorem
FOOO filtered curved A-infinity identity theorem
Definition
Blocked construction interface for multi-object Fukaya categories and polygon morphisms
Definition
Weak versus strict bounding cochains
Theorem
Exact regular polygon-count A-infinity branch
Lemma
A compact Riemann surface homeomorphic to the sphere has a global complex coordinate
Sources
- Chris Wendl, Lectures on Holomorphic Curves in Symplectic and Contact Geometry, version 3.2
- Urs Frauenfelder and Kai Zehmisch, Gromov compactness for holomorphic discs with totally real boundary conditions, v2
- John Pardon, An Algebraic Approach to Virtual Fundamental Cycles on Moduli Spaces of Pseudo-Holomorphic Curves
- Aleksey Zinger, Notes on J-Holomorphic Maps, arXiv:1706.00331v1
- Urs Frauenfelder and Kai Zehmisch, Gromov compactness for holomorphic discs with totally real boundary conditions
- Dietmar Salamon, Lectures on Floer Homology
- Chris Wendl, Lectures on Holomorphic Curves in Symplectic and Contact Geometry, version3.2
- Aleksey Zinger, Notes on J-Holomorphic Maps
- Chris Wendl, Lectures on Holomorphic Curves in Symplectic and Contact Geometry
- Fukaya, Oh, Ohta, and Ono, Lagrangian Floer Theory on Compact Toric Manifolds I
- John Pardon, An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves
- Stephen Smale, An Infinite Dimensional Version of Sard's Theorem
- Aleksey Zinger, Compactness of J-holomorphic maps, arXiv:1706.00331v1
- Bernhard Keller, A brief introduction to A-infinity algebras
- Katrin Wehrheim and Chris Woodward, Orientations for pseudoholomorphic quilts, arXiv v7, 2 November 2026
- Fukaya, Oh, Ohta, and Ono, Lagrangian Floer Theory on Compact Toric Manifolds I, arXiv:0802.1703v3
- Maxim Kontsevich, Homological Algebra of Mirror Symmetry, arXiv:alg-geom/9411018v1
- Paul Seidel, Graded Lagrangian submanifolds