Alphabeta MathHMS and Landau–Ginzburg Categories
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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
- Finite Fields and Cyclotomic Extensions
- 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
- Solvability by Radicals and Kummer Theory
- 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
- The Group Algebra and Representations of Finite Groups
Algebraic Geometry
- Affine Algebraic Sets and Coordinate Rings
- Algebraic Differentials Separability and Smooth Local Presentations
- Birational Morphisms, Contractions, and Surface Singularities
- Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps
- Coherent Duality on Projective Cohen-Macaulay Schemes
- Complex Algebraic Stratified Topology
- Dimension Constructible Images and Dimensions of Fibres
- Morphisms Local Rings and Rational Maps of Affine Varieties
- Normal Varieties, Normalization, and Zariski's Main Theorem
- Smooth-Projective Serre Duality and Flag-Variety Line Bundles
- Zariski Tangent Spaces, Regular Points, Smoothness, and Bertini
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
- 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
- Henselian Rings and Equicharacteristic Cohen Structure
- 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
- Compact Kahler Hodge Foundations
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Conformal Mapping, Branches, and the Schwarz Lemma
- 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
- Normal Families and Montel's Theorem
- 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 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
- 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
- Bounded Linear Operators and Quotient Spaces
- 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
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
- Grothendieck Spectral Sequences and Computations
- Kunneth Exactness and Splittings over Principal Ideal Domains
- 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
- 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
- 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
- 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
- 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
Quantum Field Theory Mathematics
- ADHM Linear Data, Positive-FI Quotients and Framed Monads
- 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
- 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 Formal Laurent Series Field ℝ((t⁻¹)): Cauchy Complete, Non-Archimedean, Not Complete
- 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
- Affine Schemes and the Structure Sheaf
- Algebraic Spaces, Stacks, and Derived Algebraic Geometry Foundations
- 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
- Hilbert Functors and Projective Hilbert Schemes
- 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
- Residues Serre Duality for Curves and the Full Riemann Roch Theorem
- Riemann Roch for Curves via Euler Characteristics
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Cohomology Cech Cohomology and Comparison
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
- Smooth Proper Curves Divisors Genus and Ramification
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
- Analytic Compactness, Transversality, Floer and Fukaya Operations
- Bridgeland Stability and DT Frontiers
- Classical and Derived Deformation Data
- Complex, Kähler and Hodge Foundations
- Differential Cohomology, Deligne Cohomology and Gerbes
- Mirror Geometry, SYZ and Log-Calabi–Yau Data
- Symplectic, Floer and Fukaya Foundations
Topology
- 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
Fully faithful functor with thickly generating image is an equivalence
Definition
Toric fan and Laurent potential data for the line-bundle case
Definition
General Landau–Ginzburg matrix-factorization data
Definition
Weinstein Lefschetz fibration data interface
Lemma
The matrix-factorization Hom differential is a dg differential
Lemma
Exterior smash Hochschild calculation and equivariant transfer for the named Fermat models
Lemma
Constructive equivariant q-adic one-parameter comparison
Lemma
Coherent generators and Fermat deformation for the quartic and quintic mirrors
Definition
Log HMS surface data with distinguished complex structure
Lemma
Named Fermat compact relative and first-order mixed Fukaya construction
Lemma
Named Fermat graph algebra, compact grading, and geometric Fermat class
Lemma
Compact rational geometric Dehn twist and actual evaluation-cone triangle
Definition
Landau–Ginzburg pair and distinct categories
Definition
Directed and wrapped Fukaya category interfaces
Lemma
The named Fermat graph lifts identify the actual vanishing cycles
Lemma
Negative monodromy and full compact generation for the named Fermat pencils
Theorem
Homological mirror symmetry for a quartic K3
Theorem
Homological mirror symmetry for the quintic threefold
Definition
Relative Fukaya pre-category interface for the toric mirror
Lemma
Surface polygon corner divisor, automatic regularity and actual compactification
Theorem
Toric line-bundle subcategory mirror theorem
Lemma
Elliptic geometric copy Homs and parallel Floer products
Lemma
Elliptic theta, coherent classification and all-arity transversal comparison
Lemma
Elliptic Hamiltonian-copy completion and the full derived dg comparison
Theorem
Categorical mirror equivalence for elliptic curves
Sources
- Paul Seidel, Homological mirror symmetry for the quartic surface, arXiv:math/0310414v4
- Mohammed Abouzaid, Morse Homology, Tropical Geometry, and Homological Mirror Symmetry for Toric Varieties
- Simon Telen, Introduction to Toric Geometry, arXiv v1, 3 March 2022
- Alexander Polishchuk and Arkady Vaintrob, Chern characters and Hirzebruch–Riemann–Roch formula for matrix factorizations, arXiv:1002.2116v3
- Paul Hacking and Ailsa Keating, Homological mirror symmetry for log Calabi–Yau surfaces
- Emmanuel Giroux and John Pardon, Existence of Lefschetz fibrations on Stein and Weinstein domains
- Paul Seidel, Homological Mirror Symmetry for the Quartic Surface
- Yuichi Nohara and Kazushi Ueda, Homological Mirror Symmetry for the Quintic 3-Fold
- Nick Sheridan, On the homological mirror symmetry conjecture for pairs of pants
- Nick Sheridan, Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space
- Yuichi Nohara and Kazushi Ueda, Homological mirror symmetry for the quintic 3-fold
- Paul Seidel, Homological mirror symmetry for the quartic surface
- Paul Seidel, A long exact sequence for symplectic Floer cohomology
- Maxim Kontsevich, Homological Algebra of Mirror Symmetry
- Denis Auroux, Mirror Symmetry, Lecture 18
- Mohammed Abouzaid and Paul Seidel, An open string analogue of Viterbo functoriality
- Wei-Dong Ruan, Lagrangian torus fibration of quintic Calabi-Yau hypersurfaces II
- Polishchuk, A-infinity structures on an elliptic curve
- Paul Seidel, Graded Lagrangian submanifolds
- Denis Auroux, A beginner's introduction to Fukaya categories
- Alexander Polishchuk and Eric Zaslow, Categorical mirror symmetry for the elliptic curve
- Polishchuk and Zaslow, Categorical Mirror Symmetry: The Elliptic Curve
- Alexander Polishchuk, A-infinity structures on an elliptic curve
- Alexander Polishchuk and Eric Zaslow, Categorical Mirror Symmetry: The Elliptic Curve
- Denis Auroux, Mirror Symmetry, Lecture 1