Alphabeta MathSt Protected Wrapped M2 Quantum Sectors
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Averaging and Character-Theory Prerequisites
- 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
- 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
- Tensor Products of Modules
- The Group Algebra and Representations of Finite Groups
Algebraic Topology
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- Fibrations Fiber Bundles and Homotopy Exact Sequences
- Orientations Poincare Lefschetz and Alexander Duality
- Real Clifford Modules, KO Theory, and ABS Orientations
- 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
- Topological Vector Bundles and Grassmannian Classification
Category Theory
- Abelian Categories
- 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
- 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
- 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
- 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
- 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
- The Gauss Bonnet Theorem for Riemannian Surfaces
- Vector Fields Flows and Lie Derivatives
- Whitney Embedding Tubular Neighbourhoods and Approximation
Functional Analysis
- Banach Valued Integration and the Radon Nikodym Property
- Bounded Linear Operators and Quotient Spaces
- Compact Operators and Riesz Schauder Theory
- 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
- Schwartz Space and the Plancherel Theorem
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
- Unbounded Self Adjoint Operators and Stones Theorem
Group Theory
Homological Algebra
Lie Theory
Linear Algebra
- Complexification, Realification and Real Structures
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- 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
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 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
Partial Differential Equations
Quantum Algebra and Representation Theory
Quantum Field Theory Mathematics
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
- 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
- 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
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
- 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 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
M2 Protected Block Index Comparison
Definition
M2 Regulated Protected Sector Data
Definition
M2 Curve Comparison Realization Data
Remark
M2 Protected Quantization Construction Obligation
Sources
- Stefan Haller, The Atiyah–Singer Index Theorem, full lecture notes (2012/13)
- Bergshoeff, Sezgin, and Townsend, Supermembranes and Eleven-Dimensional Supergravity (1987)
- Paul K. Townsend, Four Lectures on M-theory
- Timo Weigand, TASI Lectures on F-theory, arXiv:1806.01854v1
- Bergshoeff, Sezgin, and Townsend, Properties of the Eleven-Dimensional Supermembrane Theory (1988)