Alphabeta MathSwampland Conjectures And Cosmological Interfaces Examples
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‡ 12 results recorded, not proved here · all sources checked
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- 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
Category Theory
Complex Analysis
Differential Geometry
- 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
- Jacobi Fields, Conjugate Points, and the Cut Locus
- Manifolds with Boundary Collars and Orientations
- Rank Theorems and Embedded Submanifolds
- Riemann Curvature and Riemannian Submanifolds
- Riemannian Comparison Theorems
- Riemannian Metrics Length Distance and Volume
- 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
- Vector Fields Flows and Lie Derivatives
Functional Analysis
General Relativity
Homological Algebra
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
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- 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
- Further Trigonometric Identities and Inverse Functions
- 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
- 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 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
- Volumes of Elementary Solids and Solids of Revolution
Set Theory
Special Relativity
String Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- 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 Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Open problem
Swampland distance conjecture
Open problem
Electric weak-gravity conjecture in four dimensions
Open problem
Single-axion weak-gravity bound for instantons
Open problem
Charge-completeness conjecture
Open problem
Emergence proposal for EFT kinetic data and cutoffs
Open problem
Refined de Sitter conjecture for positive scalar potentials
Open problem
Anti-de Sitter distance conjecture
Open problem
Non-supersymmetric AdS instability conjecture
Open problem
Parametric AdS scale-separation conjecture
Open problem
No exact global symmetries in quantum gravity
Open problem
Cobordism conjecture for quantum-gravity structures
Open problem
Trans-Planckian censorship conjecture in cosmology
Sources
- On the Geometry of the String Landscape and the Swampland
- Distance and de Sitter Conjectures on the Swampland
- Hitchhiker’s Guide to the Swampland: The Cosmologist’s Handbook to the string-theoretical Swampland Programme
- Eran Palti, The Swampland: Introduction and Review, arXiv:1903.06239v3
- Nathan Benjamin Agmon, Alek Bedroya, Monica Jinwoo Kang, and Cumrun Vafa, Lectures on the string landscape and the Swampland, arXiv:2212.06187v2
- The String Landscape, Black Holes and Gravity as the Weakest Force
- Di Ubaldo, Iliesiu, Lin and Yan, Positivity of the gravitational path integral implies the axionic weak gravity conjecture
- Andriolo et al., Duality and Axionic Weak Gravity
- Axion Cosmology
- Monopoles, Duality, and String Theory
- Constraints on symmetry from holography
- Emergence and the Swampland Conjectures
- AdS and the Swampland
- De Sitter Space and the Swampland
- Non-supersymmetric AdS and the Swampland
- Anastasi et al., Relative quantum gravity: localized gravity and the swampland
- Cobordism Classes and the Swampland
- Trans-Planckian Censorship and the Swampland
- Michele Cicoli, Joseph P. Conlon, Anshuman Maharana, Susha Parameswaran, Fernando Quevedo, and Ivonne Zavala, String Cosmology: from the Early Universe to Today, arXiv:2303.04819v2