Exploration Map
A map of ideas rather than a timetable. Link new concepts here when they become useful landmarks; a note may belong to several areas.
Algebra
- Algebra concepts
- Constructing number systems: Adjoining a Root and Tower Extensions
- Relations as construction rules: Why Quotient by a Polynomial?
- Algebraic complexity as dimension: Algebraic Numbers and Minimal Polynomials
- Symmetry and fixed information: Field Automorphisms and Fundamental Theorem of Galois Theory
- The limits of formulas: Why the Quintic Changed Mathematics
Analysis
- Analysis concepts
- Continuity through open sets: Continuity Without Epsilon-Delta
- Finite-cover reasoning and existence: Compactness
- Derivatives without an ambient space: Differential Maps and Pushforwards
- Integration as geometry: Differential Forms and Integration and Generalized Stokes Theorem
- Deterministic equations from random motion: Brownian Motion and the Heat Equation
Topology
- Topology concepts
- Structure without measurement: From Metric Spaces to Topology and Strange Topologies
- Sameness and invariants: Homeomorphism and Connectedness
- Gluing: Quotient Spaces
- Deformation and loops: Homotopy and Fundamental Group
- Turning spaces into algebra: A Glimpse of Algebraic Topology
Geometry
- Geometry concepts
- Local descriptions and global spaces: From Surfaces to Manifolds and Charts and Atlases
- First-order motion and measurement: Tangent Spaces and Cotangent Spaces and Differential Forms
- Alternation, integration and local change: Differential Forms and Integration and Exterior Derivative
- Intrinsic geometry: Curvature and Geodesics
- Boundary laws: Generalized Stokes Theorem
Probability
- Probability concepts
- Models of uncertainty: What Is Randomness Mathematically?
- Aggregate stability and fluctuations: Law of Large Numbers and Central Limit Theorem
- Counting and limiting shapes: Combinatorics Behind the Gaussian
- Random paths and scaling: Random Walks and From Random Walk to Brownian Motion
- Long-term random dynamics: Markov Chains
- Uncertainty and information: What Is Entropy? and Shannon Information
Mathematical Physics
- Intrinsic descriptions of motion and fields: Geometry Meets Physics
- Random motion and diffusion: Brownian Motion and the Heat Equation
- Microscopic states and macroscopic entropy: Statistical Mechanics Connection
- Keep durable cross-disciplinary ideas in Other concepts.
Connections Between Areas
- Field symmetries ↔ group theory: From Automorphisms to Groups
- Topology ↔ group theory: Fundamental Group
- Algebraic relations ↔ topological gluing: Why Quotient by a Polynomial? and Quotient Spaces
- Differential forms ↔ vector calculus: Exterior Derivative and Generalized Stokes Theorem
- Random walks ↔ heat equation: Brownian Motion and the Heat Equation
- Entropy ↔ information: What Is Entropy? and Shannon Information
- Geometry ↔ mechanics: Geometry Meets Physics