A ↔ B
A ↔ B The Connection Why It Is Surprising Mathematical Mechanism Example Related Notes
The knowledge base
All notes
Browse the directory or narrow the list below.
Subject
86 entries
Algebra
02 Concepts · Exploration Map
Analysis
02 Concepts · Exploration Map
Compactness
Compactness turns a possibly infinite covering problem into a finite one.
Concept Name
Concept Name Idea Definition Intuition Canonical Example Counterexample Why It Matters Connections Questions Sources
Concepts
Durable standalone knowledge, organized by area. Extract an idea only when you want to return to it independently; link it to its exploration context.
Connections
Notes about bridges, not another collection of definitions. Use Connection when a relationship deserves its own explanation.
Continuous Images of Compact Spaces
Continuous maps carry compactness forward.
Experiment
Experiment Question Prediction Work Result What This Shows New Questions
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.
First Isomorphism Theorem
A homomorphism first identifies the elements it cannot distinguish, then embeds the resulting quotient into its target.
Geometry
02 Concepts · Exploration Map
Group Actions
An action translates the abstract multiplication of a group into transformations of a set.
Group Theory
A few connected starting points for studying groups.
Inbox
Capture a question, calculation, reference, or unfinished thought here. A new Markdown file is enough; metadata can wait.
Mathematical Curiosities
Small facts, surprising examples, observations, and questions. A short note is enough; not every curiosity needs a long explanation.
Mathematics Exploration Library
Mathematics A personal exploration of mathematical structures, ideas, connections, and curiosities. Exploration Map Month 01 — Modern Mathematics 00 Inbox 02 Concepts 03 …
Mathematics Exploration Library
A personal mathematical garden: explore why concepts exist, what they reveal, and how areas connect. This is not an exam checklist or a textbook summary. Ordinary Markdow…
Month 01 — From Familiar Mathematics to Modern Mathematics
Move from familiar undergraduate mathematics toward some of the organizing ideas of modern mathematics: constructions, symmetry, invariants, intrinsic geometry, and emerg…
Month NN — Exploration Theme
Month NN — Exploration Theme Purpose Organizing Ideas Explorations Reflections
Monthly Explorations
The chronological learning journey. Permanent ideas live in 02 Concepts and are linked from sessions rather than duplicated.
Normal Subgroups
Normality is the condition that lets multiplication descend from a group to a group of cosets.
Orbit–Stabilizer Theorem
How many distinct places can a point go? Count transformations, then account for those that leave the point unchanged.
Other
02 Concepts · Exploration Map
People and History
Follow the problems that made new concepts necessary. Keep people, sources, and mathematical developments connected to the ideas themselves.
Probability
02 Concepts · Exploration Map
Problems and Experiments
A place to calculate, prove, simulate, or visualize an idea. Begin with a prediction and keep failed attempts when they clarify the question.
Question
Question Observation Explanation Connections
Quotient Groups
A quotient group lets us regard elements as the same whenever they differ by an element of a chosen subgroup.
Reading
Keep references, useful chapters, talks, papers, and your own reading notes here. Link a source to the idea it helped you understand.
Session 01 — Adjoining a Root
How can we construct a number system in which has a root? has a root?
Session 01 — From Metric Spaces to Topology
Which parts of analysis survive when we retain open sets but discard numerical distance?
Session 01 — From Surfaces to Manifolds
How can a curved space be locally Euclidean without being globally a Euclidean space?
Session 01 — What Is Randomness Mathematically?
How does a mathematical model represent uncertainty without confusing outcomes, variables and distributions?
Session 02 — Charts and Atlases
How can several imperfect coordinate systems describe one coherent smooth space?
Session 02 — Law of Large Numbers
How can unpredictable individual outcomes produce a predictable average?
Session 02 — Strange Topologies
What can unusual choices of open sets teach us about convergence and continuity?
Session 02 — Why Quotient by a Polynomial?
Why does imposing the relation create a root, and when does it create a field? create a root, and when does it create a field?
Session 03 — Central Limit Theorem
Why do many different microscopic distributions produce the same shape of fluctuations?
Session 03 — Continuity Without Epsilon-Delta
Why is continuity expressed through inverse images of open sets?
Session 03 — Tangent Spaces
What is a tangent vector when there is no preferred ambient space in which to draw an arrow?
Session 03 — When Different Constructions Give the Same Field
When are differently presented number systems genuinely the same as fields?
Session 04 — Algebraic Numbers and Minimal Polynomials
How does one polynomial capture an algebraic element and determine the size of its field extension?
Session 04 — Combinatorics Behind the Gaussian
How can counting discrete paths reveal the bell-shaped profile of aggregate randomness?
Session 04 — Differential Maps and Pushforwards
How does a smooth map transport first-order motion from one space to another?
Session 04 — Homeomorphism
What does it mean for two spaces to have the same topology?
Session 05 — Connectedness
How can open sets express that a space cannot be split into separate pieces?
Session 05 — Cotangent Spaces and Differential Forms
Why should measurements of tangent vectors be treated as geometric objects in their own right?
Session 05 — Random Walks
What structure appears when simple independent choices are accumulated into a path?
Session 05 — Tower Extensions
How does the size of a field extension change when we adjoin several elements in stages?
Session 06 — Compactness
Why is the finite-subcover property the right abstraction of a manageable space?
Session 06 — Differential Forms and Integration
Why are alternating multilinear objects the quantities that can be integrated consistently over oriented spaces?
Session 06 — From Random Walk to Brownian Motion
How can a jagged discrete path approach a continuous random process?
Session 06 — Splitting Fields
What is the smallest field in which a polynomial reveals all of its roots?
Session 07 — Brownian Motion and the Heat Equation
Why does the probability distribution of random motion evolve according to a deterministic PDE?
Session 07 — Exterior Derivative
What operation generalizes gradient, curl and divergence while respecting the geometry of forms?
Session 07 — Field Automorphisms
Which movements of algebraic numbers preserve every field operation and fix the base field?
Session 07 — Quotient Spaces
How can gluing points produce a space with a rigorously defined topology?
Session 08 — Curvature
Which aspects of bending can be detected by someone who never leaves the surface?
Session 08 — From Automorphisms to Groups
Why do the symmetries of a field extension form familiar abstract groups?
Session 08 — Homotopy
When should two continuous maps be considered essentially the same?
Session 08 — Markov Chains
How can dynamics with no memory generate persistent long-term statistical structure?
Session 09 — Fundamental Group
How do loops become a group that records information about a space?
Session 09 — Geodesics
What replaces a straight line when straightness must be defined intrinsically?
Session 09 — My First Genuine Galois Group
How can we discover a polynomial’s symmetry group by constructing its splitting field?
Session 09 — What Is Entropy?
What properties should a numerical measure of uncertainty satisfy, and why do logarithms enter?
Session 10 — Fundamental Theorem of Galois Theory
Why should intermediate fields and subgroups encode the same information in opposite directions?
Session 10 — Generalized Stokes Theorem
Why can so many integration theorems be read as the same statement about a boundary?
Session 10 — Shannon Information
How does uncertainty become a statement about communicating and compressing messages?
Session 10 — Why the Circle Produces the Integers
Why does winding around a circle produce an integer rather than a more complicated invariant?
Session 11 — A Glimpse of Algebraic Topology
What becomes possible when spaces are studied through several kinds of algebraic invariants?
Session 11 — Geometry Meets Physics
Why do the structures used for intrinsic calculus also describe motion, fields and spacetime?
Session 11 — Statistical Mechanics Connection
Why do information theory and thermodynamics use closely related entropy expressions?
Session 11 — Why the Quintic Changed Mathematics
How can a question about formulas for roots become a question about groups?
Session 12 — Week 1 Synthesis
Can I reconstruct the path from a polynomial to its symmetries and the obstruction to radical formulas?
Session 12 — Week 2 Synthesis
Can I explain why open sets, gluing, deformation and algebraic invariants belong to the same story?
Session 12 — Week 3 Synthesis
Can I explain how local coordinates give way to intrinsic calculus, curvature and boundary laws?
Session 12 — Week 4 Synthesis
Can I connect averaging, fluctuations, random paths, diffusion and information without conflating their different limits?
Session Title
Session Title Central Question Why This Matters Learning Prompt Ideas Examples Questions Connections One Thing That Surprised Me Further Rabbit Holes
Topology
02 Concepts · Exploration Map
Week 01 — Field Extensions and Galois Theory
Follow the need to construct new number systems into the symmetries of polynomial roots and the obstruction to radical formulas.
Week 02 — Topology
Strip away measurements, keep continuity, and then use deformation and algebra to ask what a space is really like.
Week 03 — Geometry and Manifolds
Reorganize multivariable calculus around local descriptions, intrinsic derivatives, forms, and the geometry added by a metric.
Week 04 — Probability Information and Emergence
Follow aggregate randomness through limiting laws and diffusion, then ask how uncertainty connects with information and physical equilibrium.
Week NN — Theme
Week NN — Theme Central Question The Story Sessions Connections Beyond This Week
What a kernel remembers
A homomorphism's kernel records exactly which elements become indistinguishable from the identity.
No entries match these filters. Try another idea or reset the filters.