Everything here, in the order it is built
Each subject holds films and documents on the same topics — the film opens the question, the document settles it. The tiers below are a dependency order, not a difficulty ranking: later subjects assume the earlier ones, and nothing assumes anything outside this page.
Foundations
What mathematics is made of, and what counts as knowing something.
Logic & Proof
What makes an argument binding?
4 films · 4 documents · 3 problem sets
- Statements›
- Quantifiers›
- Methods of proof›
Set Theory
How big can infinity get?
5 films · 5 documents · 4 problem sets
- Sets›
- Cardinals›
- Ordinals›
Number
What has to exist between the marks?
6 films · 6 documents · 5 problem sets
- Naturals & integers›
- Rationals›
- The real numbers›
The Continuous
Everything that follows from taking limits seriously.
Analysis
What does it mean to approach without arriving?
9 films · 9 documents · 7 problem sets
- Sequences›
- Continuity›
- Derivative & integral›
Measure
Can every set be assigned a size?
6 films · 6 documents · 5 problem sets
- Outer measure›
- The Lebesgue integral›
- Measurable functions›
Complex Analysis
Why are complex functions so much better behaved?
5 films · 5 documents · 4 problem sets
- Holomorphic functions›
- Contour integration›
- Residues›
Structure
What is left when you stop asking how much and start asking how it fits together.
Algebra
When does structure reveal more than quantity?
8 films · 8 documents · 6 problem sets
- Groups›
- Rings›
- Fields›
Linear Algebra
What survives a change of viewpoint?
7 films · 7 documents · 6 problem sets
- Vector spaces›
- Linear maps›
- Eigenvalues›
Number Theory
Why are the simplest questions the hardest?
7 films · 7 documents · 6 problem sets
- Divisibility›
- Congruences›
- Distribution of primes›
Space
Geometry after distance stops being the point.
Topology
What is preserved when distance is forgotten?
7 films · 7 documents · 5 problem sets
- Open sets›
- Compactness›
- Connectedness›
Geometry
What if the parallel postulate is false?
5 films · 5 documents · 4 problem sets
- Euclid's axioms›
- Non-Euclidean space›
- Curvature›
Manifolds
When does curved space look flat?
6 films · 6 documents · 5 problem sets
- Charts & atlases›
- Tangent spaces›
- Differential forms›
The Limits
Where mathematics turns on itself and asks what it cannot do.
Incompleteness
Can a system prove its own consistency?
4 films · 4 documents · 2 problem sets
- Formal systems›
- Gödel numbering›
- The two theorems›
Category Theory
What if the maps matter more than the objects?
In preparation — needs Algebra and Topology first
Probability
How do you measure what has not happened?
In preparation — follows Measure