All lessons

What survives a change of representation?

Number & structure

From quantity to rules, symbols and proof. No assumed comfort with notation.

  1. 01 · What is a number?
  2. 02 · Why enlarge the number system?
  3. 03 · Sets: the grammar of selection
  4. 04 · Functions and what they forget
  5. 05 · From seeing a pattern to proving it

Space & transformations

Vectors, independence, lost information and the signatures of quadratic forms.

  1. 06 · Coordinates are not the arrow
  2. 07 · Independence: when a choice adds something
  3. 08 · A matrix is an action
  4. 09 · Rank and the information that disappears
  5. 10 · Inner products and projection
  6. 11 · Eigenvectors and the sign of a geometry

Combinatorics & matroids

Count possibilities, find cycles, and discover one structure behind graphs and matrices.

  1. 12 · Counting without listing everything
  2. 13 · Graphs: connection and redundancy
  3. 14 · The first exact bridge: graph → matrix
  4. 15 · Matroids: independence without coordinates
  5. 16 · Bases, circuits, rank and flats
  6. 17 · Graph coloring becomes a polynomial
  7. 18 · The lattice and characteristic polynomial

The geometric bridge

Log-concavity, mixed areas, polytopes, a real Chow-ring calculation and the Hodge argument.

  1. 19 · Log-concavity: why a sequence forms a hill
  2. 20 · Mixed area: a geometric inequality you can prove
  3. 21 · Matroid polytopes: selections become points
  4. 22 · Rings: a language for combining constraints
  5. 23 · Build one actual Chow ring
  6. 24 · The Hodge bridge: signature → inequality

Patterns & intelligence

Latent states, harmonic motion, compression, inference and testable claims about intelligence.

  1. 25 · Hidden states and observable shadows
  2. 26 · Harmony as a path, not a frozen shape
  3. 27 · PCA and the geometry of compression
  4. 28 · Probability: uncertainty is not geometry alone
  5. 29 · What would “intelligence in the patterns” mean?

Three small worlds

Capstones in graphs, geometric inequalities and hidden-state reconstruction.

  1. 30 · Capstone I · One triangle, five languages
  2. 31 · Capstone II · Why the four-cycle has a hill
  3. 32 · Capstone III · Recover the hidden world