Reading Paths

Lie theory

From root angles to the Dynkin list

Follow the classification from rank-two integrality through positive-definite graph forms to the exceptional boundary.

1
The graphical alphabet
Crystallographic integrality restricts pairwise angles, edge multiplicities, and root-length ratios.
2
The positive Cartan form
Symmetrization converts the root problem into a constraint on principal minors and adjacency eigenvalues.
3
The ADE inequality
A Schur complement reduces every three-armed tree to one reciprocal inequality.
4
The affine wall
Equality produces positive null vectors and explains why extending E8 fails.
5
Double and triple edges
Leading principal minors force B, C, F4, and G2.
6
Existence and reconstruction
Coordinate models build every survivor, after which reflections recover the full root system.

Algebraic geometry

Follow a tangent intersection from its coordinate ring to infinitesimal geometry and arithmetic families.

1
The lost multiplicity
The Nullstellensatz identifies the exact information erased when an ideal is replaced by its zero set.
2
The affine machine
Prime ideals become points, localizations become functions on opens, and the structure sheaf restores local algebra.
3
Contravariance and intersection
Ring maps reverse into geometric maps, turning tensor products into fiber products and scheme-theoretic intersections.
4
Infinitesimal probes
Dual-number-valued points recover derivations, tangent spaces, and first-order deformation data.
5
Points as families
The functor of points changes resolution by changing the test ring, and Yoneda says no information is lost.
6
Geometry over a base
Fibers over Spec Z place characteristic zero, reduction modulo primes, and bad reduction in one family.