The Mathematics of Emmy Noether: symmetry and conservation in physics, and finiteness in algebra.
In 1918 Emmy Noether proved that every continuous symmetry of a Lagrangian comes with a conserved quantity. Because the laws of motion are the same from one moment to the next, energy is conserved; because they are the same from place to place, momentum is. A few years later she rebuilt commutative algebra around a single finiteness condition on ideals.
The series has three acts. Act I builds the tools: group actions, invariants, Lagrangians and the first theorem. Act II runs the theorem on energy, momentum, angular momentum, the Kepler problem and electric charge, and ends with her second theorem. Act III follows her algebra through ideals, Noetherian rings, Hilbert's basis theorem and Noether normalization.
Group actions, orbits and stabilizers, the D4 Cayley table, a Burnside count of colourings, and symmetric polynomials reduced step by step.
How the discriminant and the quartic invariants I and J scale by a power of the determinant under substitution, the question Noether's 1907 dissertation studied, checked in exact arithmetic.
A bead race to the cycloid, a knotted path relaxing onto the true motion, and the Euler-Lagrange equation that follows; least action is sometimes only a saddle.
Continuous symmetries, infinitesimal generators, and the change δL that the theorem is built on.
Five lines of calculus that turn a continuous symmetry into a conserved quantity, checked numerically.
Time shifts, space shifts and rotations; a kicked pendulum, a driven oscillator and a spinning hoop, and how to tell integrator drift from physics.
The Laplace-Runge-Lenz vector, its precession under a perturbation, the SO(4) bracket algebra of bound orbits, and the degeneracies of hydrogen.
Phase rotation of a Schrödinger field gives conserved charge, checked as current through the walls of a box; making the phase local forces a gauge field on the lattice.
Local symmetries give identities instead of conservation laws: gauge on a lattice, Maxwell's identity on a grid, underdetermined motion, and why energy in GR is subtle.
Unique factorization fails in ℤ[√−5]; Dedekind restores it by factoring ideals.
Rings where every ascending chain of ideals stabilizes: chains in ℤ and k[x], a monomial staircase that must close, and a ring of all algebraic integers where the chain never stops.
If R is Noetherian, so is R[x]. Gordan's "theology" complaint, and Buchberger's algorithm, a later constructive tool for ideals that halts by the same chain condition.
Every affine variety is a finite cover of affine space; fiber counts over sampled base points, a substitution that stops roots escaping, and dimension read from Hilbert-function growth.