Math Mapgenerated 2026-10-25 · a354234
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EX-02

the model compact group — U(1); Fourier analysis and Pontryagin duality at their self-dual, rank-one base case

exemplar_kind
object
state
current
created
2026-09-15
topic
algebra
source
../exemplars/EX-02-circle-as-compact-group.md

Coordinatesthe ground of the symmetric-spaces / dual-group axis: compact type, rank one, self-dual under Pontryagin duality

Model forcompact Lie groups, Pontryagin self-duality, U(1) gauge symmetry, periodicity / Fourier series

The circle is the minimal compact group and the home for everything periodic: its representation theory is Fourier series, and it is its own Pontryagin dual. When a problem carries a compact symmetry, the circle is the case understood completely — the one to specialize to for what "compact" buys, before noncommutativity or higher rank arrive.

generated 2026-10-25the lab index