From Axioms to Open Questions

The starting premise every system needs and cannot itself prove; Euclid's postulates are the oldest surviving systematic use.

Built to handle the sets that defy plain intuition, after Vitali's 1905 construction showed some subsets of the real line cannot be measured at all.

The theorem that made calculus one subject instead of two; who first fully grasped it is the wave's own Newton-Leibniz dissent.

Worked out the method of fluxions from 1666 but published only decades later, then presided over the Royal Society inquiry that found in his own favor.

Published first, in 1684, and gave calculus the notation still used today; the modern consensus credits both men with an independent invention.

Ties the shape of physical law to the geometry of symmetry: every continuous symmetry a system has yields a quantity conserved in time.

Proved the theorem in 1918 while barred, as a woman, from an ordinary paid professorship at Gottingen for years afterward.

The best-known of the seven Millennium Prize Problems, open since 1859, about where the zeros of a single complex function fall.

A question this atlas shares with Computing: whether every quickly verified answer is also quickly findable.

The trail's last stop: a conjecture linking the solutions of certain equations to the behavior of an associated function, still unproved.