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River of knowledge

The River of Topology

What remains of shape after distance and angle disappear?

From Euler's bridges through Riemann, Poincaré, to Perelman's proof — chasing the essence of shape.

1736–20034 city-to-city segments

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What changed between the cities?

  1. 1

    1736–1750

    scholars and schools

    Saint PetersburgBerlin

    Euler — Bridges of Königsberg

    Euler proved in 1736 that no walk crosses each of Königsberg’s seven bridges exactly once, launching graph theory and an early form of topology.

    Solutio problematis ad geometriam situs pertinentis
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  2. 2

    1854–1880

    scholars and schools

    GöttingenParis

    Riemann — Manifolds and Curvature

    In his 1854 habilitation lecture, Riemann developed a general theory of higher-dimensional manifolds.

    Riemannian manifolds
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  3. 3

    1895–1912

    scholars and schools

    ParisPrinceton

    Poincaré — Analysis Situs and the Conjecture

    Poincaré founded algebraic topology and posed his conjecture in 1904, beginning a century-long search.

    Analysis Situs and the Poincaré conjecture
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  4. 4

    2002–2003

    scholars and schools

    PrincetonSaint Petersburg

    Perelman — Proof After 100 Years

    Perelman in Saint Petersburg proved the Poincaré conjecture using Ricci flow, then declined both the Fields Medal and a million-dollar prize.

    Three papers on Ricci flow posted to arXiv
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How to read the lines

Each line is an editorial route through problems, texts, and practices. It does not imply one book moving in a straight line, a sole invention, or identical adoption everywhere.