Poincare founds algebraic topology

In 1895 the French mathematician Henri Poincare published Analysis Situs, Latin for analysis of position, followed by five supplements through 1904. It treats shapes by the properties that survive stretching, twisting or bending without tearing. A coffee mug and a doughnut count as the same shape in this sense because each has one hole, while a sphere differs. Poincare built tools to count and compare such features, including the fundamental group, which records loops in a space that can or cannot be shrunk to a point, and a systematic form of homology that generalized the hole counts of Enrico Betti.¹

The ground had been prepared. Euler's 1736 solution of the Konigsberg bridge problem and his formula linking the vertices, edges and faces of a polyhedron showed that connection patterns can matter more than distances. Riemann's work in the 1850s on surfaces with many holes and on spaces of many dimensions supplied the objects to classify. Poincare was also pushed by his own research on differential equations and the three-body problem of celestial mechanics, where understanding the overall shape of the space of possible motions was essential.

The fifth supplement to Analysis Situs, in 1904, ended with a question that became the Poincare conjecture: must a closed three-dimensional space with no holes be a three-dimensional sphere? It stood for a century. Grigori Perelman's proof, posted in 2002 and 2003 and building on Richard Hamilton's Ricci flow, was accepted, and the Clay Institute offered him its Millennium Prize in 2010, which he declined. Algebraic topology now underlies much of geometry and number theory, and it feeds into modern physics and data analysis.¹²