Poincare’s last geometric theorem: history and drama of ideas

H. Poincaré in 1912, the last year of his life, studying the three-body problem, published an unproved theorem (H.Poincaré. Sur un théorème de géométrie. Rend. Circ. Mat. Palermo, 33 (1912), 375–407), known as “Poincaré’s last geometric theorem.” Roughly speaking, it asserts that an area-preserving homeomorphism of the planar circular annulus onto itself admits at least two fixed points if the points of the boundary circles are advanced by this homeomorphism along these boundaries in opposite angular directions. Poincare proved the theorem in special cases. He expressed the hope that mathematicians would be interested in this result. The Poincaré’s hope was justified. Till now his theorem is the source of many interesting results in the theory of dynamical systems and topology. G. Birkhoff was the first who responded to the appeal of Poincaré. In 1913 he proved the theorem using his ingenious method, different from Poincaré’s reasoning, but the existence of the second fixed point was not correctly justified. The dramatic history of the proof and extensions of the Poincaré’s last geometric theorem is traced in the report.