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Seminar on the History of Mathematics
September 7, 2017 18:00, St. Peterburg
 


Poincares last geometric theorem: history and drama of ideas

A. N. Kirillov

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A. N. Kirillov



Abstract: 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), 375407), known as Poincarés 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 Poincarés 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 Poincarés reasoning, but the existence of the second fixed point was not correctly justified. The dramatic history of the proof and extensions of the Poincarés last geometric theorem is traced in the report.

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