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Qualitative Theory of Differential Equations
Through a series of memoirs, Poincare developed an entirely new field of mathematics called "qualitative theory of differential equations". Through these memoirs, he showed that even if a know solution for a differential equation cannot be obtained, a large amount of information about the equations properties and fundamental behavior can be. -
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Development of Chaos Theory
Through his attempted solution of the 3-body problem, a classical problem in the field of orbital mechanics, Poincare developed the basis for mathematical "chaos theory".
Gray, Jeremy John. “Henri Poincaré.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., 25 Apr. 2019, www.britannica.com/biography/Henri-Poincare. -
Poincare Conjecture
Poincare establish a theorem characterizing a 3-sphere that later became a millennium prize problem. Of the 7 Millenium Prize Problems, it is the only one that has been given a definitive solution to this day. Providing a correct solution to any of the listed prizes awards $1,000,000USD (and a lot of fame to go with it).
Milnor, John. “THE POINCARE CONJECTURE .” Clay Mathematics Institute, 2004, www.claymath.org/sites/default/files/poincare.pdf. -
Poincare Writes a paper on the motion of the electron
Poincare wrote a paper analyzing the motion of the electron. This was his primary achievement in the field of mathematical physics. His findings seemed to directly contradict Isaac Newton's laws of mechanics (Poincare addressed this himself). He was very close in theory to Einstein's theory of special relativity but never combined space and time and never made the same assumptions.
"Translation:On the Dynamics of the Electron (July)." Wikisource . Wikisource , 9 Dec. 2013. Web. 26 May. 2019. -
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Attempted Solutions of the Poincare Conjecture
From the early 1930's until it's ultimate solution in 2002 and 2003, the Poincare Conjecture was broken down and heavily analyzed by many scientists and mathematicians in an attempt to provide a meaningful solution. The final solution presented by Grigori Perelman, written among 3 separate papers, was made up of bit and pieces of many of the attempted solutions that came before him. Through them proving themselves wrong, he was able to combine much of the already existing correct information.
