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First steps started with his education.
Not the best in P.E or music, Henri excelled in math department that interested him enough to go a prestige university, where he received his doctorate's degree in Mathematics because of his new theories in math involving resonance of eigenvalues, and algebroid functions. He was appraised for this and almost immediately became a professor at the University of Caen where he taught Mathematical Analysis.
Resource:
https://mathshistory.st-andrews.ac.uk/Biographies/Poincare/ -
The Qualitative Theory and the Chaos equation
One of Henri's theory, The Qualitative theory, was able to extract info from insoluble equations. This later then helped with celestial mechanic's, and was able to help prove the solar system was stable. Later, he discovered the 3-body problem, but had issues due to it being sensitive to conditions. This then became the basis of the Chaos Theory, or the butterfly effect. Resource:
https://www.famousscientists.org/henri-poincare/ -
Topology / Poincare Conjecture
Poincare was responsible for leaving a mathematic question that was big for almost a century, and that was the challenge of describing our 3D universe. Using abstract algebra, he tried to investigate Topology in space to explain how the universe is shaped. This question was later solved in 2002 by Grigori Perelman.
Resource: https://www.storyofmathematics.com/19th_poincare.html Short video: https://www.youtube.com/watch?v=YPuOt0JUZBY -
The Relative Theory Runner-up
Before Einstein's theory, Poincare had a theory of his own. At a talk, he gave his theory that is similar to Einstein's theory of Relativeness. The key point that crumbled his theory was that in his version of Relative theory, is that the relativity is the end result of another force due to electromagnetic inertia. While this was close, Einstein later said that everything has an equal, but opposite reaction, instead of another force,
Resource:https://www.naturelovesmath.com