His three major contributions to math/physics are the Poincare conjecture, the three body problem, and his work on relativity.
- The Poincare conjecture is about something called a closed 3-d manifold, which is a space that locally looks like normal 3-d space, but it's all connected, finite, and has no boundary. The theorem says that if the space has the property that if you draw a loop at any point and you tighten the loop, it converges to a point (see image), then the space is just a sphere. A Russian mathematician named Grigori Perelman recently proved this conjecture, and was offered the Fields medal (most prestigious prize in mathematics) but refused to accept it (he is very weird and I think he lives with his mother). Solving this problem was also one of the Millennium Prizes, and it is unclear whether Perelman will receive any of the one million dollar prize money for solving it.
- The three body problem is just Newton's laws of gravity applied to three objects, which is an extremely difficult problem to solve mathematically without a computer. Poincare did not solve the entire problem, but his work on it had many ideas which lead to the development of chaos theory.
- Poincare worked on relativistic transformations with a guy named Lorentz (who, for anyone who knows about transformations, is the one whose name is on the equations) on time dilation and reference frames, etc. All this helped Einstein develop his theory of special relativity not too long after (Einstein was working on his stuff right around the same time, but was much younger than Poincare).
Anyway, sorry to those of you who don't care about this stuff, but I think it's cool. This guy's name comes up a lot and I just wondered what he did. Hope it wasn't too boring! :)