Jules Henri Poincaré

1. Poincaré Recurrence Theorem

States that certain systems will eventually return to a state very close to the initial one.

$$ T^n (A) \cap A \neq \emptyset $$

2. Poincaré-Hopf Index Theorem

Relating vector fields to the Euler characteristic.

$$ \sum_i \text{index}_{x_i}(v) = \chi(M) $$