Morse theory and Floer homology

Morse homology studies the topology of smooth manifolds by examining the critical points of a real-valued function defined on the manifold, and connecting them with the negative gradient of the function. Rather surprisingly, the resulting homology is proved to be independent of the choice of the real-valued function and metric defining the negative gradient. This leads to a topological lower bound on the number of critical points. In the 1980s, the construction of Morse homology served as a prototype to define a homology spanned by $1$-periodic Hamiltonian diffeomorphisms on symplectic manifolds. The resulting homology, introduced by Andreas Floer, spectacularly revolutionized the area of symplectic topology and leaded to a proof of the famous Arnold conjecture. Floer theory still is the subject of a lot of active and exciting research and is nowadays an essential technique in symplectic topology.