Morse theory generalizes calculus of variations by relating the stationary points of a smooth real-valued function on a manifold
to the global topology of the manifold. For example,
if a compact manifold admits a function whose
only stationary points are a maximum and a minimum, then the manifold is a sphere.
Technically speaking, Morse theory applied to a function 
on a manifold 
with 
and 
shows that every bordism
can be realized as a finite sequence of surgeries. Conversely,
a sequence of surgeries gives a bordism.
There are a number of classical applications of Morse theory, including counting geodesics on a Riemann surface and determination of the topology of a Lie group (Bott 1960, Milnor 1963). Morse theory has received much attention in the last two decades as a result of the paper by Witten (1982) which relates Morse theory to quantum field theory and also directly connects the stationary points of a smooth function to differential forms on the manifold.
