The principle of least action is an elegant mathematical model for
a wide range of phenomena in physics. From a very limited number of
concepts, it mathematically implies conservation laws (energy, momentum,
rotational momentum), Newton laws (F=ma), and is the starting point of Euler's mathematical modeling of fluids.
After a brief introduction, I will give an overview of some of its connections
with different domains, including mathematics (optimal transport theory)
[Brenier], [Benamou and Brenier], computational geometry (power diagrams)
[Aurenhammer Hoffman Aranov], [Alexandrov], [Merigot], [L] and new
methods for computational physics [Merigot], [Desbrun]. I will also present some
efficients algorithms to solve the Monge-Ampère equation, a fundamental
building block for this family of numerical methods [Jordan, Kinderlehrer, Otto], [Merigot] that may also have applications in geometry processing / shape analysis / shape correspondence.