Symplectic geometry rant part 3:
Recap of previous parts --
a) Symplectic geometry is the geometry of optimization, of light, of caustics, of the large-scale structure of the universe
b) Optimal variations of paths give complex analytic surfaces, and a mirror world of algebra (I could rant for a long time about b -- about translating problems from one side to the other, etc.)

Part 2 of my symplectic geometry rant.
Previously: "Symplectic geometry is the geometry of optimization". The intrinsic geometry of the Legendre transform. Area & "symplectic size" is about variation of the function we are optimizing over the space of paths.

But then: GROMOV. As I said the basic objects are paths over which we optimize functionals. Area is about the variation of this functional. Good boundary conditions give special objects (Lagrangians, Legendrians) and describe caustics. Q: What is the **optimal variation**?

A thread Twitter recently asked `what is symplectic geometry about'. I am pretty sure that this is something that people who actually work on the field ask themselves repeatedly. I certainly was confused by this Q as a grad student.
@tarunchitra @chessapig @taz_chu There are not many fundamental `geometries', and each geometry corresponds to a meaningful aspect of the world. Thus Riemannian geometry is the geometry of distance, etc.

Symplectic geometry is the *geometry of the calculus of variations*.