Steady fluid flows and symplectic geometry

Filtering Techniques for Complex Geometry Fluid Flows

We develop a class of lters based upon the numerical solution of high-order elliptic problems in l R d that allow for independent determination of order and cut-oo wave number and that default to classical Fourier-based lters in homogeneous domains. However, because they are based on the solution of a PDE, the present lters are not restricted to applications in tensor-product-based geometries a...

Symplectic Geometry

1 Symplectic Manifolds 5 1.1 Symplectic Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Symplectic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Cotangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Moser’s Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Darboux and Moser Theorems . . . . . . . . . . . . . . . ....

Symplectic Geometry

Symplectic Geometry

Symplectic Geometry

These notes are written for a ten week graduate class on symplectic geometry. Most of the material here is included in Michèle Audin’s book Torus actions on symplectic manifolds, which I used heavily in preparing these notes. There are a few topics that we cover that are not in Audin’s book, such as polygon spaces, cohomology of flag manifolds, and CS/GKM theory. More significant, however, are ...