Mathematics 797DS - ST-InfiniteDimensnalIntegrlSys
Spring
2020
01
3.00
Franz Pedit
M W 2:30PM 3:45PM
UMass Amherst
51162
Lederle Grad Res Tower rm 1114
pedit@math.umass.edu
Hitchin once said that there is no real definition of what an infinite dimensional integrable system should be, but if you encounter one, you know it is one.
This being said, there are a number of descriptive aspects: infinite hierarchies (e.g. KdV, KP, non-linear Schroedinger, etc.) of flows and functionals whose symplectic gradients are those flows; loop Grassmannians; actions of loop groups and loop algebras; zero curvature equations; Lax pairs; spectral curves and linear flows on Jacobians/Prim varieties etc. The objective of the course is to discuss the interrelations among those various aspects by studying some of the classical examples such as KdV or non-linear Schroedinger. A novelty (when compared to the existing literature) is the geometric approach: the (pre)symplectic (or Poisson) manifolds will be spaces of geometric objects, such as manifolds of curves or surfaces in a certain target manifold rather than function spaces.
Prerequisites:
basic knowledge of manifolds, symplectic geometry, Lie algebras, Riemann surfaces.
This being said, there are a number of descriptive aspects: infinite hierarchies (e.g. KdV, KP, non-linear Schroedinger, etc.) of flows and functionals whose symplectic gradients are those flows; loop Grassmannians; actions of loop groups and loop algebras; zero curvature equations; Lax pairs; spectral curves and linear flows on Jacobians/Prim varieties etc. The objective of the course is to discuss the interrelations among those various aspects by studying some of the classical examples such as KdV or non-linear Schroedinger. A novelty (when compared to the existing literature) is the geometric approach: the (pre)symplectic (or Poisson) manifolds will be spaces of geometric objects, such as manifolds of curves or surfaces in a certain target manifold rather than function spaces.
Prerequisites:
basic knowledge of manifolds, symplectic geometry, Lie algebras, Riemann surfaces.
https://spire.umass.edu