Named after the Irish physicist, astronomer, and mathematician W. R. Hamilton (1805-1875), Hamiltonian systems are an important class of dynamical systems with certain conservation laws and rigidity features. A well-known classical example is the n-body problem (`movement of the planets around the sun'). In fact, Hamiltonian systems appear in many shapes throughout mathematics, physics, chemistry, biology, and engineering.
Classical Hamiltonian problems are formulated as systems of ordinary differential equations on finite dimensional spaces. Nevertheless, there are also equations that can be reformulated as Hamiltonian systems, but this time on infinite dimensional spaces. Such systems are called Hamiltonian partial differential equations, in short Hamiltonian PDEs. Examples are the Korteweg-de Vries equation, the Sine-Gordon equation, the nonlinear Schrödinger equation, nonlinear sigma models etc.
This project starts out from a `triholomorphic' Dirac-type equation on a so-called hyperkähler manifold. It can be transformed it into a Hamiltonian PDE on the infinite dimensional loop space of the manifold. For this new equation, we expect to show conservation laws, integrability (`extra symmetries'), features from modern symplectic geometry (`non-squeezing' properties, symplectic capacities etc.), and estimates on the number of periodic solutions (`infinite dimensional Arnold conjecture').
So far, only three people studied Hamiltonian PDEs with modern symplectic methods. Hamiltonian PDEs with hyperkähler background having symplectic features, as in this project, have never before been investigated. Due to its link to the Cauchy-Riemann-Fueter equation in the supersymmetric sigma model, our results may also be of interest to physicists.