In 2019 Niek started his research as a PhD student with Dr. Oliver Fabert, extending the celebrated use of pseudoholomorphic curves from symplectic geometry to the setting of field equations.
Pseudoholomorphic curves were introduced in the 80s by the famous mathematician Mikhail Gromov and in the following years were used by Andreas Floer to prove a conjecture by Vladimir Arnold about the number of periodic solutions that a Hamiltonian system (at least) must have. Such systems describe the dynamics of a classical physical system and one is interested periodic motions. Rather than counting periodic solutions of such dynamical systems directly, one can count how many pseudoholomorphic curves exist that connect two such periodic solutions. This has proven to be a very fruitful way of viewing such counting problems.
Since then the field has developed into many different directions and the techniques have been used in different areas of mathematics and physics. However, the techniques were only applied to finite-dimensional Hamiltonian systems, describing the motion of some object (a particle, or planets) through space. Infinite-dimensional Hamiltonian systems on the other hand describe the evolution of fields, such as the Maxwell equations describing the behavior of electromagnetic fields.
During his PhD, Niek worked together with his supervisor to develop a way in which the pseudoholomorphic curve techniques could be used to prove the existence of periodic solutions of a class of infinite-dimensional Hamiltonian systems describing, for example, the behavior of a charged particle coupled with an electromagnetic field. This research combines the field of symplectic geometry, from which the pseudoholomorphic curve techniques stem, with the field of partial differential equations.
Currently Niek works with Dr. Fabert to include a more general class of equations in the aforementioned results. Developing similar techniques but from another point of view, namely that of polysymplectic geometry, Niek also works together with his supervisor and Ronen Brilleslijper, Dr. Fabert’s other PhD student, to prove similar results about the number of periodic solutions, to another class of dynamical systems.
Niek’s thesis is available from this link: https://research.vu.nl/en/publications/pseudoholomorphic-curve-methods-for-infinite-dimensional-hamilton