Principal Investigator: Helmut Hofer
The field of symplectic geometry has a large interface to other mathematical disciplines, like algebraic geometry, differential topology (particularly in small dimensions) and dynamical systems to name a few. Dr. Hofer's project is concerned with the study of fundamental aspects of symplectic geometry, its applications to dynamical systems, as well as the development of mathematical technology to address analytical problems arising in the field. One part of the project is devoted to the study of Symplectic Field Theory (SFT) which currently is the most general and most comprehensive theory of symplectic invariants. Another part develops a general approach for studying certain classes of nonlinear elliptic partial differential equations as they occur in SFT. These methods potentially should have other applications in nonlinear analysis as well. A third part is devoted to the applications of the theory to dynamical systems. The aim is the development of mathematical infrastructure, based on a combination of Floer theory and the theory of finite energy foliations due to Dr. Hofer and his collaborators. This research aims at the understanding of long-term behavior of iterated area-preserving disk maps with its numerous applications.
Many physical systems like the flow of an incompressible ideal fluid, the movement of a satellite under the gravitational forces of celestial bodies, or the movement of charged particles in a magnetic field, to name a few, are examples of so called dynamical systems. The mathematical theory of dynamical systems provides tools to understand their complex behavior and allows to make predictions. The particular examples mentioned above are of so-called Hamiltonian nature and have an intricate structure leading to extreme complicated dynamical behavior. Stabilizing a beam of particles in a partic= le accelerators, or sending a probe on an interstellar journey, or understanding the dynamics of a stationary flow of an incompressible ideal fluid are problems whose mathematical underpinnings are touched by the research proposed in this project. Some of the methods developed potentially have application to larger classes of partial differential equations of relevance in physics.
