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.
The time evolution of a physical system is very often described by a differential equation, which may be finite-dimensional or even infinite-dimensional. Among these differential equations the so-called Hamiltonian systems play an important role, and occur very frequently for problems arising in physics. Due to some particular preserved quantity these systems have more structure and are related to an interesting geometry called "Symplectic Geometry". Unlike the usual geometries, which have as basic notions length and distance, the main concept of Symplectic Geometry is area. Symplectic Geometry provides a tool to study an important class of dynamical systems by geometric means. Since the beginning of the eighties, the understanding of symplectic geometry accelerated in an unprecedented way. Also it became clear that there are relations between symplectic geometry and string theory, and to other fields in mathematics like algebraic geometry. One of the basic ideas in studying questions in symplectic geometry is that certain classes of surfaces in an otherwise high-dimensional space should play an important role. These surfaces are in particular minimal surfaces, which are two-dimensional surfaces, which locally have the smallest area possible. The occurrence of minimal surfaces, given the fact that the geometry has area as its basic notion is very natural. Many of the questions in symplectic geometry can be reduced to a counting of the number of such minimal surfaces in "good position". It is quite involved to give an accurate definition of what "good position" means. However, it can be shown that in most situations a good position does not exist. To overcome this problem the idea is to study a larger class of problems where counting still leads to the desired results and where in all situations a "good position" can be achieved. Here the PI and his collaborators introduce a new theory, called "Polyfold Theory" which overcomes these problems. Polyfold Theory leads to a generalization of differential geometry, in particular it is possible to develop calculus on spaces with varying dimensions. It also generalizes what is called "Nonlinear Fredholm Theory", and leads to a new method for studying nonlinear equations on rather badly behaved infinite-dimensional spaces. As an end result one ends up with a method and "tool box" for studying problems in symplectic geometry. Another part of the project was concerned with combining the new tools with problems arising in the study of Hamiltonian systems. It led to new methods allowing to solve known and longstanding questions for which up to now any workable idea seemed elusive. This approach has contributed heavily to the creation of a new field called "Symplectic Dynamics" which studies problems in Hamiltonian Dynamics with an integrated viewe point of dynamical systems and symplectic geometry ideas.
