Principal Investigator: Weimin Chen
Abstract: This project will center on three topics in symplectic topology and geometry. In the first part the principal investigator (joint with Y. Ruan) will establish the Gromov-Witten invariants for symplectic orbifolds. This work aims at computing the Gromov-Witten invariants of a symplectic manifold by decomposing it into two pieces. Such an operation will in general introduce orbifold singularities, so the establishment of Gromov-Witten invariants for symplectic manifolds with orbifold singularities will lay the foundation for a program of computing Gromov-Witten invariants, which has great potential applications in symplectic topology as well as other related fields such as birational geometry. Other motivations come from mirror symmetry and string theory of theoretical physics, in which there is a demand to consider spaces with controlled singularities such as orbifold singularities. In the second part of this project the principal investigator will continue his work on the 3-dimensional Reeb dynamical systems by exploiting a potential connection between Seiberg-Witten Floer homology and the contact homology. The last part concerns symplectic structures on smooth 4-manifolds with vanishing second homotopy group. Currently very less is known about them.
In recent years, one has witnessed some great interactions between different branches of mathematics in the area of geometry and topology, with the input of ideas from theoretical physics. This project seeks to exploit the intimate interplay between these different fields to investigate some very interesting or fundamental questions in such areas as quantum cohomology, birational geometry, the existence of periodic orbits of Reeb dynamics on a 3-dimensional space, and symplectic 4-manifolds. The principal investigator believes that a successful outcome of this project will make a significant advancement of knowledge in the areas listed above, which are important not only within mathematics but also in real life problems. For example, the importance of Reeb dynamics is seen in the following example. The motion of a satellite in the presence of the gravitational forces of the sun, the planets and the moon is described mathematically as a Reeb dynamics. The relevant part of this project aims at solving the 20-year-old Weinstein conjecture for Reeb dynamical systems on a 3-dimensinal space, which is one of the most important conjectures in the field.
