The Principal Investigator plans to continue his work in gauge theory and symplectic, low-dimensional topology. More concretely, he plans to study the following problems: (1) general properties of finite groups of symplectomorphisms of a symplectic 4-manifold, (2) whether a symplectic circle action on a 6-dimensional manifold is Hamiltonian if its fixed-point set is non-empty and finite, and (3) general, global restrictions on the occurrence of quotient singularities in an algebraic surface. A unifying theme in these studies is a theory, yet to be developed in this project, on the Seiberg-Witten and Gromov invariants of a symplectic 4-orbifold, which will be built on and substantially extend the related fundamental work of Taubes on symplectic 4-manifolds.
A 4-dimensional orbifold is a space which locally looks like the space-time we all live in, except that there is a certain degree of ambiguity caused by a finite symmetry occured locally. Many examples of this kind of space can be found easily and naturally in Mathematics (e.g. topology and geometry) as well as models in Theoretical Physics (e.g. orbifold string theory). The proposed research seeks to understand a fundamental duality in a 4-dimensional orbifold between a certain type of fields that are distributed all over the space and a certain type of worldsheet that is wiped out by a collection of strings in the space which is condensed in a 2-dimensional subspace. When there is no such ambiguity of finite symmetries, such a duality is well-understood, and is a fundamental piece of mathematics.
