RESEARCH STATEMENT

My research lies in the general mathematical area of geometry and topology, and more specifically in the fields of low-dimensional topology and symplectic geometry. Recently I have been collaborating intensively with physicists as well, as we have discovered that my work is closely related to current developments in string theory and mirror symmetry, and I anticipate that a fair amount of my future work will point in this direction. Low-dimensional topology is the study of three- and four-dimensional geometric spaces, as well as knots in three-dimensional spaces. This field has rapidly developed in the past three decades, in part because of powerful new approaches to the subject through algebra, analysis, and physics (especially gauge theory). Since then, ideas have continued to pour into the subject from other areas of mathematics. One of the currently hottest approaches to low-dimensional topology, and an area that I have devoted much of my research to, comes from symplectic geometry. Symplectic geometry, a venerable subject with origins in physics and specifically classical mechanics, can be thought of as a branch of differential geometry and involves much interplay between geometry and analysis. It studies symplectic manifolds, which are even-dimensiona