My research interests are in symplectic and contact geometry. I work on developing constructions for analyzing 4-dimensional symplectic manifolds and 3-dimensional contact manifolds via differential topology. I am interested in developing tools for symplectic birational geometry in dimension 4. My work takes inspiration from classical algebraic geometry, low dimensional topology, and complex analytic geometry.
My dissertation is focused on the theory of contact 3-manifolds associated to symplectic divisors with normal crossings, which are symplectic analogs of normal crossing divisors from algebraic geometry. These contact 3-manifolds arise as the boundaries of regular symplectic neighborhoods of a special class of divisors known as concave divisors. Concave divisors are the symplectic analog of compactifying divisors in projective geometry. I have used my study of these manifolds to produce tools for generating interesting examples (of both contact 3-manifolds and symplectic 4-manifolds). With my results, I hope to give examples of non-affine symplectic manifolds which are symplectic manifolds that cannot be symplectomorphic to any affine algebraic variety. I am particularly interested in finding exotic non-affine manifolds i.e. symplectic manifolds which are non-affine but are nonetheless diffeomorphic to some affine variety.
While my research focuses are in contact and symplectic structures on low dimensional manifolds, I consider myself a geometer, broadly defined.