Kähler-Ricci Flow is Symplectic Deflation Plus and Exact Symplectic Deformation (Submitted) We demonstrate that Kähler-Ricci flow on a Fano surface decomposes as a symplectic deflation along a plurianticanonical divisor plus an exact symplectic deformation in the divisor complement. We apply this decomposition to the study of certain projective bundles. We establish the optimal decay rate in the setting of Song, Székelyhidi, and Weinkove for a family of toric metrics which match the cohomological profile of Kähler-Ricci flow. In doing so, we identify a C2-obstruction to proving the optimal decay rate in the setting of ruled toric surfaces. We present these ideas as a new symplectic-topological approach to the study of Kähler-Ricci flow. [ pdf ][ arXiv ]
Non Affine Stein Manifolds and Normal Crossing Divisors (Submitted): In this paper, it is demonstrated that there are Stein manifolds that admit normal crossing divisor compactifications despite being neither affine nor quasi-projective. This is achieved via the study of contact boundaries of neighborhoods of symplectic normal crossing divisors via a contact-geometric analog of W. Neumann's plumbing calculus. In particular, conditions are given under which the neighborhood is determined by the contact structure on its boundary. [ pdf ][ arXiv ]
Stein Compactifications and Lefschetz Fibrations (in preparation): Joint with Sierra Knavel-Runner. We show that a Positive Allowable Lefschetz Fibration (PALF) may be compactified to a Lefschetz fibration over S^2 if a certain monodromy condition holds. This compactification can be achieved by attaching the neighborhood of a symplectic normal crossing divisor, effectively completing the PALF. We apply our results to the study of compactifications of Stein manifolds.
My research interests are in advancing the understanding of complex birational geometry from the perspective of symplectic topology, contact geometry, low-dimensional topology, and geometric analysis.
I study symplectic manifolds such as Kahler varieties through the symplectic topology of normal crossing divisors. These are symplectic generalizations of normal crossing divisors from algebraic geometry. I have contributed to the understanding of birational geometry by demonstrating that there are Stein manifolds which admit normal crossing divisor compactifications despite being neither affine nor quasi-projective. I have developed a divisor-centered approach to Kahler-Ricci flow and work to contribute to the field via the symplectic topology of divisor complements, which are exact symplectic manifolds.
Generally, I work on developing constructions to analyze 4-dimensional symplectic manifolds and 3-dimensional contact manifolds via differential topology. I am interested in developing tools for symplectic birational geometry and the analytic minimal model program in dimension 4. 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. My work takes inspiration from classical algebraic geometry, low dimensional topology, and complex analytic geometry.
While my research focuses are in contact and symplectic structures on low dimensional manifolds, I consider myself a geometer, broadly defined.
My dissertation focused on the theory of contact 3-manifolds associated to symplectic normal crossing divisors. 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.