I am interested in problems in contact and symplectic topology and geometry. I am exploring connections of these fields with other fields, such as smooth low-dimensional topology, knot theory, and cluster algebras.
Contact and symplectic geometry are structures on odd- and even- dimensional manifolds, respectively. These geometries have no local invariants, unlike Riemannian geometry, where one has curvature. In some sense, contact structures on a (2n+1)-dimensional manifold are compatible with symplectic structures on (2n) and (2n+2) - dimensional manifolds. These manifolds have distinguished submanifolds called Legendrian submanifolds (in contact) and Lagrangian submanifolds. Symplectic manifolds are often studied using a gadget called pseudoholomorphic curves -- thus they behave like complex manifolds in some cases. Some of the questions I like thinking about are:
Given a contact manifold M, what are all the symplectic manifolds of one higher dimension that have M as their boundary?
Can one understand all the Legendrian submanifolds of a given contact manifold?
What are the interfaces between the smooth, symplectic, and complex worlds?
If a particle is moving around in space, we may be interested in knowing how its position and velocity will change throughout its motion. In an ideal setting, given initial conditions, Newton's laws can predict exactly how the particle will move. However, even then, understanding this prediction in practice can be extremely difficult, especially if there are multiple interacting forces acting on the particle (imagine the Solar system and trying to predict how a planet will move). These are known as the three-body problem or many-body problems, and we don't yet know if there can be a stable solution to these situations -- i.e., whether, given an initial condition, will the planetary system be stable or will it eventually always collapse or disperse?
These were the original motivations out of which symplectic and contact geometry were born. Since then, the questions in this field have evolved and become a little more abstract. There is plenty of research happening along symplectic dynamics. However, there is a lot of motivation towards understanding contact and symplectic structures as shapes that exhibit certain rigidity properties -- i.e. unlike smooth topology where one can stretch and deform as one likes, symplectic and contact objects are more rigid. This makes studying contact and symplectic phenomena quite interesting!
Please click on the title to be directed to a page with the paper abstract.
(2025) Legendrian Doubles, Twist Spuns, and Clusters, joint with James Hughes. (arxiv)
(2024) Spinal Open Books and Symplectic Fillings with Exotic Fibers, joint with Hyunki Min, Luya Wang. (arxiv)
(2024) Anchored Symplectic Embeddings, joint with Michael Hutchings, Morgan Weiler, Yuan Yao.(arxiv)
(2023) Small Symplectic Caps and Embeddings of Homology Balls in Complex Projective Space, joint with John Etnyre, Hyunki Min, Lisa Piccirillo, Selecta Math (accepted). (arxiv) (slides)
(2022) Constructions and Isotopies of Higher Dimensional Legendrian spheres. (arxiv), (slides)
(2021) Symplectic Fillings and Cobordisms of Lens Spaces, joint with John Etnyre, Transactions of the AMS. arxiv link