My research investigates two intertwined themes. What can we learn about two-dimensional dynamics and four-dimensional symplectic embeddings via three-dimensional contact geometry (specifically, from the invariants of embedded contact homology)? And how can we leverage low-dimensional topology to compute these contact invariants? Key tools I use are open book decompositions, torus and circle actions on three- and four-manifolds, and symplectic fillings/cobordisms.
- The ECH of Seifert Fiber Spaces, with Jo Nelson. We compute the ECH of Seifert fiber spaces, extending our work on prequantization bundles.
- With Maria Bertozzi, Tara Holm, Nicole Magill, Dusa McDuff, Grace Mwakyoma, and Ana Rita Pires, I am extending my work on classifying infinite staircases of Hirzebruch surfaces.
- Knot filtered embedded contact homology.
- Infinite staircases for Hirzebruch surfaces (2020), with Maria Bertozzi, Tara Holm, Emily Maw, Dusa McDuff, Grace Mwakyoma, and Ana Rita Pires. We identify several infinite families of infinite staircases of symplectic embeddings of four-dimensional ellipsoids into one-point blowups of the complex projective plane.
- Symplectic embeddings of four-dimensional polydisks into half integer ellipsoids (2020), with Leo Digiosia, Jo Nelson, Haoming Ning, and Yirong Yang. We obstruct embeddings of polydisks into certain rational ellipsoids, extending the techniques of . This work was part of a SIREN run by Jo Nelson for which I served as a mentor in Summer 2020.
- Embedded contact homology of prequantization bundles (2020), with Jo Nelson. In his 2011 thesis, David Farris computed the ECH of prequantization bundles over Riemann surfaces, relating it as a Z_2-graded theory to the exterior algebra of the homology of the base. We extend the grading on his isomorphism to Z, and complete several techical proofs using Seiberg-Witten Floer cohomology and the asymptotics of pseudoholomorphic curves.
- Mean action of periodic orbits of area-preserving annulus diffeomorphisms (2018), published version in the Journal of Topology and Analysis, nearly final version on the arXiv. I use knot-filtered embedded contact homology to understand the mean action spectrum of annulus symplectomorphisms. In particular I prove a quantitative criterion for such symplectomorphisms to have no periodic orbits.
- Pattern avoidance in poset permutations, with Sam Hopkins, published version in Order 33. We extend the concept of permutation pattern avoidance to partially ordered sets.