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.

### In Preparation

*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*.

### Papers

*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.