Research

Research Overview

My research is in four-dimensional symplectic geometry, with a particular focus on symplectic embedding problems. These problems ask when one symplectic space can fit inside another while preserving its symplectic structure. I use tools including exceptional curves, almost toric fibrations, and embedded contact homology capacities to study the geometric and arithmetic structures underlying these questions.

The figure below visualizes the classification of infinite staircases for Hirzebruch surfaces. Figure by Ana Rita Pires.

Symplectic Embeddings and Infinite Staircases

Given a symplectic manifold X, one can study the smallest scaling of X into which an ellipsoid of a given eccentricity symplectically embeds. The resulting embedding function is continuous but often only piecewise smooth. In some examples, it has infinitely many corners accumulating at a finite point; this phenomenon is called an infinite staircase.

With Dusa McDuff, Ana Rita Pires, and Morgan Weiler, I classified which Hirzebruch surfaces have infinite staircases. This work reveals a recursive structure governed by symmetries of embedding functions and exceptional classes. It also shows that infinite staircases occur for a highly structured, nongeneric set of parameters.

Almost Toric Fibrations and Mutations

Almost toric fibrations provide geometric models of symplectic manifolds in which embedding constructions can be seen directly in two-dimensional diagrams. Mutating these diagrams produces new fibrations of the same symplectic manifold and can reveal recursive families of embeddings and obstructions.

My recent work develops a dictionary between mutations of quadrilateral almost toric base diagrams and recursive algebraic data associated to exceptional classes. This helps explain how the obstruction classes forming infinite staircases can arise from explicit geometric constructions. In earlier work, I used almost toric fibrations to construct full fillings at the accumulation points of certain staircases.

Convex Toric Domains and ECH Capacities

Embedded contact homology capacities associate a sequence of numerical invariants to a four-dimensional symplectic space. These capacities give powerful obstructions to symplectic embeddings, and their asymptotic behavior can also reflect finer geometric properties of the space.

With Dan Cristofaro-Gardiner and Dusa McDuff, I study how features of a convex toric domain—such as its perimeter and the geometry of its boundary—are reflected in the asymptotic behavior and complexity of its ECH capacities.