research

Research summary

My research interests are centered around symplectic geometry. Symplectic geometry is the study of manifolds equipped with a closed, non-degenerate 2-form known as a symplectic form. It was originally formulated to model the phase space of classical physical systems and study their dynamics, but since its beginnings in mechanics, symplectic geometry has expanded in many directions - especially with the advent of J-holomorphic techniques, Floer homology, and symplectic capacities in the 1980s. The main goal of my research is to improve the understanding of and explore the relationships between several rapidly developing fields related to symplectic geometry.

In particular, I study integrable systems, symplectic & Hamiltonian group actions, (immersed) Floer homology, and dynamics. There is a particular focus in my work on integrable systems with an underlying torus action - which roughly can be thought of as physical systems which admit rotational symmetries. Furthermore, I often study the space of all integrable systems admitting a fixed torus action. That is, starting with a (Hamiltonian) torus action, I am interested in questions like: 

This last question naturally leads to the study of bifurcations of integrable systems with a fixed torus action.


I have 17 papers: 12 accepted/published articles and 5 additional preprints. 

I have given about 60 research talks: click here to see a (mostly) complete list of the talks I have given. 

Slides & Videos

Here are a few recorded talks of mine:

Here are some slides from talks I've given:

Undergraduate research

I've also hosted a total of six undergraduate research projects through the Illinois Geometry Labl (IGL) program:

Various Figures

Figure: The semitoric polygon and focus-focus fibers of the system in [Hohloch-Palmer 2018] with parameters R1=R2. This figure is from my preprint with Pelayo and Tang, where we extend the semitoric classification to systems which include multipinched tori as their fibers.

Figure: Also from my preprint with Pelayo and Tang, this figure shows the process of 'straightening out' the momentum map image of a (non-simple) semitoric system to obtain a polygon.

Primary Collaborators

(this list is a bit out of date...)

Jaume Alonso (University of Antwerp, Belgium)

Alessio Figalli (ETC Zurich, Switzerland)

Sonja Hohloch (University of Antwerp, Belgium)

Daniel M. Kane (UC San Diego, USA)

Yohann Le Floch (University of Strasbourg, France)

Melvin Leok (UC San Diego, USA)

John Man Shun Ma (Rutgers University, USA)

Álvaro Pelayo (UC San Diego, USA)

Xiudi Tang (University of Toronto, Canada)

Susan Tolman (University of Illinois Urbana Champaign, USA)

Christophe Wacheux (Center for Geometry and Physics, South Korea)

Christopher Woodward (Rugters University, USA) 

Thesis

PhD thesis: Symplectic invariants and moduli spaces of integrable systems

My thesis is mostly a combination of early versions of some of my papers that have since been published, with an introduction to the symplectic geometry of integrable systems.