Research

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My primary interests are symplectic geometry and completely integrable systems. In particular, I am interested in integrable systems (especially the semitoric case), symplectic group actions, singular Lagrangian fibrations, (immersed) Floer cohomology, and geometric mechanics.

I have 9 accepted/published articles and 3 additional preprints.

Click here to see a list of the talks I have given.

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Brief summary of current work:

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, due to Floer, Gromov, Ekeland, Hofer, Zehnder, and others. 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, Floer homology, and dynamics. There is a particular focus in my work on integrable systems of semitoric type - a type of four dimensional integrable system with an circle symmetry which generalize symplectic toric manifolds in dimension four and were classified in 2011 by Pelayo-Vu Ngoc.

To this end, I've done the following so far:

• Studied the invariance of immersed Floer cohomology with Chris Woodward under Maslov flow (i.e. reverse mean curvature flow) in [10] and under Lagrangian surgery to remove a self-intersection point in [12];


• Produced new examples of semitoric systems and a strategy to construct systems with desired properties with Yohann Le Floch in [11]. Constructed an example of a semitoric system with two focus-focus points with Sonja Hohloch in [9]. These are related to finding explicit examlpes of the minimal models studied in [8];


• classified the possible minimal models of semitoric systems (in the sense that they do not admit any blowdowns of toric type) in [8], expanding on techniques we developed in [6] (joint with Á. Pelayo and D.M. Kane);


• defined a G-equivariant analogue of symplectic capacites for any Lie group G [7] (joint with A. Figalli and Á. Pelayo).


• Used the classification of semitoric systems to describe a metric on their moduli space [4].

Someday I'll put a more detailed research summary here, but until then take a look at my papers or contact me if you find this interesting.

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Primary Collaborators:

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)
Christophe Wacheux (Center for Geometry and Physics, South Korea)
Christopher Woodward (Rugters University, USA)

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Papers

Most of these papers can be found on my arXiv profile.

Preprints:

12. Invariance of immersed Floer cohomology under Lagrangian surgery
    (with C. Woodward) 96 pages
    arXiv:1903.01943


11. Semitoric families
    (with Y. Le Floch) 85 pages
    arXiv:1810.06915


10. Immersed Floer cohomology and Maslov flow
    (with C. Woodward) 71 pages
    arXiv:1804.06799