Department of Mathematics & Statistics
Hamilton Hall, Room 218
1280 Main Street West
Hamilton, Ontario, L8S 4K1, Canada
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My research focuses on integrable systems in symplectic geometry. In layman's terms, integrable systems are physical systems that are as symmetrical as they can possibly be. They occur throughout mathematical physics. Integrable spinning tops are a great example of integrable systems in classical mechanics (aka symplectic geometry) but there are many others.
From the perspective of geometers, integrable systems are interesting because they often relate the geometry of the symplectic manifold to the geometry of a convex polytope. The most well-known instance of this is Delzant's Theorem which says a certain family of integrable systems on symplectic manifolds called "toric manifolds" are classified up to isomorphism by a certain family of convex polytopes (which are now called "Delzant polytopes").
The study of integrable systems is dominated by examples, but new examples of integrable systems are quite rare. My most significant contribution is a recent paper which constructs many new examples of integrable systems. These systems are a generalization of a family of examples known as Gelfand-Zeitlin systems which were introduced by Guillemin and Sternberg in the early 1980's. The problem of generalizing them to the fullest possible extent was long standing and our work is the first to achieve this goal. Our construction uses a novel combination of techniques from algebraic geometry, stratified spaces, and symplectic geometry. As a by-product, we also prove a general result about integrable system constructed by toric degeneration that expands on earlier work of Harada and Kaveh from 2015.
In other recent work, I studied the fibers of moment maps of classic Gelfand-Zeitlin systems. These fibers are interesting from several perspectives, including Floer theory, quantization, and integrable systems theory. They are also an interesting family of topological spaces: they share features of Bott towers and their topology can be neatly encoded in combinatorial gadgets called "Gelfand-Zeitlin patterns." Although these fibers had been studied extensively in recent years by other authors, surprisingly little about their topology was known. In our paper we give very explicit descriptions of their diffeomorphism types, cohomology rings, and homotopy groups. We express our results in terms of the combinatorics of the associated Gelfand-Zeitlin patterns. In current work we are studying these fibers in relation to geometric quantization (see this twitter thread where I describe a fun part of that project).
A full list of my research works can be found over on Google Scholar or my CV. Openly available pdfs can be found on arXiv.
I don't know, a proof is a proof. What kind of...proof, it's a proof! A proof is a proof and when you have a good proof it's because it's proven.
- Jean Chrétien,
20th Prime Minister of Canada and amateur philosopher of mathematics.