Research
The Hopf fibration, which can be understood in terms of a contact structure on the three-dimensional sphere.
Overview
I study a kind of topology that comes in two flavors: symplectic topology and contact topology. Symplectic structures arise from the mathematical abstraction of Hamiltonian mechanics, and symplectic manifolds are even-dimensional objects that generalize phase space. Contact manifolds are the odd-dimensional siblings of symplectic manifolds --- they have their own physical motivation, but also, for example, arise naturally as hypersurfaces in symplectic manifolds.
Symplectic and contact topology are deep subjects in their own right with many interesting problems. They also have a wide range of applications in other areas of math and physics, both pure and applied. For example, symplectic/contact structures can be used to define powerful invariants of smooth manifolds.
Topics of interest in my research include convex hypersurface theory in contact topology and the study of Liouville domains and Weinstein domains in symplectic topology. Currently, I am particularly interested in contact topology in arbitrary dimensions.
Preprints and publications
Folded symplectic forms in contact topology.
To appear in Journal of Geometry and Physics (2024). (arXiv link)
The Giroux correspondence in arbitrary dimensions.
(Joint work with K. Honda and Y. Huang.)
arXiv preprint (2023). (arXiv link)
Mitsumatsu's Liouville domains are stably Weinstein.
(Joint work with A. Christian.)
arXiv preprint (2021). (arXiv link)
Morse-Smale characteristic foliations and convexity in contact manifolds.
Proc. Amer. Math. Soc. 149 (2021), 3977-3989. (arXiv link) (Journal link)
On the sign characteristic of Hermitian linearizations in DL(P).
(Joint work with M. Bueno, S. Ford, and S. Furtado.)
Linear Algebra and its Applications, 519 (2017), 73-101. (Journal link)