# Confoliations

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## Summer 2019

This summer was busy; I figure it’s about time for an update.

My research this summer was primarily to continue ongoing work on H-type foliations that I presented last year in Grenoble and Hannover. I’ve also been continuing to write about connections in foliated manifolds, in the same vein as my earlier posts on the blog.

I’ve also been looking into confoliations (Eliashberg and Thurston), which act as an interpolation between foliations and contact structures. That is, we can understand both foliations and contact structures on 3-dimensional manifolds as being generated by a plane field $$\xi$$ determined by a nonvanishing one-form $$\alpha$$ obeying the Pfaffian equation $$\alpha(\xi) = 0$$. If the condition $$\alpha \wedge d\alpha \equiv 0$$ is satisfied then $$\xi$$ is a foliation, while the nonvanishing condition $$\alpha \wedge d\alpha > 0$$ implies that $$\xi$$ is a positive contact structure. To generalize this perspective, positive confoliations are defined as plane fields $$\xi$$ generated by a one form $$\alpha$$ satisfying the condition $$\alpha \wedge d\alpha \geq 0$$. Observing that foliations and contact structures are extreme cases of confoliations, they can be used to unify the two (historically disjoint) theories.

I spent a good portion of my summer running an REU at UConn (with oversight by Oleksii Mostovyi) in financial mathematics (a first for me!) We had three students, Sarah Boese, Tracy Cui, and Sam Johnston, who worked intensely all summer learning about hedging by sequential regression; they were able to show several interesting results about the Follmer-Schweizer decomposition in discrete models including a new result on asymptotic stability of the decomposition.

At UHart I taught a course in Multivariable Calculus; I made it my goal to focus on Stokes’ Theorem and its various special cases in $$\mathbb{R}^2$$ and $$\mathbb{R}^3$$, which went very well. Some comments from a very active student have led me to add a lecture on the relationship between Clairaut’s Theorem, Fubini’s Theorem, and Leibniz’s integral rule the next time I cover the material. At UConn I also TA’d a course in Linear Algebra.

On a personal note, I was able to travel to Ecuador with my family. It was my first trip that far south, and an absolutely gorgeous country. We spent most of the time in Guayaquil, and had the chance to visit Cuenca and Quito.