"Scattering-symplectic geometry" (video ) is a 60 minute talk I gave in July 2017 at a Conference on Poisson Geometry and Stacks at The Fields Institute. Scattering-symplectic manifolds are manifolds equipped with a type of minimally degenerate Poisson structure that is not too restrictive so as to have a large class of examples, yet restrictive enough for standard Poisson invariants to be computable. My talk is an introduction: defining scattering-symplectic geometry, providing plenty of examples, and discussing connections with contact geometry.
"Rescaled tangent bundles in Poisson geometry" (video or slides) is a short talk I gave in December 2016 at the Casa Matematica Oaxaca workshop: Geometric and Spectral Methods in Partial Differential Equations. Poisson manifolds are rich geometric objects generalizing symplectic manifolds. Unfortunately it can be quite challenging to say anything meaningful about these manifolds in general. In this talk I discussed how rescaled tangent bundles, a familiar tool of geometric microlocal analysis, allow us to gain traction and understand nice classes of Poisson manifold.
3 Weird Tricks Using The Determinant That Even Your Math Teacher Didn't Know!
An Invited talk I gave in the Graduate Student Colloquium at the University of Illinois on November 2, 2016. The slides can be viewed here.
Abstract: What can the determinant of a 2 by 2 matrix tell you about the twisting of a mobius strip or the crunchiness of a turtle brownie? Come find out! In this talk I will introduce Poisson geometry in a fun, easy to follow fashion with tons of examples.
Exploding trousers and computing the intractable
A research poster I presented on January 6, 2017 at the AWM Workshop Poster Session, Joint Math Meetings in Atlanta, Georgia. Note that this poster was designed for an audience of mathematicians not in my specific area of research. (Click image to enlarge.)
Abstract: The study of symplectic manifolds is equivalent to studying non-degenerate Poisson manifolds. However, Poisson mani- folds are much, much more general than the symplectic case. Our goal is to study Poisson structures that are degenerate, but which live close enough to the symplectic world that we can understand them using symplectic tools. We introduce scattering-symplectic manifolds, such a class of Poisson structure. Employing standard symplectic machinery, we con- struct a new type of cobordism and compute a usually intractable invariant: Poisson cohomology!
Image of research: The Banana of My Eye.
The Image of Research is an annual multidisciplinary competition celebrating graduate student research at the University of Illinois at Urbana-Champaign. Graduate students are invited to submit an image and a brief text narrative that articulates how the image relates to their research. (Click image to enlarge.)
Title: The Banana of My Eye.
Narrative: "The apple of my eye" is something I cherish above all others. As a mathematician I study minimally degenerate Poisson manifolds. You can think of a Poisson manifold as the skin of a fruit. Imagine an apple. You probably imagine a fruit with a shiny bright red facade. This pristine and perfect skin corresponds to a non-degenerate Poisson structure. As is true when picking out fruit at the grocery store, many Poisson manifolds aren't quite perfect, but come with dents and bruises. Given a fruit (Poisson manifold), with a minor blemish, I use calculus to try and understand the damage. Sometimes I can model this degeneracy and incorporate the blemish into the geometry. For instance, I can handle the depicted banana, splitting it apart to understand subtle nuances in the geometry that other mathematicians weren't able to see before. However, I am unable to grapple with the bruise on the pear. Through the lens of Poisson geometry, I come to see flawed fruit as perfect and pristine. While this fruit may appear bruised or defective to another mathematician, it becomes the apple - or perhaps I should say banana - of my eye.