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Research Interests

Here is a summary of my research interests. For more, you can view my research statement here.

Radial points

Much of my graduate research has focused on understanding pseudodifferential operators which have radial points. I have focused on the common case (one might say 'free particle' case, motivated by microlocal scattering theory) in which the operators have Lagrangian submanifolds of radial points. András Vasy and I have proved several propagation of singularities results for such operators. More recently, I constructed a normal form for such operators. I have constructed microlocal parametrices for such operators, and am working on a microlocal Poisson operator in this context (in preparation).

Index Theory

I am interested in applications of a framework due to Vasy. In particular, the Vasy framework identifies Fredholm problems via a relation between certain operators on several spaces, and I am interested in seeing what this says about index theory on these spaces.

Second microlocalization

Second microlocalization treats Lagrangian (and more generally, coisotropic) regularity analogously to how standard microlocal analysis treats smoothness. My interest in it begins with its potential applications to the study of radial points. My above mentioned paper with András Vasy gives a propagation of Lagrangian regularity statement in the context of a Lagrangian submanifold of radial points, and I am interested in how this and related situations look with second microlocalization.

Foundations of quantum field theory

I am interested in, quite broadly, the foundations of quantum field theory, and, even more broadly, communication between mathematics and theoretical physics as practiced today. The study of radial points began as an attempt to understand the microlocal structure of resolvants of Pauli-Fierz Hamiltonians. I have interest in recent works in the foundations of quantum field theory, particularly that of Borcherds and Costello. While my interests in this haven't yet led to anything concrete yet, I'd certainly like to chat about these works!

Papers

A normal form around a Lagrangian submanifold of radial points, 14 pages, preprint, 2012.

Propagation of singularities around a Lagrangian submanifold of radial points, joint work with András Vasy, 39 pages, preprint, 2011.