# Employment

2016-present: Assistant Professor of Mathematics, New College of Florida.

2013-2016: Research Instructor, Mathematics Department, Northeastern University.

2010-2013: Tamarkin Assistant Professor, Mathematics Department, Brown University.

PhD 2010, MIT. Student of Richard Melrose.

# Interests

Global, geometric and microlocal analysis; gauge theory, moduli spaces and loop spaces.

# Research

My research broadly concerns geometric moduli spaces and topological invariants, especially those involving noncompact and singular spaces, with an approach through the analysis of partial differential equations. I specialize in the methods of geometric microlocal analysis (pseudodifferential and Fourier integral operators on manifolds), index theory and analysis on manifolds with corners. I am especially interested in problems set within the intersection of analysis, geometry and topology, and in problems arising from mathematical physics, particularly gauge theory and string theory.

Here is a (slightly out of date) research statement, and a curriculum vitae.

My current research projects include

- The study of the moduli spaces of magnetic monopoles, on R^3 (with an aim of compactifying the moduli spaces and proving Sen’s conjecture for their L^2 cohomology) and more general 3-manifolds with asymptotically conic ends. Some of this work is joint with R. Melrose and M. Singer. I co-organized a workshop The Sen conjecture and beyond which took place at UCL in June 2017.
- A construction of the Dirac operator on the free loop space of a compact manifold, with the goal of being able to treat it seriously as a differential operator and eventually to understand Witten’s index formula for the elliptic genus. Joint with R. Melrose.
- A resolution theory in the category of manifolds with (possibly generalized) corners, which is related to the algebro-geometric theory of logarithmic geometry (itself a generalization of toric geometry and toroidal embeddings). Some of this work is joint with R. Melrose.

Previously, I did some work in applied mathematics on perturbation theory for anisotropic dielectric interfaces, and before that, on large scale parallel numerical simulation of fluid dynamics.

# Publications and preprints

- Functorial compactification of linear spaces.

arXiv:1712.03902 (2017), 13 pages. - Partial compactification of monopoles and metric asymptotics.
(With
M. Singer)

arXiv:1512.02979 (2015), 113 pages. - Blow-up in manifolds with generalized corners.

*International Mathematical Research Notices*, vol. 2018, no. 8, (2018), pp. 2375-2415.

arXiv:1509.03874 - Dimension of monopoles on asymptotically conic 3-manifolds.

*Bulletin of the LMS*, vol. 47, no. 5, (2015), pp. 818-834.

arXiv:1310.2974 - Equivalence of string and fusion loop-spin structures.
(With
R. Melrose)

arXiv:1309.0210 (2013), 48 pages. - Loop-fusion cohomology and transgression.
(With
R. Melrose)

*Mathematical Research Letters*, vol. 22, no. 4, (2015), pp. 1177-1192.

arXiv:1309.7674 - A Callias-type index theorem with degenerate potentials.

*Communications in PDE*, vol. 40, no. 2, (2015), pp. 219-264.

arXiv:1210.3275 - Generalized blow-up of corners and fiber products.
(With
R. Melrose)

*Transactions of the AMS*, vol. 367, no. 1, (2015), pp. 651-705.

arXiv:1107.3320 - An index theorem of Callias type for pseudodifferential operators.

*Journal of K-Theory*, vol. 8, no. 3, (2011), pp. 387-417.

arXiv:0909.5661 - Accurate finite-difference and time-domain simulation of anisotropic media by subpixel smoothing.
(With
A.F. Oskooi and
S. Johnson)

*Optics Letters*, vol. 34, no. 18, (2009), pp. 2778-2780.

- Perturbation theory for anisotropic dielectric interfaces, and application to subpixel smoothing of discretized numerical methods.
(With
A.F. Oskooi and
S. Johnson)

*Phys. Rev. E*, vol. 77, no. 3, (2008), pp. 6611-6621.

arXiv:0708.1031 - Vortex core identification in viscous hydrodynamics.
(With
L. Finn and
B. Boghosian)

*Phil. Trans. R. Soc. A*, vol. 363, no. 1833, (2005), pp. 1937-1948.

# Notes & other writings

*Index Theory*, notes from the 2010 Talbot Workshop of Loop Groups and Twisted K-theory. This is a brief introduction to the Atiyah-Singer families index theorem for topologists, emphasizing the role of the index as a Gysin map in K-theory, including a discussion of spin^c structures, orientation and Dirac operators.*Extension off the boundary in a manifold with corners*. This is a short note about extending vector bundles, sections and connections smoothly from the boundary of a manifold with corners into the interior.