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Here is a (slightly out of date) research statement, and a curriculum vitae.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

List of my arXiv 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.