Interests

Global analysis and topology of moduli spaces, geometric microlocal analysis, mathematical physics.

Research

I work primarily in geometric analysis and gauge theory, with a focus on noncompact and singular spaces. I specialize in the methods of microlocal analysis (pseudodifferential operators on manifolds), index theory, and analysis on manifolds with corners. I am particularly 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.

My current research projects include

• Analysis of noncompact hyperkahler manifolds. Examples of these include moduli spaces of magnetic monopoles on R^3, Higgs bundles on surfaces, and Nakajima quiver varieties, including Hilbert schemes of points on C^2. In collaboration with F. Rochon, I have developed a framework for analysis on many of these spaces as ‘quasi-fibered boundary (QFB)’ manifolds, which has led to progress (and in some cases resolution) of long standing conjectures for their L^2-cohomology.
• As part of the above, constructing compactifications of the spaces in which the metrics have good behavior (i.e., asymptotic expansions) near infinity. My work on the compactification of magnetic monopole moduli spaces is variously joint with K. Fritzsch, M. Singer, and R. Melrose.
• 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 of a string manifold. This is joint with R. Melrose. As part of this project we developed an alternative approach to higher gerbes called ‘bigerbes’ (‘multigerbes’ in general).
• Various foundational projects involving categories of manifolds with corners and their generalizations. These include manifolds with ‘generalized corners’ (as defined by D. Joyce), which bring together ideas from logarithmic algebraic geometry and manifolds with corners, and manifolds with ‘fibered corners’, which are equivalent to a suitable category of stratified spaces. In these contexts I am interested in questions about resolution of singularities as well as the existence of categorical constructions like pullbacks, with a goal of bringing problems on singular spaces into a framework suitable for geometric analysis.

Long ago, as a graduate student I did some work in applied mathematics on perturbation theory for anisotropic dielectric interfaces, and before that as an undergraduate, on large scale parallel numerical simulation of fluid dynamics.

Publications and preprints

• Products of manifolds with fibered corners. (With F. Rochon)
  Annals of Global Analysis and Geometry, 64(9):1-61, 2023.
  arXiv:2206.07262
• L2-cohomology of quasi-fibered boundary metrics. (With F. Rochon)
  Inventiones Mathematicae, 236:1083-1131, 2024.
  arXiv:2103.16655
• Quasi-fibered boundary pseudodifferential operators. (With F. Rochon)
  Asterisque, to appear.
  arXiv:2103.16650 127 pages, 2021.
• Low energy limit of the resolvent of some fibered boundary operators. (With F. Rochon)
  Communications in Mathematical Physics, 390:231-307, 2022.
  arXiv:2009.10108
• Bigerbes. (With R. Melrose)
  Algebraic and Geometric Topology, 21(7):3335-3399, 2021.
  arXiv:1905.03081
  Note: The published version contains an error which is corrected by the current version on arXiv.
• Monopoles and the Sen Conjecture: Part I. (With K. Fritzsch and M. Singer)
  arXiv:1811.00601 28 pages, 2018.
• Functorial compactification of linear spaces.
  Proceedings of the AMS, 147(9):4067-4081, 2019.
  arXiv:1712.03902
• Partial compactification of monopoles and metric asymptotics. (With M. Singer)
  Memoirs of the AMS, 280(1383):124, 2022.
  arXiv:1512.02979
• Blow-up in manifolds with generalized corners.
  International Mathematical Research Notices, 2018(8):2375-2415, 2018.
  arXiv:1509.03874
• Dimension of monopoles on asymptotically conic 3-manifolds.
  Bulletin of the LMS, 47(5):818-834, 2015.
  arXiv:1310.2974
• Loop-fusion cohomology and transgression. (With R. Melrose)
  Mathematical Research Letters, 22(4):1177-1192, 2015.
  arXiv:1309.7674
• Equivalence of string and fusion loop-spin structures. (With R. Melrose)
  arXiv:1309.0210 48 pages, 2013.
• A Callias-type index theorem with degenerate potentials.
  Communications in PDE, 40(2):219-264, 2015.
  arXiv:1210.3275
• Generalized blow-up of corners and fiber products. (With R. Melrose)
  Transactions of the AMS, 367(1):651-705, 2015.
  arXiv:1107.3320
• An index theorem of Callias type for pseudodifferential operators.
  Journal of K-Theory, 8(3):387-417, 2011.
  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, 34(18):2778-2780, 2009.
  
• 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, 77(3):6611-6621, 2008.
  arXiv:0708.1031
• Vortex core identification in viscous hydrodynamics. (With L. Finn and B. Boghosian)
  Phil. Trans. R. Soc. A, 363(1833):1937-1948, 2005.
  

Links to conferences organized

• The Sen conjecture and beyond June 2017
• Celebrating Singularity: a conference in honor of Richard Melrose January 2024

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.