Info

Tamarkin Assistant Professor, Mathematics Department, Brown University.

Interests

PDE, analysis on singular spaces, gauge theory and mathematical physics.

Research

My thesis, written under the direction of Richard Melrose, concerned the moduli space of SU(2) monopoles on asymptotically conic manifolds. These are solutions modulo gauge action of the Bogomolny equation on sections of an su(2) bundle over the manifold. Monopoles constitute a gauge theory on noncompact 3-manifolds; they are related by dimensional reduction to instantons on 4-manifolds.

Here is a full research statement, and my curriculum vitae.

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

• Generalized blow-up of corners and fiber products. With Richard Melrose.
  arXiv:1107.3320.
• Index Theorems and Magnetic Monopoles on Asymptotically Conic Manifolds. MIT Ph.D. thesis, available here.
• An index theorem of Callias type for pseudodifferential operators. Journal of K-Theory, Vol. 8, Issue 03, 2011.
  JKT online
  arXiv:0909.5661.
• Perturbation theory for anisotropic dielectric interfaces, and application to subpixel smoothing of discretized numerical methods.
  With Ardavan Farjadpour and Steven G. Johnson.
  Phys. Rev. E 77, 036611 (2008).
• Vortex core identification in viscous hydrodynamics.
  With L. Finn and B. Boghosian.
  Phil. Trans. R. Soc. A, vol. 363 no. 1833 (2005).

Notes & other writings

• Index Theory, notes from the 2010 Talbot Workshop on 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.