Info

Tamarkin Assistant Professor, Mathematics Department, Brown University.

I was a student of Richard Melrose.

Interests

Global analysis, topology of singular spaces, gauge theory and mathematical physics.

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 full research statement, and a curriculum vitae.

My current research projects include

• The study of the moduli space of magnetic monopoles on 3-manifolds with asymptotically conic ends, which I began in my PhD thesis. Some of this work is joint with R. Melrose and M. Singer.
• 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 resolvent construction for the Witten deformation of the Laplacian by a Morse-Bott or generalized Morse function, having applications to Morse theory and to analytic torsion in the equivariant and stratified singular settings. Joint with P. Albin and R. Mazzeo.
• A resolution theory in the category of manifolds with corners which leads to a systematic treatment of fiber products in that category and which is related to the algebro-geometric theory of logarithmic geometry (itself a generalization of toric geometry and toroidal embeddings). Part of this work is joint with R. Melrose and part is joint with W. Gillam and S. Molcho.

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

• Log differentiable spaces and manifolds with corners. (With W. Gillam and S. Molcho)
  In preparation.
• A Dirac operator on loop space. (With R. Melrose)
  In preparation.
• Morse-Bott-Witten deformation on compact manifolds. (With P. Albin and R. Mazzeo)
  In preparation.
• Callias' index theorem and monopole deformation.
  arXiv:1210.3275, Submitted.
• Generalized blow-up of corners and fiber products. (With R. Melrose)
  arXiv:1107.3320, Submitted.
• An index theorem of Callias type for pseudodifferential operators.
  Journal of K-Theory, 8, Issue 03 (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, Issue 18 (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, 036611, 2008.
• Vortex core identification in viscous hydrodynamics. (With L. Finn and B. Boghosian)
  Phil. Trans. R. Soc. A, 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.