I am an L. E. Dickson instructor and Simons fellow at the University of Chicago.
My email address can be found in the university's official directory here.
I have a broad interest in number theory and its interaction with other areas of mathematics. My research includes two-dimensional class field theory and adelic methods in arithmetic geometry, especially the relations to K-theory, Arakelov theory, and the geometric and arithmetic Langlands programmes.
My CV.
Published papers and preprints
The most recent versions of my papers are generally those provided here, not on the arXiv:
Integration on product spaces and GL_n of a valuation field over a local field Communications in Number Theory and Physics, vol. 2, no. 3, 563-592, 2008, available here if your institute subscribes to CNTP.
Fubini's theorem and non-linear changes of variables over a two-dimensional local field 2008, arXiv:0712.2177. See also chapter 4 of my thesis.
An explicit approach to residues on and dualizing sheaves of arithmetic surfaces New York Journal of Mathematics, vol. 16, 2010, available here.
Grothendieck's trace map for arithmetic surfaces via residues and higher adeles 2010, awaiting publication with the Journal of Algebra and Number Theory, arXiv:1101.1883, available here.
Continuity of the norm map on Milnor K-theory: a local proof 2010, available here, awaiting publication with the Journal of K-theory.
Constructing higher dimensional local fields A note, with complete proofs, describing the successive completion and localisation process by which one constructs higher dimensional local fields.
Thesis
The final version of my Ph.D. thesis, which essentially consists of some of the papers above together with a chapter on Hrushovksi-Kazhdan style integration for two-dimensional local fields, is available here.
Video!
The clever organisers of the LMS Durham symposium 'New directions in the model theory of fields' filmed all the talks and put them online here. Mine is a general introduction to integration over two-dimensional local fields with some model-theoretic flavour.
Lecture notes
The eventual aim of these sets of notes is to write an entirely self-contained description of higher local class field theory:
An introduction to Milnor K theory, covering basics of Milnor K-theory, a careful construction of the norm map and proofs of all its important properties (including continuity for discrete valuation fields), and relations to differential forms. Some gaps.
An introduction to higher local fields, covering basics of higher local fields, their construction, and material on sequential topologies and topological Milnor K-groups. The most important parts of these notes are better covered by the paper Constructing higher dimensional local fields above.
Some teaching notes
REU. Incomplete notes for the REU course "Number theory: Reciprocity and Polynomials".
MATH 242. Notes for MATH 242: Algebraic Number Theory.
Awards, etc.
Gibbs Proxime Accessit Prize in Mathematics, University of Oxford, July 2006.
Junior Mathematical Prize, University of Oxford, July 2006. The Cecil King Travel Scholarship, London Mathematical Society, April 2008. The report is available in the LMS December 2009 newsletter.
EPSRC Ph.D. Plus research fellowship, October 2009 - September 2010. Simons fellowship.
