My research at the moment is in algebraic K-theory and (topological) cyclic homology for applications to algebraic cycles on algebraic and arithmetic varieties, particularly using infinitesimal methods. I remain interested in higher dimensional local fields and adèles in arithmetic geometry.
Published papers and preprints
The definitive versions of my papers are generally those provided here, not on the arXiv:
Pro excision and h-descent for K-theory, submitted.
K-theory of one-dimensional rings via pro-excision,
awaiting publication with the Journal de l'Institut de Mathématiques de Jussieu.
A singular analogue of Gersten's conjecture and applications to K-theoretic adèles, submitted.
K_2 of localisations of local rings, submitted.
Continuity of the norm map on Milnor K-theory: a local proof Journal of K-Theory, vol. 9, issue 03, 565-577, 2012.
Grothendieck's trace map for arithmetic surfaces via residues and higher adèles
Journal of Algebra and Number Theory, vol. 6, no. 7, 1503-1536, 2012. pdf
An explicit approach to residues on and dualizing sheaves of arithmetic surfaces
New York Journal of Mathematics, vol. 16, 2010.
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.
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.
Integration on valuation fields over local fields
The Tokyo Journal of Mathematics, vol. 33, 2010, 235-281.
An introduction to higher dimensional local fields, providing an introduction to the theories of higher dimensional local fields and higher dimensional adèles, including a sketch of higher dimensional local class field theory. These supersede other introductory texts I have written on the subject.
Ph.D. thesis. This essentially consists of some of the papers above together with a chapter on Hrushovksi-Kazhdan style integration for two-dimensional local fields.
Video. My talk at the LMS Durham symposium 'New directions in the model theory of fields', providing a a general introduction to integration over two-dimensional local fields with some model-theoretic flavour, was filmed and is available at the link.
Workshop: Towards a local proof of the local Langlands correspondence.
Some teaching notes
REU. Incomplete notes for the REU course "Number theory: Reciprocity and Polynomials".
MATH 242. Notes for MATH 242: Algebraic Number Theory.