Arithmetic geometry, especially algebraic K-theory, (topological) cyclic homology, algebraic cycles, and p-adic Hodge theory.
Mini-course notes
Integral p-adic Hodge theory – notes. Notes of a mini-course, detailing some of the main results of the paper Integral p-adic Hodge Theory, with Bhargav Bhatt and Peter Scholze, which was given at Jussieu. (Updated on 28 March to include outline of the first part of the proof of the p-adic Cartier isomorphism.)
Preprints
The following versions are often more up-to-date than those found on the arXiv.
A note on higher direct images in crystalline cohomologypdf
Integral p-adic Hodge theory, with Bhargav Bhatt and Peter Scholze pdf
K-theory and logarithmic Hodge-Witt sheaves of formal schemes in characteristic parXiv
A Variational Tate Conjecture in crystalline cohomologypdf
Publications
Analogues of Gersten's conjecture for singular schemes, with Amalendu Krishna pdf
Selecta Mathematica, to appear.
Finite generation and continuity of topological Hochschild and cyclic homology, with Bjørn Ian Dundas pdf
Annales Scientifiques de l'École Normale Supérieure, to appear.
Integral p-adic Hodge theory – announcement, with Bhargav Bhatt and Peter Scholze pdf
Mathematical Research Letters, to appear.
Zero cycles on singular varieties and their desingularisationspdflink
Documenta Mathematica, extra volume dedicated to A. Merkurjev.
Pro unitality and pro excision in algebraic K-theory and cyclic homologypdflink
Journal für die reine und angewandte Mathematik (Crelle's Journal), to appear.
A case of the deformational Hodge conjecture via a pro Hochschild–Kostant–Rosenberg theorempdflink
Comptes Rendus Mathématique, vol. 352, issue 03, 173-177, 2014.
Pro cdh-descent for cyclic homology and K-theorypdflink
Journal de l'Institut de Mathématiques de Jussieu, to appear.
K-theory of one-dimensional rings via pro-excisionpdflink
Journal de l'Institut de Mathématiques de Jussieu, vol. 13, issue 02, 225-272, 2014.
A singular analogue of Gersten's conjecture and applications to K-theoretic adèlespdflink
Communications in Algebra, vol. 43, issue 11, 4951-4983, 2015.
K_2 of localisations of local ringspdflink
Journal of Algebra, vol. 399C, 190-204, 2014.
Continuity of the norm map on Milnor K-theory: a local proofpdflink
Journal of K-Theory, vol. 9, issue 03, 565-577, 2012.
Grothendieck's trace map for arithmetic surfaces via residues and higher adèlespdflink
Journal of Algebra and Number Theory, vol. 6, no. 7, 1503-1536, 2012.
An explicit approach to residues on and dualizing sheaves of arithmetic surfaceslink
New York Journal of Mathematics, vol. 16, 2010.
Integration on valuation fields over local fieldspdf
The Tokyo Journal of Mathematics, vol. 33, 2010, 235-281.
Integration on product spaces and GL_n of a valuation field over a local fieldlink
Communications in Number Theory and Physics, vol. 2, no. 3, 563-592, 2008.
MATH 242. Complete notes for MATH 242, Algebraic Number Theory, at the University of Chicago.
Investigations in two dimensional arithmetic geometry, Ph.D. thesis. This is superseded by my three earliest publications, except for an unpublished development of Hrushovksi--Kazhdan's motivic integration for two-dimensional local fields. Available upon request.