# Research

I am interested in Algebraic Geometry, Number Theory, Topology, and Homological Algebra. Here you can find the latest versions of my papers.
 Hochschild cohomology of Azumaya algebras and the Atiyah class. We compute the Hochschild cohomology of the ring of differential operators in characteristic p. Last update: October 2012. pdf On the center of the ring of differential operators on a smooth variety over Z/p^nZ. Joint with Allen Stewart. Last update: May 2012. To appear in Compositio Math. pdf Some applications of weight structures of Bondarko. We construct a functor from the triangulated category of Voevodsky motives to a certain derived category of mixed Hodge structures enriched with an integral weight filtration (as opposed to the rational weight filtration introduced by Deligne). We use this construction to study 1-motives. To appear in IMRN. Last update: December 2011. pdf Motivic integral of K3 surfaces over a non-archimedean field. Joint with Allen Stewart. Advances in Math. (2011). Erratum added (July, 2012). pdf Geometric construction of the canonical coordinates. This unfinished draft suggests a construction of the canonical coordinates appearing in the Mirror Symmetry which makes their integrality properties (studied in joint papers with Maxim Kontsevich and Albert Schwarz) transparent. Last update: April 2012. pdf Hodge realizations of 1-motives and the derived Albanese. To appear in the Journal of K-theory. Last update: December 2011. pdf On the derived DG functors. This paper extends some foundational results of Grothendieck on derived functors to the DG setting. Math. Research Letters. (2010). pdf Integrality theorems in the theory of topological strings. Joint with Albert Schwarz. Nuclear Physics B. (2009). pdf On the N-integrality of instanton numbers. This paper gives proofs of the results announced in my paper "Integrality of the instanton numbers and p-adic B-model" joint with Maxim Kontsevich and Albert Schwarz. Last update: 2008 pdf A DG guide to Voevodsky's motives. Joint with Alexander Beilinson. Updated version of the paper published in GAFA. Last update: 2008. pdf P-adic Cartier Transform. This paper extends the nonabelian Hodge Theory in characteristic p, as developed by Arthur Ogus and myself, to the case of schemes over Z/p^nZ. Last update: 2008. pdf Some applications of Goodwillie's Theorem. This unfinished draft describes algebraic K-groups of a flat infinitesimal deformation of a smooth projective variety. Last update: 2008. pdf Nonabelian Hodge Theory in Characteristic p. Joint with Arthur Ogus. Publ. Math. IHES (2007). pdf Motivic sheaves over a curve. In this rough draft joint with Anna Otwinowska we, following a suggestion of Beilinson, construct a triangulated category of motivic sheaves over a curve gluing the categories of Voevodsky motives over the points. of the curve. Last update: 2007. pdf Frobenius transformation, mirror map and instanton numbers. Joint with Albert Schwarz. Physics Letters B. (2008). pdf Integrality of the instanton numbers and p-adic B-model. Joint with Maxim Kontsevich and Albert Schwarz. Physics Letters B. (2006). pdf Non-commutative compactifications and elliptic curves. Joint with Yan Soibelman. IMRN (2003). pdf Hodge structure on the fundamental group and its application to p-adic integration. This is my Ph.D. Thesis. Mosc. Math. J. (2003). pdf $\gamma$-functions of representations and lifting. Main text by A. Braverman and D. Kazhdan. I am the author of an appendix. GAFA (2000). pdf On measure of K-points of an abelian variety, where K is a local field. Appendix to "An Algebraic Integration" by D. Kazhdan, in "Mathematics: frontiers and perspectives", (2000). pdf