Research

Research Interests: Algebraic geometry and Geometric representation theory.

Most of my research is related to the geometric Langlands Program and to principal bundles under actions of reductive groups. More precisely, I am interested in principal bundles on smooth affine schemes, principal bundles with connections on projective curves (in particular, connections with singularities), the Hitchin system, the geometry of affine Grassmannians, moduli spaces of principal bundles, Higgs bundles, and principal bundles with connections, and their motivic classes.

I am also interested in Hodge structures, Hodge modules, and arithmetic algebraic geometry.

You can download my CV here.

You can download my publication list here.

Work in progress

• Satake equivalence for Hodge modules on affine Grassmannians (joint with D. Arinkin).

Publications and Preprints

• On the purity conjecture of Nisnevich for torsors under reductive group schemes. arXiv:2109.10332.
• On the Grothendieck–Serre conjecture on principal bundles in mixed characteristic. arXiv:1501.04224. Accepted for publication in Transactions of the AMS.
• On the Grothendieck–Serre Conjecture about principal bundles and its generalizations. arXiv:1810.11844. Accepted for publication in Algebra and Number Theory.
• Motivic Donaldson–Thomas invariants of parabolic Higgs bundles and parabolic connections on a curve (joint with A. Soibelman and Y. Soibelman). SIGMA 16(70):1–49, 2020. Available at arXiv:1910.12348.
• Two circles and only a straightedge (joint with A. Akopyan). Proc. AMS, 147(1):91–102, 2019. A preprint is available at arXiv:1709.02562.
• Motivic classes of moduli of Higgs bundles and moduli of bundles with connections (joint with A. Soibelman and Y. Soibelman). Communications in Number Theory and Physics, 12(4):687–766, 2018. A preprint is available at arXiv:1705.04890.
• Variations of Hodge structures for hypergeometric differential operators and parabolic Higgs bundles. Int. Math. Res. Not., 2018(18):5583–5608, 2017. A preprint is available at arXiv:1505.01704.
• Partial Fourier–Mukai transform for integrable systems with applications to Hitchin fibration (joint with D. Arinkin). Duke Math. Journal 165(15):2991––3042, 2016. A preprint is available at arXiv:1408.0984.
• Affine Grassmannians of group schemes and exotic principal bundles over affine lines. Amer. Journal of Math., Vol. 138, no. 4, pp. 879–906. A preprint is available at arXiv:1308.3078.
• A proof of GrothendieckSerre conjecture on principal bundles over regular local rings containing infinite fields (Joint with I. Panin). Inst. Hautes Études Sci. Publ. Math., Vol. 122 (2015), no. 1, pp. 169–193. DOI 10.1007/s10240-015-0075-z. A preprint is available at arXiv:1211.2678.
• An example of Langlands correspondence for irregular rank two connections on P¹ (joint with D. Arinkin). Advances in mathematics, Vol. 230 (2012), no. 3, pp. 1078–1123. DOI 10.1016/j.aim.2012.03.003. A preprint is available at arXiv:1003.6112.
• Frobenius manifold structures on the spaces of abelian integrals. Journal of Geometry and Physics, Vol. 61 (2011), no. 2, pp. 485–497. DOI 10.1016/j.geomphys.2010.10.015. A preprint is available at arXiv:math/0701581.
• Irregular Wakimoto modules and the Casimir connection. Selecta Mathematica (N.S.), Vol. 16 (2010), no. 2, pp. 241–266. DOI 10.1007/s00029-010-0019-x. A preprint is available at arXiv:0812.4472.
• Algebraic and hamiltonian approaches to isostokes deformations, Transformation Groups, Vol. 11 (2006), no. 2, pp. 137–160. A preprint is available at arXiv:math/0412121.
• Upper bounds of the number of orbital topological types of planar polynomial vector fields "modulo limit cycles", Proceedings of the Steklov Institute of Mathematics, Vol. 254, pp. 238–254. A preprint is available at arXiv:math/0402214. Russian original: Верхние оценки числа орбитальных топологических типов полиномиальных векторных полей на плоскости "по модулю предельных циклов", Труды Математеского института им. В.А. Стеклова, 2006, Том 254, стр. 254–271.
• Upper bounds of the number of orbital topological types of polynomial vector fields on the plane "modulo limit cycles" Russian Mathematical Surveys, 2004, Vol. 59, No. 3, pp. 569–570. Russian original: Верхние оценки числа орбитальных топологических типов полиномиальных векторных полей на плоскости "по модулю предельных циклов", Успехи Мат. Наук, 2004, Том 59, Номер 3, стр. 183–184.
• Lower bounds for the number of orbital topological types of planar polynomial vector fields "modulo limit cycles", Moscow Math. J., 2001, Vol. 1, No. 4. pp. 539550.

Selected Invited Talks

• September 2019. University of Pittsburgh. Department of Mathematics Colloquium: Hitchin system and Langlands duality. Slides.
• August 2019. CIRM, Luminy, France, Summer school "Buildings and Affine Grassmannians". Affine Grassmannians of group schemes and exotic principal bundles over A¹.
• July 2019. Bergische Universität Wuppertal. Workshop of the GRK 2240. A conjecture of Grothendieck and Serre on principal bundles in mixed characteristic.
• May 2019. Heinrich Heine Universität Düsseldorf. Mathematical Colloquium. Principal bundles in algebraic geometry.
• April 2019. University of North Carolina, Chapel Hill. Mini-course. Geometric Langlands Duality.
• January 2019. Pennsylvania State University. Algebra and Number Theory Seminar. A conjecture of Grothendieck and Serre on principal bundles.
• December 2018. RIMS, Kyoto, Japan. Conference "D-modules, Quantum geometry, and related topics". Coherent Satake Functor.
• September 2018. University of Pittsburgh. Undergraduate Mathematics seminar. Two circles and only a straightedge.
• May 2018. ENS Lyon, France. Conference "Algebraic Groups and Geometrization of the Langlands programm". Counting points on the global nilpotent cone.
• May 2018. Claude Bernard University of Lyon, France. Mini-course: The conjecture of Grothendieck and Serre on principal bundles.
• February 2018. University of Wisconsin, Madison. Algebra and algebraic geometry seminar. A conjecture of Grothendieck and Serre on principal bundles in mixed characteristic.
• June 2017. Universitäat Duisburg–Essen, Germany. Oberseminar. Motivic classes of moduli of Higgs bundles on curves.
• January 2016. University of Pittsburgh, USA. Colloquium Principal Bundles, Moduli Spaces, Landlands Duality and Motives.
• January 2016. EPFL, Lausanne, Switzerland. Workshop — Higgs bundles and Hitchin system - VBAC2016.
• June 2015. United Kingdom, Durham. Colloquium talk: "Principal bundles in algebraic geometry".
• February 2015. Max Planck Institute for Mathematics, Bonn, Germany. Hodge structures on hypergeometric D-modules.
• May 2014. University of Ottawa, Canada. Fields workshop on Algebraic and Geometric Invariants of Linear Algebraic Groups and Homogeneous Spaces. Mini-course: Introduction to principal homogeneous spaces.
• June 2013. The Fields Institute, Toronto. Conference on Torsors, Nonassociative Algebras and Cohomological Invariants. A conjecture of Grothendieck and Serre on principal bundles. Video of this talk.
• June 2013. The Fields Institute, Toronto. Conference on Torsors, Nonassociative Algebras and Cohomological Invariants. A conjecture of Grothendieck and Serre and affine Grassmannians. Video of this talk.
• December 2012. Université Paris Sud 11. On a conjecture of Grothendieck and Serre concerning principal bundles.
• December 2012. University of Strasbourg. On a conjecture of Grothendieck and Serre concerning principal bundles.
• November 2011. The University of Miami, Miami, Florida. Categorical geometric Langlands duality.
• April 2011. Korteweg-de-Vries Institute for Mathematics, Amsterdam, the Netherlands. Categorical geometric Langlands duality.
• April 2011. Trinity College, Dublin, Ireland. Categorical geometric Langlands duality.
• March 2011. The University of Strasbourg, France. Categorical geometric Langlands duality.
• December 2010. Kansas State University. Categorical geometric Langlands duality.
• March 2009: MIT, Noncommutative Algebra Seminar, Non-highest weight representations of affine Kac-Moody algebras, DMT connection, and irregular Wakimoto modules.
• April 2007. Northeastern University, "Isomonodromic deformations and affine Lie groups", talk at the Geometry-Algebra-Singularities-Combinatorics Seminar.
• February 2007. Massachusetts Institute of Technology, "Frobenius manifold structures on the spaces of abelian integrals", talk at the Infinite-Dimensional Algebra Seminar.

Seminars

• Boston University Geometry Seminar
• Geometric Langlands Program
• Riemann Surfaces, Lie algebras, and Mathematical Physics