Research Interests
My current research interests lie in the interactions between algebraic geometry and number theory. Currently, I'm working on four broad projects. One deals with understanding a geometric analog of the classical theory of epsilon factors. Another (joint w/ M. Strauch and T. Schmidt) project deals with a geometric localization of codamissible modules arising in p-adic representation theory. This will eventually be applied to understanding certain locally analytic representations of central interest to the p-adic Langlands program. In joint work w/ M. Nori, we study the structure and periods of various motives arising from higher homotopy theory. Finally, in joint work w/ G.V. Ravindra, we invesitgate motivic versions of the classical weak Lefschetz theorem.

Publications/Preprints:

1. Power weak mixing does not imply multiple recurrence in infinite measure and other counterexamples.
    With Kate Gruher, Fred Hines, Cesar Silva, and Robert Waelder.
    New York Journal of Mathematics, Volume 1, 2003, pp. 1-23

2. De Rham epsilon factors. (pdf)
    Inventiones Mathematicae, Volume 190, Number 2, 2012, pp. 299-355

3. Towards connectivity theorems for codimension 2 cycles: Infinitesimal Deformations. (pdf)
    With G.V. Ravindra.
    Journal of Algebra, 399, 2014, pp. 407-422

4. Motivic structure on higher homotopy of hyperplane arrangements. (pdf)
    Mathematische Annalen, 366(1), 2016, pp. 279-300

5. K-theory of microdifferential operators and de Rham epsilon factors. (pdf)
    (Submitted.)

6. Locally analytic representations and sheaves on the Bruhat-Tits building. (pdf)
    With Tobias Schmidt and Matthias Strauch.
    Algebra and Number Theory, 8, 2014, no. 6, pp. 1365-1445

7. Weak Lefschetz for Chow groups: Infinitesimal Lifting. (pdf)
    With G.V. Ravindra.
    Homology Homotopy and Applications, 16, 2014, no. 2, pp. 65-84.

8. Integral models of P^{1} and analytic distribution algebras for GL(2). (pdf)
    With Tobias Schmidt and Matthias Strauch.
    Munster Journal of Mathematics, 7, 2014, pp. 241-271

9. A relative version of the Beilinson-Hodge cojecture. (pdf)
    With Rob de Jeu and James Lewis.
    Recent advances in Hodge theory (period domains, algebraic cycles, and arithmetic), 241-263, Lecture Note Series
    London Mathematical Society, 247, Cambridge University Press, 2016.

10. Arithmetic differential operators on a semistable model of P^{1}. (pdf)
     With Tobias Schmidt and Matthias Strauch.
     (Submitted.)

11. Locally analytic representations of GL(2) via semistable models of P^{1}. (pdf)
      With Tobias Schmidt and Matthias Strauch.
      (to appear Journal of the Institute of Math. of Jussieu.)

12. D^{\dagger}-affinity of formal models of flag varieties. (pdf)
      With C. Huyghe, T. Schmidt, and M. Strauch.
      (Submitted.)

13: An infinitesimal Noether-Lefschetz theorem for Chow groups. (pdf)
      With G.V. Ravindra.
      Journal of Pure and Applied Algebra, 221, 2017, no. 8, pp. 1926-1933

14: A localization formula for Global de Rham Epsilon factors of tensor products. (pdf)
      With Tomoyuki Abe.
      (to appear Annals of K-theory.)

15: Enriched Hodge Structures on thickenings. (pdf)
      With M. Nori and V. Srinivas.
      (In preparation.)

16: Lefschetz theorems for torsion algebraic cycles in codimension 2. (pdf)
      With G.V. Ravindra.
      (to appear Advances in Math.)

17: Specialization of cycles and the K-theory elevator. (pdf)
      With C. Doran, J. Iyer, M. Kerr, J. Lewis, S. Muller-Stach, P. Paleaz.
      (Submitted)

18: Semi-Abelian Motives.
      With M. Nori.
      (In preparation)