Research

Research Interests

Most of my research involves p-adic methods applied to algebraic geometry. Recently I have been applying methods from p-adic cohomology to study L-functions on curves in characteristic p>0. In particular, I have resolved a conjecture of Deligne, that the irregular Hodge filtration of an exponential differential equation on a Riemann surface forces bounds on the p-adic valuations of Gauss sums on curves (see the articles starting with "p-adic estimates".) This conjecture gives concrete p-adic estimates for exponential sums and character sums on curves. The methods developed in this work have been further applied to Daqing Wan's program on the behavior of p-adic slopes along p-adic towers of curves. I have also done a good amount of work on p-adic representations that arise from slope filtration of F-isocrystals. This encompasses representations arising from p-adic Tate modules and p-adic etale cohomology for varieties in characteristic p. While these representations do not share the same nice properties as their l-adic counterparts (e.g. p-adic representations can have infinite monodromy), they satisfy some mysterious and miraculous properties. One is naturally leads to the concept of F-crystals with logarithmic decay, which is the subject of several of my articles. My thesis was on p-adic modular forms and p-adic L-functions. Although my research has strayed from these topics, I still have a lingering interest.

Published or Accepted Papers

• 1. Some unlikely intersections between the Torelli locus and Newton strata in A_g, Journal de Théorie des Nombres de Bordeaux.

  2. p-adic estimates of exponential sums on curves, Algebra & Number Theory.

  3. Log-decay F-isocrystals on higher dimensional varieties, Math. Res. Lett.

  4. The monodromy of unit-root F-isocrystals with geometric origin, to appear in Compositio Mathematica.

  5. p-adic L-functions on Hida families, to appear in Annales de l’Institut Fourier.

  6. p-adic estimates of abelian Artin L-functions on curves, to appear in Forum of Math, Sigma.

Preprints or unpublished manuscripts

1. Newton Polygons of Sums on Curves I: Local-to-Global Theorems (joint with James Upton), submitted.

2. Newton Polygons of Sums on Curves II: Variation in p-adic Families (joint with James Upton), submitted.

3. The monodromy of F-isocrystals with log-decay, submitted.

4. Structural Properties of Power Digraphs Modulo n, unpublished manuscript.

Research Talks

Here is a list of selected research talks. For a complete list see my CV.

• 1. February 2021, RAMpAGe (Recent Advances in Modern p-adic Geometry) Seminar, Zoom talk (upcoming)

  2. February 2021, Number Theory Seminar, University of Rochester (upcoming Zoom talk)

  3. January 2021, Automorphic Forms and Representation Theory Seminar, Purdue University (Zoom talk)

  4. January 2021, Dwork Theory Seminar, Zoom talk

  5. Spring 2020, Upstate Number Theory Conference, New York. (Plenary Speaker, Canceled due to the COVID-19)

  6. December 2019, Number Theory Seminar, University of California, San Diego.

  7. November 2019, Math Department Colloquium, University of Minnesota.

  8. April 2019, FRAGMENT Seminar, Colorado State University, Colorado.

  9. April 2019, Arithmetic Geometry and Number Theory Seminar, University of Califronia, Berkeley.

  10. April 2019, Joint Columbia-CUNY-NYU Number Theory Seminar, New York University.

  11. April 2019, Geometric Applications of the Langlands Correspondence Working Seminar, Institute of Advanced Study/Princeton.

  12. February 2019, Algebra and Geometry Seminar, California Institute of Technology.

  13. November 2018, UCLA Number Theory Seminar, University of Califronia, Los Angeles.

  14. November 2018, p-adic Cohomology and Arithmetic Geometry, Tohoku University.

  15. October 2017, p-adic Cohomology and Arithmetic Applications, Banff International Research Station.

  16. February 2017, Séminaire d'arithmétique à Lyon, École normale supérieure de Lyon.

  17. November 2016, 30th London Number Theory Seminar, Imperial College.

  18. August 2016, Galois Representations and Automorphic Forms, Stefan Banach International Mathematical Center.

  19. December 2015, Palmetto Number Theory Series (PANTS), Clemson University (Graduate Plenary Speaker).