Mathematical Interests:  

    Number Theory: Galois Representations, Modular Forms, p-adic Hodge Theory

  Research:

Preprints:


1. The Buzzard-Diamond-Jarvis conjecture for unitary groups (with Toby Gee and Tong Liu)
   (pdf)

   Published papers:
   

2. Crystalline extensions and the weight part of Serre's conjecture (with Toby Gee and Tong Liu)
   To appear, Algebra and Number Theory. (pdf)

3. Serre weights for mod p Hilbert modular forms: the totally ramified case (with Toby Gee)
   J. Reine Angew. Math. 660 (2011), 1-26. (pdf)

4. Serre weights for quaternion algebras (with Toby Gee)
   Compositio Mathematica 147 (2011), Issue 04, 1059-1086. (pdf)

5. Poids de l'inertie modérée de certaines représentations cristallines (avec Xavier Caruso)
   Journal de Théorie des Nombres de Bordeaux 22 (2010), 79-96. (pdf)
6. Polygones de Hodge, de Newton et de l'inertie modérée des répresentations semi-stables (avec Xavier Caruso)
   Mathematische Annalen 343 (2009), no. 4, 773-789. (pdf)
7. Polynomials, meanders, and paths in the lattice of noncrossing partitions
   Transactions of the A.M.S 361 (2009), 3083-3107. (pdf ps)

8. Solutions to x³=y² in PSL(2,F_q), 3-page appendix to "Generalised knot groups distinguish the square and granny knots" by Christopher Tuffley
   Journal of Knot Theory and its Ramifications 18 (2009), no. 8, 1129-1157. (pdf)

9. Breuil modules for Raynaud schemes
   Journal of Number Theory 128 (2008), 2939-2950. (pdf dvi)

10. Harmonic algebraic curves and noncrossing partitions (with Jeremy Martin and Ted Singer).
    Discrete and Computational Geometry 37 (2007), 267-286. (pdf ps)

11. On a Conjecture of Conrad, Diamond, and Taylor.
    Duke Mathematical Journal 128 (2005), no. 1, 141-197.
    (published version / corrigendum / corrected version)

12. Modularity of some potentially Barsotti-Tate Galois representations.
    Compositio Mathematica 140 (2004), no. 1, 31-63. (pdf ps)

13. The maximum number of points on a curve of genus 4 over F₈ is 25 (with an appendix by Kristin Lauter).
    Canadian Journal of Mathematics 55 (2003), no.2, 331-352. (pdf ps). Some examples accompanying the paper: (pdf ps)

14. A note on the symmetric powers of the standard representation of S_n (with Richard P. Stanley).
    Electronic Journal of Combinatorics 7 (2000), R06, 8 pages. (pdf ps)

    Books edited:
    

15. p-adic Geometry: Lectures from the 2007 Arizona Winter School (co-edited with Dinesh Thakur)
    University Lecture Series 45, American Mathematical Society, Providence, RI, 2008. 203 pp.

    Thesis:
    

16. Modularity of some potentially Barsotti-Tate Galois representations, Ph.D. thesis, Harvard, 2001. Advisor: Richard Taylor. (pdf ps)
    (Note: don't read this. Instead, read the Compositio paper with the same title.)

  Expository:

• The Mathematics of Gauss. Ongoing project. (pdf ps)
• Cohomology on Grothendieck Topologies (with Andrew Archibald). (pdf ps) Warning: this paper still requires some revision.
• A site-theoretic characterization of points in a topological space (with Andrew Archibald and Pete Clark). (pdf ps)
• Associated Primes and Primary Decomposition. Clarifies differences among the treatments of this subject by Atiyah-MacDonald, Bourbaki, and Matsumura. (pdf ps)
• Noncommutative Algebra for Part III of Serre's Linear Representations of Finite Groups. Harvard University minor thesis. (pdf)

  Problem solving:

• From January 1998 to December 2001, I edited the Challenge Board problems section of Crux Mathematicorum with Mathematical Mayhem. Those volumes are still subscribers-only content at the Crux website, so I cannot post them all, but here are two sample columns:
  • Volume 27, Issue 1: problems (pdf ps)
  • Volume 27, Issue 6: solutions (pdf ps)
• Russell Mann and I wrote the problems for the 2001 Konhauser Problemfest. Here are the problems, together with a brief history of the Konhauser Problemfest by Stan Wagon of Macalester College: (pdf ps)