Mathematical Interests:  

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

  Research:

Preprints:


1. Serre weights for quaternion algebras (with Toby Gee)
   Submitted for publication. (pdf)

2. Serre weights for mod p Hilbert modular forms: the totally ramified case (with Toby Gee)
   Submitted for publication. (pdf)

   Published papers:
   

3. Poids de l'inertie modérée de certaines représentations cristallines (avec Xavier Caruso)
   Journal de Théorie des Nombres de Bordeaux, to appear. (pdf)
4. 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)
5. Polynomials, meanders, and paths in the lattice of noncrossing partitions
   Transactions of the A.M.S 361 (2009), 3083-3107. (pdf ps)

6. 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)

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

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

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

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

11. 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)

12. 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:
    

13. 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:
    

14. 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 ps)

  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)