Student Research

See my research blog for project ideas.

Current Students (who I mentor)

All students listed below are at University of Washington, unless otherwise noted.

• Alyson Deines (Ph.D. student):
  • Kolyvagin's conjecture over totally real fields
• Joanna Gaski (Masters thesis student)

Recent Students (who I played a nontrivial role mentoring)

• Robert Bradshaw (Ph.D. student, expected Ph.D. 2010):
  • Thesis: Computing L-functions
  • Rigorous computation with complex-analytic L-functions (building on work of Dokchitser).
  • Explicit investigations into p-adic analogues of the Birch and Swinnerton-Dyer conjecture for jacobians of genus 2 curves
  • Coleman integration
  • p-adic heights and Monsky-Washnitzer cohomology
  • Symbolic algebra (automatic coercion)

  • Compilers (http://www.cython.org)

  • Asymptotically fast exact linear algebra
• Robert Miller (Ph.D. student, expected Ph.D. 2010):
  • Thesis: Computational Verification of the Birch and Swinnerton-Dyer Conjecture
  • Construction of rational points on high-rank elliptic curves using congruences and Galois cohomology
  • Computation of canonical labelings of graphs
• Tom Boothby (Masters student):
  • Computing L-functions to high precision
  • Multiplication of matrices over GF(3^k)
  • Fast polynomial evaluation
  • Tables of modular forms
• Jennifer Balakrishnan (Ph.D. student at MIT):
  • Explicit computation of Kolyvagin classes
• Christopher Swierczewski (undergraduate):
  • Senior Thesis on the relationships between the Convergence Sato-Tate Conjecture and the Generalized Riemann Hypothesis. In particular, a focus on the implication of the latter conjecture from the former. Examining different formulations of Sato-Tate.
  • Computations related to the Sato-Tate conjecture
• Bobby Moretti (undergraduate):
  • Weierstrass equations for elliptic curves [mostly done]
  • Very high-precision evaluation of L-series of elliptic curves of high rank

  • See http://sage.math.washington.edu/home/moretti/vigre.pdf

• Dimitar Jetchev (Ph.D., Berkeley)
  • Ph.D. thesis improving on Kolyvagin's Sha bounds
  • Explicit computation of Heegner points and Kolyvagin classes
• Emily Kirkman (undergraduate)
  • Senior thesis
• Koopa Koo (Ph.D. student of Greenberg, UW)
  • Irreducibility of Hecke polynomials

  • for higher degree newforms

• Ifti Burhanuddin (Ph.D, USC, 2007)
  • Thesis: "Some computational problems motivated by the Birch and Swinnerton-Dyer Conjecture"