# 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

- 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"