department of mathematics
Ph.D. Stanford University
Honors and Awards
- Liftoff Fellowship, Clay Mathematics Institute, 2002
- Alfred P. Sloan Research Fellowship, 2007-2009
My primary research interests lie in arithmetic geometry, especially elliptic curves and abelian varieties, their moduli spaces, and connections with the theory of modular forms. There are two main directions of research that I am currently pursuing. The first is Iwasawa theory, which deals with the behavior of number theoretic objects such as elliptic curves, Galois representations, and modular forms as they vary in families. The second deals with connections between intersection theory on Shimura varieties and the theory of modular forms. Such connections were exploited by Gross and Zagier in low dimensional cases to prove results toward the Birch and Swinnerton-Dyer conjecture, and the development of the theory in higher dimensions has seen tremendous advances in recent years.
- Intersection theory on Shimura surfaces, Compositio Mathematica 145, No. 2 (2009), 423-475.
- Central derivatives of L-functions in Hida families, Mathematische Annalen 339, No. 4 (2007), 803-816.
- Variation of Heegner points in Hida families, Inventiones Mathematicae 167, No. 1 (2007), 91-128.
- Anticyclotomic Iwasawa theory of CM elliptic curves (joint with A. Agboola), Annales de L'Institut Fourier 56, No. 4 (2006), 1001-1048.
- The Iwasawa theoretic Gross-Zagier theorem, Compositio Mathematica 141, No. 4 (2005), 811-846.
- Iwasawa theory of Heegner points on abelian varieties of GL2-type, Duke Mathematical Journal 124, No. 1 (2004), 1-45.
- The Heegner point Kolyvagin system, Compositio Mathematica 140, No. 6 (2004), 1439-1472.