Heuristics for the growth of Mordell-Weil ranks in big extensions of number fields. BIRS, June 2016.

Elliptic curves and Hilbert's Tenth Problem. MAA sectional meeting, UCI, October 16 2007.

Right triangles and elliptic curves. Ross Program Reunion, OSU, July 20 2007.

The solving of Fermat's Last Theorem. The story of Fermat's Last Theorem and its solution by Andrew Wiles. "Discover the Physical Sciences" breakfast lecture, UCI, March 20 2007.

Applications of number theory and algebraic geometry to cryptography.
Plenary lecture at *Global KMS Day*, Seoul, October 28, 2006.

Rational points on abelian varieties and The Birch & Swinnerton-Dyer conjecture. Lectures at the Introductory Workshop to the MSRI program "Rational and integral points on higher-dimensional varieties", January 2006.

Elliptic curves and Iwasawa theory. ICM lecture, Beijing, August 2002.

Ranks of elliptic curves. Fields Institute 10th anniversay celebration, Toronto, June 2002.

Ranks of elliptic curves. AMS-MAA-SIAM lecture, Washington DC, January 20 2000.