Spring 2011, Thursdays, Milner 216, 4:00-4:50 PM

Date: February 17

"  mini-talks "
* Prof. Matthew Young

* Prof. Alexei Poltoratski


M. Young: "Average ranks of elliptic curves"

: A rational elliptic curve can be thought of as the equation y^2=x^3 + ax + b where a and b are integers. One of the most basic questions we can ask about such equations is if they have any solutions with rational values for x and y. This turns out to be a very difficult problem. The Birch and Swinnerton-Dyer conjecture gives precise information about the arithmetic of the elliptic curve (including whether it has infinitely rational points or not) in terms of an associated L-function (which is a Dirichlet series that packages some of the information about the elliptic curve mod p for all primes p). This connection allows us to investigate the following question: Suppose one picks a and b at random many times; how often do we expect the elliptic curve to have rational points? I will discuss some recent developments in this area.

A. Poltoratski: "Applications of Harmonic Analysis in Spectral Theory"

: I will discuss recently found connections between classical problems of Harmonic Analysis on completeness of exponential functions in L2-spaces and spectral problems for the Schroedinger equation.