## Seminar:

Spring 2011, Thursdays, Milner 216, 4:00-4:50 PM
__ Date:__ ** February 17 **

**" mini-talks "**

* Prof. Matthew Young

* Prof. Alexei Poltoratski

* Prof. Matthew Young

* Prof. Alexei Poltoratski

__Abstract__ *M. Young*:** "Average ranks of elliptic curves" Abstract**: 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"**

Abstract: 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.

Abstract