Title: An introduction to elliptic curves

Abstract: A positive integer is called "congruent" if it occurs as the 
area of a right triangle with rational sides.  For example, 6 is 
congruent as it is the area of the 3-4-5 triangle.  An ancient problem 
in number theory (1000+ years old) is to determine which integers are 
congruent, and certainly this problem falls into the category of number 
theory problems that are easy to state but extremely difficult to solve, 
at least by elementary means.  In this talk we will discuss an almost 
complete solution of the "congruent number problem" due to Tunnell from 
the 1980's.

Tunnell's solution draws on the theory of elliptic curves.  We will set 
up the connections between congruent numbers and elliptic curves and 
determine how certain elliptic curves can tell you which numbers are 
congruent.  During the talk we will also discuss a way to win $1 million.