For thousands of years, various people have tried to find a simple way
to determine whether or not a given positive whole number is the area
of a right triangle with integer side lengths.  Such a number is
called a "congruent numbers".  It is easy to write down examples of
such numbers; for example, since 3^2 + 4^2 = 5^2, there is a right
triangle with sides of lengths 3, 4, and 5.  The area of this triangle
is 1/2 x (base) x (height) = 6, so 6 is a congruent number.  It turns
out that 1 is NOT a congruent number, but this is surprisingly tricky
to prove.

Amazingly, this problem is now mostly solved!  The "near solution"
involves beautiful objects called elliptic curves, which are central
in much of modern number theory, and the Birch and Swinnerton-Dyer
conjecture, which purports to describe their "arithmetic behavior".
If you come to my talks, perhaps someday you will understand this
conjecture well enough to give a full solution to the congruent number
problem.

I will give three lectures.  In the first, I will give a
historically-motivated description of the congruent number problem.
Then, in the second lecture, I will introduce elliptic curves and tell
you why they play a central roll in the congruent number problem.  In
my third lecture, I will describe a conjecture, which, if somebody
could only prove it, then we would finally have a solution to the
ancient congruent number problem.