Call a number $n$ congruent when it is the area of a right triangle with all rational sides; multiplying the sides through by $s$ adjusts the area by the factor $s^2$, thus we can always adjust the triangle so that the area is a squarefree positive integer. This leads to the congruent number problem: determine all squarefree positive integers that are congruent numbers; more specifically, find an algorithm that, given $n$, determines whether $n$ is congruent.
An equivalent definition of congruent number is that $n$ is congruent if and only if there exists a rational number $x$ such that the three numbers $x,x+n,x-n$ are each squares of rational numbers.
By listing all Pythagorean triples, one can easily produce a listing of all congruent numbers, but there is no guarantee how soon a particular number will appear on this list; so this is not the sought algorithm.
Tunnel's theorem gives a necessary condition for $n$ to be congruent. Furthermore, if the Birch & Swinnerton-Dyer conjecture (from the theory of elliptic curves) holds, then Tunnel has a sufficient condition. So this would solve the problem. So elliptic curves have something to do with congruent numbers.
Given a congruent number $n$ arising from the triangle with sides $X<Y<Z$, we obtain the three squares $x=(Z/2)^2$, $x+n$, $x-n$, and in fact we obtain a rational point $(x,y)$ on the elliptic curve $y^2=x^3-n^2x$. This rational point has the following three properties: (1) $x$ is a square; (2) the denominator of $x$ is even; (3) the numerator of $x$ is relatively prime to $n$. Conversely, if the elliptic curve $y^2=x^3-n^2x$ has a rational point $(x,y)$ satisfying (1),(2),(3), then $n$ is a congruent number. Thus the congruent number problem has become: given a positive squarefree integer $n$, does the elliptic curve $y^2=x^3-n^2x$ have a rational point $(x,y)$ satisfying (1),(2),(3)?
There were two or three more talks, but I didn't write up summaries at the time, and no longer have a record of when I spoke or what I spoke about.
Writing an integer as a sum of two cubes, for example $1729=1^3+12^3$ or $1729=9^3+10^3$, amounts to finding integer points on the family of elliptic curves $x^3+y^3=m$.
I'll explain the significance of the modular group SL(2,Z) in general, what this has to do with elliptic curves, how modular functions and modular forms come into play, and what it means for an elliptic curve to be modular. (The Taniyama-Shimura-Weil conjecture claims all curves to be modular... Wiles's proof of this in special cases led to Fermat's Last Theorem.)
[I'm adding this 1 Aug 98 from memory - ARP] Gary discussed some theorems that help you calculate the torsion subgroup of the group of rational points on an elliptic curve. Some basic results were proved. Then Mazur's theorem was quoted, which states precisely which torsion subgroups are possible. Gary gave examples of an elliptic curve with each possible torsion subgroup, and for a few of them, we calculated the torsion subgroup together.
Here is a Mathematica notebook I wrote (August 1998) based on the printout of a Mathematica file handed to us by Gary.