Abstract of Michael Stoll's talk
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Taking up an old idea of Hermite and Julia, we show how to define
a reasonable notion of being reduced for a binary form over the
integers. In this way, we can select a particular form in each
SL(2,Z)-orbit (SL(2,Z) acts by linear substitutions on the variables)
having (fairly) small coefficients. This can be used to find a smaller
equation for a given hyperelliptic curve y^2 = f(x). (This is joint
work with John Cremona.)

Abstract of William Stein's talk
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The elliptic curves 54B, 431A, and 5389A star in Three Curves, the
story of a small group of adventurous elliptic curves who are determined to
provide counterexamples to three tempting assertions. Finding a map they
believe will take them to the gold, they embark on a journey that leads to 
unexpected discoveries, enabling them to rise to heroic challenges that 
drastically change their lives.