The lace expansion and the enumeration of self-avoiding walks

The lace expansion is an elegant combinatorial construction that provides a recursion relation for the number of self-avoiding walks. We first give an introduction to the lace expansion, and then explain how it has been used recently (in joint work with Nathan Clisby and Richard Liang) to enumerate self-avoiding walks on the hypercubic lattice up to n=30 steps in dimension 3, and up to n=24 steps in all dimensions above 3. Major improvements to the 1/d expansion for the connective constant have also been obtained. In addition, an algorithmic improvement called the two-step method will be described.