Macaulay 2 is a specialized computer algebra system for computing in algebraic geometry and commutative algebra (and related fields). The system is under active development, is freely available for linux, MacOS, Windows, and other systems, and is being used by many researchers. Dan Grayson (of University of Illinois) and I have been working on this project since 1993.
    We show some ways that Macaulay 2 can be used to aid research in algebraic geometry. One of our examples will be a "mystery curve", and we will attempt to understand the structure of this curve. Along the way, we will see how to compute with ideals, Groebner bases, syzygies, divisors, and projective geometry.
    One goal of the talk is to give some idea of what can currently be computed in algebraic geometry, in general, and also to give an indication of what Macaulay 2 is capable of.