Computational algebraic geometry and Macaulay2

Mike Stillman (Math, Cornell University)

Computational algebraic geometry is the study of algebraic sets: the zero sets of a collection of multi-variate polynomials. This field contains algebraic methods (e.g. “Groebner bases”) to understand features of an algebraic set, as well as numerical methods (which tend to have a more geometric flavor) to solve and understand the same set (homotopy continuation, monodromy, etc). There is software for using these techniques in practice (e.g. my open source system “Macaulay2”, joint with Dan Grayson, and David Eisenbud, and now with well over a hundred user contributed packages).

Algebraic sets arise in many different settings. In this two part talk, we will first start with a problem from dynamical systems and oscillators, and consider the set of equilibrium points of the system. We translate this set to an algebraic geometry setting. Then, we will show how Macaulay2 can be used to understand certain properties of this system. After that, we will discuss some of the key aspects of the theory, including Groebner bases, and their most important applications. In the second week, we will introduce the main methods of numerical algebraic geometry, and show their use on the same problem, using Macaulay2.

Our goal in these two talks is to present the needed background in both the algebraic and numerical side of computational algebraic geometry, as well as enough of the Macaulay2 language and concepts, in enough detail to allow these three topics to become part of your toolbox. We will not assume any previous knowledge of algebraic geometry or Macaulay2!