What is the point of numerical algebraic geometry?
Dan Bates Computations in algebraic geometry have become much more sophisticated over the last 20-30 years, thanks largely to the development of software packages like Macaulay, Singualar, and CoCoA. However, many of the methods implemented in such packages are symbolic and have certain complexity drawbacks. Numerical algebraic geometry consists of a set of numerical (and numerical-symbolic) methods for computing some of the same information that can be computed symbolically. These numerical methods parallelize nicely, and they have a different set of drawbacks. In this talk, I will describe some of the basic maneuvers in numerical algebraic geometry. As an indication of their potential value within algebraic geometry, I will describe how to compute certain genera numerically. I will also briefly describe an open problem in the field coming from a joint project with a group of engineers at ETH (A. Beccuti, I. Fotiou, M. Morari, and P. Rostalski).
