Theory and Applications of Numerical Algebraic Geometry

Polynomials are everywhere. Designing windshield wiper mechanisms, modeling stoichiometric chemical reactions, nonlinear optimization and control, and many other scenarios, all generate polynomial systems -- and demand they be solved. The functions can be univariate, or multivariate. The systems can be structured, or dense. The solutions can be sets of points, or entire positive-dimensional spaces. And they can be computed in a variety of ways.

Numerical algebraic geometry is a field of computational mathematics which solves and manipulates polynomial systems primarily using numerical continuation. The field is highly application-driven, and continues to mature, both in terms of theoretical boundaries and software implementations. Horizons include self-tuning trackers and systems having billions of solutions. This talk will discuss numerical algebraic geometry -- its motivations, incarnations, and applications.