3264 Conics in a Second

Abstract:

Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical algebraic geometry determines these solutions for any given instance. This lecture illustrates how these two fields complement each other, especially in the light of emerging new applications. We start with a gem from 19th century geometry, namely the 3264 conics that are tangent to five given conics in the plane. This topic was featured in the January 2020 issue of the Notices of the American Mathematical Society.  We conclude with an application in statistics, namely maximum likelihood estimation for linear Gaussian covariance models.