Real Schubert calculus Frank Sottile University of Wisconsin 29 January 1999

In 1981, Brockett and Byrnes showed how the static feedback laws which control a given linear system are determined through the Schubert calculus of enumerative geometry. This talk will describe that connection and propose numerical homotopy methods to solve the resulting systems of polynomials.

The related questions of finding real feedback laws and of trying to do Schubert's calculus over the real numbers are intertwined with a precise conjecture of Shapiro and Shapiro. We discuss a new result which shows it is possible to do Schubert's calculus over the reals and proves a version of this conjecture. In a similar vein, problems of enumerating rational curves in the Grassmannian (also known as quantum enumerative geometry) may also be solved over the real numbers.