Colloquium

A Gromov-Witten invariant in the real world

by Frank Sottile, Assistant Professor of Mathematics, University of Massachusetts, Amherst

Abstract

In the last (20th) century, two different groups ofscientists were led to exactly the same question in puremathematics, a problem of counting certain rational curves on aGrassmannian, that is, determining a particular Gromov-Witteninvariant of the Grassmannian. The work of one group, theoreticalphysicists, spawned important mathematical activity includingquantum cohomology. This story of mathematics inspired by physicsis well-known, even though the physical motivation remains obscureto many mathematicians.

The other group of scientists were engineers working in systemstheory, specifically on the problem of dynamic feedback control oflinear systems. While their story is less-known, theirmotivations for studying curves on Grassmannians arestraightforward and their work leads to a very concreteunderstanding of this problem.

In this talk, I will make the second story better-known,explaining how engineers were led to study spaces of curves onGrassmannians and some further mathematics inspired by their work,particularly, a recursive construction of solutions to thisproblem which shows that all solutions may be real.