Symmetric output feedback control and isotropic Schubert calculus 

-Frank Sottile

  One area of application of algebraic geometry has been 
in the theory of the control of linear systems.  In a very precise
way, a system of linear differential equations corresponds to a
rational curve on a Grassmannian.  Many fundamental questions
about the output feedback control of such systems have been 
answered by appealing to the geometry of Grassmann manifolds.
This includes work of Hermann, Martin, Brockett, and Byrnes.

   Helmke, Rosenthal, and Wang initiated the extension of this 
to linear systems with structure corresponding to symmetric 
matrices, showing that for static feedback it is the geometry 
of the Lagrangian Grassmannian which is relevant.

  In my talk, I will explain this relation between geometry and
systems theory, and give an extension of the work of Helmke, et al.
to linear systems with skew-symmetric structure.  For static 
feedback, it is the geometry of spinor varieties which
is relevant, and for dynamic feedback it is quantum cohomology
and orbifold quantum cohomology of Lagrangian and orthogonal
Grassmannians.   This is joint work with Chris Hillar.