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 whose transfer functions (and state space 
realizations) are 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 transfer functions.
For static feedback, it is the geometry of spinor varieties which
is relevant.   I will close with some challenges to algebraic
geometry---to identify spaces of curves in the Grassmannian
corresponding to some natural symmetry classes of transfer functions.