## Laurent Gruson

It is well known that the restricted Picard group $Pic_0(S)$ of a cubic surface $S$ in $P_3$ is isomorphic to the root lattice $E_6$. The choice of such an isomorphism is a markng of the surface. The moduli space of marked cubic surfaces is thus acted on by the Weyl group $W(E_6)$. We seek a equivariant morphism of this moduli space into $P(E_6 \otimes C)$. Ellingsrud and Peskine have identified $Pic_0(S)\otimes C$ with the vector space W of dual quadrics apolar" to S. Our morphism transforms $S$ into the hyperplane of $W$ consisting of quadrics containing the faces of the Sylvester pentahedron of $S$. We give an explicit form of this parametrisation

## 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.