Commutative Algebra and Algebraic Geometry
Tuesdays, 3:45-6pm in Evans 939
Organizer: David Eisenbud
Date: Feb 26
A model of the moduli space of marked cubic surfaces
Speaker: 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 parametrization.
Symmetric output feedback control and isotropic Schubert calculus
Speaker: 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.