Talk abstract:

Linear Systems as Vector Bundles on Spheres

Clyde Martin
Texas Tech University
Lubbock, TX 79409
Clyde.Martin@mail.iehh.ttu.edu

In work that was done almost a quarter century ago it was recognized that linear systems and algebraic geometry were closely related. Robert Hermann and Clyde Martin published a series of papers exploiting this relation. The seminal result obtained was that that the Grassmannian manifolds were central to understanding the relation between geometry and linear systems. They showed that there are natural embeddings of linear systems as curves in grassmannians and as a consequence showed that there is a natural vector bundle on the sphere that is associated with each linear system. A classic paper of Grothendeik shows that any such bundle can be decomposed as sums of line bundles and that the degrees of the line bundles are a natural invariant. Herman and Martin showed that these degrees are in fact the feedback invariants of the associated linear system. In later work by Byrnes, Hazewinkel and others it was shown that Grothendeiks theorem is equivalent to the canonical forms for linear systems under feedback. David Lieberman pointed out that this construction is the 0-dimensional case of the Goppa codes. Unfortunately it seems that there is not a higher dimensional analogy between systems and Goppa codes.

Back to Codes, Systems and Graphical Models