Report on the work of Clayton Shonkwiler. In his PhD thesis, Shonkwiler provides a new set of invariants which measure the relative position, in the L2 inner product, of the harmonic fields which represent absolute and relative cohomology. These invariant angles always vanish when the manifold is closed, never vanish when it has a boundary, and appear to go to zero as the manifold closes up. They can be computed explicitly for certain subdomains of complex projective spaces and Grassmann manifolds, using invariant differ- ential forms and the solution of systems of differential equations, and in these cases do go to zero as the boundary shrinks. Shonkwiler then discovers an original and unexpected connection between these angles and the generalized Dirichlet-to-Neumann map for differential forms (higher dimensional electrical impedance tomography), and applies this towards detection of the cup product structure from boundary data, a problem proposed last year by Belishev and Sharafutdinov. Herman Glück visitará granada desde el 4 al 8 de mayo.

The Four Vertex Theorem, one of the earliest results in global differential geometry, says that a simple closed curve in the plane, other than a circle, must have at least four "vertices", that is, at least four points where the curvature has a local maximum or local minimum. In 1909 Syamadas Mukhopadhyaya proved this for strictly convex curves in the plane, and in 1912 Adolf Kneser proved it for all simple closed curves in the plane, not just the strictly convex ones. The Converse to the Four Vertex Theorem says that any continuous real-valued function on the circle which has at least two local maxima and two local minima is the curvature function of a simple closed curve in the plane. In 1971 I proved this for strictly positive preassigned curvature, and in 1997 Björn Dahlberg proved the full converse, without the restriction that the curvature be strictly positive. Herman Glück visitará granada desde el 4 al 8 de mayo