Geometry/topology Seminar
Tuesday, September 15, 2009, 4:30pm, 119 Physics
Clay Shonkwiler (Haverford College)
Poincaré duality angles on Riemannian manifolds with boundary
Abstract:
In this talk I will discuss some new invariants of compact Riemannian manifolds called Poincaré duality angles. These angles measure the relative positions in the L2 inner product of the harmonic forms which represent absolute and relative cohomology. These invariant angles always vanish when the manifold is closed, never vanish when it has boundary, and appear to go to zero as the manifold closes up. I will show how, using invariant differential forms, to explicitly compute the Poincaré duality angles of certain manifolds with boundary obtained from complex projective spaces and Grassmann manifolds. I will also mention an unexpected connection between these angles and the generalized Dirichlet-to-Neumann map for differential forms which can be applied towards detection of the cup product structure from boundary data, as proposed last year by Belishev and Sharafutdinov.

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