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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