The Kerr solutions to Einstein's equations describe rotating black holes. For the wave equation in flat-space and outside the non-rotating, Schwarzschild black holes, one method for proving decay is the vector-field method, which uses the energy-momentum tensor and vector-fields. Outside the Schwarzschild black hole, a key intermediate step in proving decay involved proving a Morawetz estimate using a vector-field which pointed away from the photon sphere, where null geodesics orbit the black hole. Outside the Kerr black hole, the photon orbits have a more complicated structure. By using the hidden symmetry of Kerr, we can replace the Morawetz vector-field by a fifth-order operator which, in an appropriate sense, points away from the photon orbits. This allows us to prove the necessary Morawetz estimate, also called a local energy decay estimate, which is a key step in proving pointwise decay estimates.

This is joint work with Lars Andersson.

At the end, I also hope to speak about current work on Maxwell's equations.

http://www.math.princeton.edu/~seminar/2010-11-sem/OttoAbstract9-20-2010.pdf

Zero schemes of exact 1-forms have received more attention recently as moduli spaces associated to Calabi-Yau threefolds; they are called gradient schemes or critical schemes. In this talk I will introduce the notion of "multi-gradient schemes" as an obvious generalization and explain their classification in the codimension one and monomial cases, as well as how they naturally arise as certain moduli spaces associated to varieties with globally generated canonical bundles.