Differential equations and geometric structures on manifolds

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Stated in general terms, my research proposal is concerned with the relationship between differential equations and geometric structures on curved spaces. The proposal has five components which explore specific aspects of this theme, partly through questions motivated by mathematical problems and conjectures coming Physics and partly through problems which are of a more fundamentally geometric nature. Progress on these questions should help improve our understanding of the behavior of solutions of differential equations on curved spaces, help shed further light on some existing conjectures, and also help open new avenues for research. We now list the five components of our proposal. The first project deals with the long term dynamics of waves in higher dimensional axisymmetric black hole geometries, with the AdS/CFT correspondence for the geometries built on certain cohomogeneity one Sasaki-Einstein metrics and with entanglement in Kerr geometry. The second is concerned with instances in which spectral problems for differential operators can be solved at least partially by algebraic methods, and the connections with orthogonal polynomial systems as well as integrable systems. The third deals with the construction of ambi-toric geometries on manifolds and orbifolds, inspired in part by their Lorentzian analogues in General Relativity. The fourth is an attempt to extend the Cartan-Kaehler existence theorem for the existence of integral manifolds of involutive analytic exterior differential systems by using Leray's theory of the ramified Cauchy problem for systems of Cauchy-Kovalevskaia type. Finally the fifth project is an exploration of the behavior of the Ricci curvature for random metrics on spaces of dimension three or higher, using the techniques on the geometry of excursion sets pioneered by Adler and Taylor. ****************************************************