Solving the Kodaira-Spencer problem using Harmonic Analysis on torus bundles.

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The goal of this research is to answer in full the Kodaira-Spencer problem, which appeared as Problem 20 in Hirzebruch's 1954 problem list. The problem concerns the extension of the Hodge numbers from complex manifolds to almost-complex manifolds. Specifically, by considering the case of the Kodaira-Thurston manifold it will be demonstrated that the property of metric-invariance which applies in the complex setting is lost when moving to the almost-complex setting. Furthermore, it can be shown that by varying the almost-complex structure the Hodge numbers can be made arbitrarily large. The main technique used to arrive at these results is derived from the Weil-Brezin transform and will be further generalised so as to apply on any torus bundle over S1. The main beneficiaries of the proposed research will be research mathematicians working in differential geometry, symplectic geometry and complex geometry.