Geometry and Analysis on Calabi-Yau and Hermitian Manifolds

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AbstractAward: DMS 1308988, Principal Investigator: Valentino TosattiThe PI proposes to investigate several problems about the geometry of complex and symplectic manifolds using nonlinear partial differential equations. The first project is about understanding the ways in which Ricci-flat Calabi-Yau manifolds can degenerate in families. Building on his previous work, the PI proposes to understand these degenerations, to explore the structure of the possible limit spaces, and to apply these results to attack a conjecture of Kontsevich-Soibelman, Gross-Wilson and Todorov related to the Strominger-Yau-Zaslow picture of mirror symmetry for hyperkahler manifolds. In the second project the PI will study the geometry of Hermitian manifolds using the Chern-Ricci flow, an extension of the Kahler-Ricci flow to all complex manifolds. This flow is intimately related to the complex structure of the manifold and will be used to widen our understanding of non-Kahler compact complex surfaces. The third project is centered on Donaldson's program to extend Yau's solution of the Calabi Conjecture in Kahler geometry to symplectic four-manifolds. This would provide a new and powerful analytic tool to construct symplectic forms on closed symplectic four-manifolds as solutions of a highly nonlinear PDE, and would allow to solve basic open questions in symplectic topology, such as: given a compact almost-complex four-manifold, when are there compatible symplectic forms?The proposed research is in the field of Geometric Analysis. In this area one studies problems of geometric nature (for example how a high-dimensional space, called a manifold, is curved), using the tools of analysis and differential equations. One of the main objects of study in the proposed research are Calabi-Yau manifolds. According to string theorists, our physical space-time is not four-dimensional but rather ten-dimensional. The remaining six dimensions are extremely small, so that we don't normally perceive them, but are crucial for understanding elementary particles. These six dimensions together form a tiny geometric space, which is a Calabi-Yau manifold, and which captures essential features of particle physics. Understanding its geometry would allow us to understand how particles are created and how they interact, and is one of the main current problems in mathematical physics.