Singularity Formation in Kahler Geometry and Yang-Mills Instantons

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This project will study singularity formation of some special geometric structures governed by non-linear partial differential equations. These structures are generalizations of solutions to the Einstein’s equation for gravity and Maxwell’s equation for electromagnetism, and play significant roles in modern theoretical physics (for example string theory and M-theory). The solution to these problems will enhance the understanding of deep interaction between various research directions in mathematics. In the meantime, there are many interesting related questions that may serve as research projects for the training of graduate students interested in this area.The PI will study problems in two specific directions. First, the PI would like to study collapsing of Calabi-Yau metrics, and connections with algebraic geometry and moduli compactification. Previous work of the PI and collaborators has yielded a satisfactory description in the case of minimal degenerations and the PI proposes to study the more general situation. This requires exploiting the existing techniques from collapsing theory, and further developing those. Secondly, the PI will study singularity analysis for Yang-Mills instantons. In the case of Hermitian-Yang-Mills connections over Kahler manifolds, recent work of the PI and Xuemiao Chen gives a complete algebro-geometric description of the tangent cones, and the PI wants to push these further by discovering more refined structures, and by investigating the case of G2 instantons. This requires building algebro-geometric framework, as well as extending some of these to the more analytic setting.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.