Analysis and PDEs in complex geometry

The proposed research work focuses on open problems and developing programs arising from geometry and physics, including singularity analysis, canonical metrics, and geometric flows. The proposed research aims to develop new conceptual frameworks and technical tools that will provide profound insights and understanding of the geometric and analytic structures of the universe. The proposed project also aims to bring in research and teaching innovation both at Rutgers and in the regional mathematical community. The PI will continue to organize and participate in the integrated research/education programs and activities that will promote the education level of the nation. The PI aims to develop the theory of geometric analysis on complex spaces with singularities. In particular, he will study Riemannian geometric properties of singular Kahler metrics and related moduli problems for Einstein manifolds with applications to nonlinear partial differential equations, algebraic geometry and physics. The PI will continue to investigate and make progress in the analytic minimal model program with Ricci flow with a focus on formation of singularities as a global and local metric uniformization by solitons. The PI will also study analytic and algebraic criteria for solving global Hessian type equations and their applications to geometric analysis on singular spaces. The deep understanding of these problems will help make fundamental progress in the study of analytic and geometric singularities from fully nonlinear partial differential equations in geometry and physics. 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.