Geometric PDEs and Algebraic Geometry

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Geometric partial differential equations are ubiquitous in the study of natural physical phenomena-- for example, Einstein's equations in general relativity, and Maxwell's equations in electrodynamics. In practice these equations do not always admit a solution. Understanding the properties of spaces in which these equations have solutions, and furthermore understanding precisely the obstructions to finding solutions sheds new light on the underlying structure of the universe. A thorough understanding of the obstructions to solving these natural equations can lead to new and exciting predictions in the fields of string theory and high energy physics.The PI proposes to investigate several problems concerning the geometry of Kahler manifolds using techniques from partial differential equations and algebraic geometry. The central theme of this research is to relate the convergence or singularity formation of several geometric heat flows, such as the Kahler-Ricci flow and the J-flow, to algebraic geometry. In the setting of the Kahler-Ricci flow it is essential to understand the algebro-geometric properties of the blow-up set for the flow at any finite time singularity. In the non-projective setting, this involves finding transcendental approaches to algebro-geometric problems-- for example, Kawamata's base point free theorem. In the setting of the J-flow, the convergence or singularity formation of the flow is expected to be determined by an algebro-geometric stability condition. The PI aims to investigate how this stability condition is related to the analytic behavior of the flow for large time both in specific examples, and in general.