Geometric PDEs and Algebraic Geometry

几何偏微分方程组在自然物理现象的研究中无处不在--例如，广义相对论中的爱因斯坦方程和电动力学中的麦克斯韦方程。在实践中，这些方程并不总是有解的。了解这些方程有解的空间的性质，并进一步准确地了解寻找解的障碍，将为了解宇宙的基本结构提供新的线索。对这些自然方程求解障碍的透彻理解可以在弦理论和高能物理领域带来新的令人兴奋的预测。PI建议使用偏微分方程组和代数几何的技巧来研究与Kahler流形有关的几个问题。这项研究的中心主题是将几种几何热流，如Kahler-Ricci流和J流的收敛或奇点形成与代数几何联系起来。在Kahler-Ricci流的背景下，了解该流在任意有限时间奇点的爆破集的代数几何性质是至关重要的。在非射影的背景下，这涉及到寻找解决代数几何问题的先验方法--例如，川田的基点自由定理。在J流的背景下，流的收敛或奇点的形成是由代数几何稳定性条件决定的。PI的目的是调查这种稳定性条件如何与流的分析行为在很长一段时间内相关，无论是在特定的例子中，还是在一般情况下。

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.