Geometric PDEs and Algebraic Geometry

几何偏微分方程和代数几何

• 批准号: 1856457
• 负责人: Tristan Collins
• 金额: $ 0.29万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1856457&HistoricalAwards=false
• 关键词: Geometric; PDEs; Algebraic; Geometry

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.

几何偏微分方程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。 实际上，这些方程并不总是有解。 了解这些方程有解的空间的性质，并进一步精确地了解寻找解的障碍，可以使我们对宇宙的基本结构有新的了解。 深入了解这些自然方程的求解障碍，可以在弦理论和高能物理学领域带来新的令人兴奋的预测。PI提出利用偏微分方程和代数几何的技术来研究Kahler流形的几何问题。 本研究的中心主题是将几种几何热流，如Kahler-Ricci流和J流的收敛或奇异性形成与代数几何联系起来。 在Kahler-Ricci流的背景下，必须理解流在任意有限时间奇点处的爆破集的代数几何性质。 在非投射设置，这涉及到寻找超越的方法来代数几何问题-例如，川俣的基点自由定理。 在J-流的设置中，流的收敛或奇点形成预期由代数几何稳定性条件确定。PI的目的是研究如何在特定的例子中，以及在一般情况下，这种稳定性条件与大时间的流的分析行为有关。