Problems in Complex Analysis, Partial Differential Equations, and Mathematical Physics

复分析、偏微分方程和数学物理问题

• 批准号: 1855947
• 负责人: Duong Phong
• 金额: $ 30.99万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855947&HistoricalAwards=false
• 关键词: Problems; Complex; Analysis; Partial; Differential

The goal of this project is to address some problems at the interface of complex geometry and unified string theories, whose solution is essential for further progress. An underlying common thread is supersymmetry, which appeared in physics a while ago. However, its importance in areas of geometry and analysis such as the theory of super Riemann surfaces, supermoduli space, and the theory of non-linear partial differential equations, has only recently been more fully appreciated, and our understanding is still very incomplete. Supersymmetry results in certain cohomological constraints. The partial differential equations which can implement these constraints are of considerable interest in their own right from the point of view of analysis. They pose many new challenges which should be very valuable for the future development of the theory. It has not been uncommon in the past for the same partial differential equation, if it is highly constrained by either geometry or physics, to emerge from very different applications of mathematics. We can expect the same from these new equations, and progress on them to be of wide value. The research project also brings together ideas and techniques from several areas of mathematics and physics, and it should provide an excellent training ground for students and young postdoctoral researchers.More specifically, the cohomological constraints arising from supersymmetry are generalizations of, but may be markedly different from, the Kahler condition of Hermitian metrics. As such, they lead on one side to non-Kahler geometry, and on the other side, to partial differential equations which can be much more complicated than the complex Monge-Ampere equation or the Kahler-Ricci flow. New difficulties arise from the facts that the equations are systems, or they may involve higher powers of the curvature tensor, or the dependence of their long-time behavior on the initial data may be more delicate. A major goal of this project is to develop a general theory for such equations, beginning with Anomaly flows. These are flows introduced by the PI and collaborators with the precise goal of implementing cohomological constraints, and which have shown their power in providing new proofs of fundamental results in Kahler geometry such as the Fu-Yau theorem and Yau's solution of the Calabi conjecture. Another goal is the development of a hybrid cohomology which can help extract holomorphic scattering amplitudes from non-holomorphic projections from supermoduli space to moduli space.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.

该项目的目标是解决复杂几何和统一弦理论界面上的一些问题，这些问题的解决对于进一步的进展至关重要。 一个潜在的共同点是超对称性，它不久前出现在物理学中。 然而，它在几何和分析领域的重要性，如超黎曼曲面理论，超模空间和非线性偏微分方程理论，直到最近才得到更充分的理解，我们的理解仍然非常不完整。 超对称性导致某些上同调约束。 从分析的观点来看，能够实现这些约束的偏微分方程本身就具有相当大的意义。 它们提出了许多新的挑战，这对该理论的未来发展非常有价值。 在过去，如果同一个偏微分方程受到几何或物理的高度约束，那么它从非常不同的数学应用中出现并不罕见。 我们可以从这些新方程中得到同样的结果，并且在它们方面的进展具有广泛的价值。 该研究项目还汇集了来自数学和物理学几个领域的思想和技术，它应该为学生和年轻的博士后研究人员提供一个很好的培训基地。更具体地说，超对称性产生的上同调约束是埃尔米特度量的卡勒条件的推广，但可能明显不同。因此，它们一方面导致非Kahler几何，另一方面导致偏微分方程，其可能比复杂的Monge-Ampere方程或Kahler-Ricci流复杂得多。 新的困难来自于方程是系统的事实，或者它们可能涉及更高的曲率张量幂，或者它们的长期行为对初始数据的依赖性可能更加微妙。 这个项目的一个主要目标是开发一个一般的理论，这样的方程，开始与异常流。这些是由PI和合作者引入的流，其精确目标是实现上同调约束，并且在提供Kahler几何中的基本结果的新证明中显示了它们的力量，例如Fu-Yau定理和Yau的Calabi猜想的解决方案。 另一个目标是开发一个混合上同调，它可以帮助从超模空间到模空间的非全纯投影中提取全纯散射振幅。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。