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.

该项目的目的是在复杂的几何学和统一的字符串理论的界面上解决一些问题，该理论的解决方案对于进一步的进展至关重要。 一个基本的共同线程是超对称性，不久前出现在物理中。 然而，它在几何和分析领域的重要性，例如超级黎曼表面，超模型空间的理论以及非线性偏微分方程的理论，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。 超对称会导致某些共同体结构。 从分析的角度来看，可以实施这些约束的部分微分方程本身具有相当大的兴趣。 它们提出了许多新的挑战，这些挑战对于理论的未来发展非常有价值。 过去，对于相同的部分微分方程而言，如果它受到几何或物理学的高度限制，则从数学的截然不同的应用中出现。 我们可以从这些新方程式中期望相同，并且它们的进步具有广泛的价值。 该研究项目还汇集了来自数学和物理学几个领域的想法和技术，它应该为学生和年轻的博士后研究人员提供良好的培训理由。更具体地说，由超对称性产生的共同体学限制是概括的，但可能与Hermitian Hermitian Metrics的Kahler条件有明显不同。因此，它们在一侧导致非kahler几何形状，而在另一侧，部分微分方程可能比复杂的Monge-Ampere方程或Kahler-Irci-ricci流程更为复杂。 新的困难是由于方程是系统的事实，或者它们可能涉及曲率张量的较高功能，或者其长期行为对初始数据的依赖性可能更加精致。 该项目的主要目标是从异常流开始为这种方程式开发一般理论。这些是PI和合作者引入的流动，其确切目标是实施共同体的限制，并且在提供Kahler几何学基本结果的新证据（例如Fu-Yau Theorem和Yau）的解决方案和Yau的解决方案方面已经显示出了他们的力量。 另一个目标是开发混合同胞，可以帮助提取从超级统一空间到Moduli Space的非孤晶性投影中提取全体形态散射幅度。该奖项反映了NSF的法定任务，并通过评估该基金会的知识分子功能和广泛的影响来审查Criteria，并通过评估进行评估。