Canonical Kahler metrics and complex Monge-Ampere equations

该项目将侧重于解决几何分析中的开放问题，并探索其在几何，拓扑和数学物理等各个领域的应用。这些问题在数学研究的活跃领域中起着核心作用，包括微分几何，偏微分方程（PDE）和高维超引力。鉴于该项目的跨学科性质，它将促进来自不同学科的研究人员之间的合作，该项目的成果将引入新的方法，并为奇异品种几何的分析研究提供有价值的见解。该项目的一个重要目标是为研究和教育的整合奠定基础，丰富数学课程，提高罗格斯-纽瓦克的数学教育。根据这一目标，首席研究员（PI）将组织研讨会和讲座，旨在促进全国数学教育的进步。PI还将在高中，本科和研究生阶段进行指导。PI将继续在复杂流形上的线性和完全非线性偏微分方程的正则性理论中开发新的方法，特别关注复杂的Monge-Ampere方程和相关的Kahler度量。这些指标的几何将从分析和几何的角度进行研究。 重点将放在研究一族Kahler度量的退化，包括Kahler-Ricci流的几何收敛和其他由几何和物理引起的流。为此，PI将推进辅助微分方程的技术，旨在分析Kahler度量族空间的紧致性。沿着这条道路，预计新的分析工具，如统一庞加莱和Sobolev不等式，以及热核估计，将被开发。此外，结合从复几何和代数几何的技术，这些工具将被用来调查的渐近行为的度量附近的奇点。此外，PI将继续探索由PI及其合作者在高维超引力中引入的抛物线方法。该奖项反映了NSF的法定使命，通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project will focus on addressing open problems in geometric analysis and exploring their applications in various fields such as geometry, topology and mathematical physics. These problems play a central role in active areas of research in mathematics, including differential geometry, partial differential equations (PDE), and high-dimensional supergravity. Given the interdisciplinary nature of this project, it will foster collaborations among researchers from various disciplines, and the outcomes of the project will introduce novel approaches and provide valuable insights into the analytic study of the geometry of singular varieties. An important objective of the project is to establish a foundation for the integration of research and education, enriching the mathematics curriculum and enhancing the mathematics education at Rutgers - Newark. In line with this objective, the Principal Investigator (PI) will organize seminars and deliver lectures, aiming to contribute towards the advancement of mathematics education nationwide. The PI will also engage in mentoring at at high school, undergraduate, and graduate levels. The PI will continue to develop novel approaches in the regularity theory for linear and fully nonlinear PDEs on complex manifolds, with a specific focus on the complex Monge-Ampere equations and the associated Kahler metrics. The geometry of these metrics will be investigated from both analytic and geometric perspectives. An emphasis will be placed on studying the degeneration of a family of Kahler metrics, including the geometric convergence of Kahler-Ricci flow and other flows arising from geometry and physics. To this end, the PI will advance the techniques of auxiliary differential equations, aiming to analyze the compactness of the space of the family of Kahler metrics. Along this path, it is expected that new analytic tools such as uniform Poincare and Sobolev inequalities, as well as heat kernel estimates, will be developed. Furthermore, combined with techniques from complex geometry and algebraic geometry, these tools will be employed to investigate the asymptotic behavior of metrics near singularities. In addition, the PI will continue to explore the parabolic approach, introduced by the PI and collaborators, in high-dimensional supergravity. This exploration aims to discover new ansatz and construct new solutions to the coupled systems, thereby deepening the understanding of the underlying 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.