Geometric Partial Differential Equations and Complex Geometry

几何偏微分方程和复几何

基本信息

批准号：
2231783
负责人：
Valentino Tosatti
金额：
$ 22.69万
依托单位：
New York University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2231783&HistoricalAwards=false
关键词：
Geometric Partial Differential Equations Complex

项目摘要

This project is concerned with the study of problems of geometric nature, often involving the curvature of a space or object, using primarily tools from partial differential equations. This is a central field in mathematics, which has ramifications and connections in physics and other sciences. One of the main themes of this research is the study of a class of spaces, known as Calabi-Yau, which play an important role in mathematics as well as high energy theoretical physics. According to string theory, our four-dimensional physical space-time possesses six extra dimensions which are extremely small, so that we don't normally perceive them, but are crucial for understanding elementary particles. These six dimensions together form a tiny Calabi-Yau space, which captures essential features of particle physics. Understanding its geometry would allow us to understand how particles are created and how they interact, and is one of the main current problems in mathematical physics. The PI will use techniques from geometric analysis and nonlinear partial differential equations to investigate problems about the geometry of complex and symplectic manifolds. The first project is about understanding limits of Ricci-flat Calabi-Yau manifolds as the Kahler class degenerates. This is closely related to the theory of mirror symmetry, which was inspired by physical considerations. The second project concerns the long-time behavior of the Ricci flow on compact Kahler manifolds, in the most difficult case when collapsing occurs at infinite time. The Ricci flow was used spectacularly to prove the Poincare and Geometrization conjectures for 3-manifolds, and understanding its behavior on higher-dimensional manifolds is a central problem in the field. The third project is centered on Donaldson's program to extend Yau's solution of the Calabi Conjecture in Kahler geometry to symplectic four-manifolds. This would provide a new analytic tool to construct symplectic forms four-manifolds as solutions of a highly nonlinear PDE, and would have striking applications in symplectic topology.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.

该项目涉及几何性质问题的研究，通常涉及空间或物体的曲率，主要使用偏微分方程的工具。这是数学的核心领域，在物理学和其他科学中具有影响和联系。这项研究的主题之一是研究一类被称为卡拉比-丘的空间，它在数学和高能理论物理中发挥着重要作用。根据弦理论，我们的四维物理时空拥有六个非常小的额外维度，因此我们通常无法感知它们，但对于理解基本粒子至关重要。这六个维度共同形成了一个微小的卡拉比-丘空间，它捕捉了粒子物理学的基本特征。了解它的几何形状将使我们能够了解粒子是如何产生的以及它们如何相互作用，这是数学物理学中当前的主要问题之一。 PI 将使用几何分析和非线性偏微分方程技术来研究复杂流形和辛流形的几何问题。第一个项目是关于了解卡勒级退化时 Ricci 平坦 Calabi-Yau 流形的极限。这与受物理考虑启发的镜面对称理论密切相关。第二个项目涉及紧凑卡勒流形上 Ricci 流的长期行为，在最困难的情况下，当无限时间发生塌陷时。 Ricci 流被广泛用于证明 3 流形的庞加莱猜想和几何化猜想，而理解其在高维流形上的行为是该领域的一个中心问题。第三个项目以唐纳森的计划为中心，将丘的卡勒几何中卡拉比猜想的解扩展到辛四流形。这将为构造辛形式四流形作为高度非线性偏微分方程的解提供一种新的分析工具，并将在辛拓扑中具有引人注目的应用。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力评估进行评估，被认为值得支持。优点和更广泛的影响审查标准。