Geometric Partial Differential Equations and Complex Geometry

这个项目主要研究几何性质的问题，通常涉及空间或物体的曲率，主要使用偏微分方程组中的工具。这是数学中的一个中心领域，在物理学和其他科学中有分支和联系。这项研究的主要主题之一是研究一类被称为Calabi-Yau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额外的维度，这些维度非常小，所以我们通常不会感知它们，但对理解基本粒子至关重要。这六个维度共同构成了一个微小的卡拉比-尤空间，它捕捉到了粒子物理学的基本特征。了解它的几何结构将使我们能够了解粒子是如何产生的，以及它们是如何相互作用的，这是当前数学物理中的主要问题之一。PI将使用几何分析和非线性偏微分方程的技术来研究关于复流形和辛流形的几何问题。第一个项目是关于理解当Kahler类退化时Ricci-Flat Calabi-Yau流形的极限。这与镜面对称理论密切相关，镜面对称理论的灵感来自于物理方面的考虑。第二个项目涉及紧致Kahler流形上Ricci流的长时间行为，在最困难的情况下，无限时间发生坍塌。Ricci流被用来证明三维流形上的Poincare猜想和几何猜想，而理解它在高维流形上的行为是该领域的一个中心问题。第三个项目是以Donaldson的程序为中心，将Kahler几何中Calabi猜想的Yau解推广到辛四维流形。这将提供一种新的分析工具来构造辛形式的四维流形作为高度非线性偏微分方程组的解，并在辛拓扑中具有显著的应用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.