Geometric Partial Differential Equations and Complex Geometry

几何偏微分方程和复几何

• 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

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的工具。这是数学的一个中心领域，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空间，称为Calabi-Yau，它在数学和高能理论物理中起着重要作用。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-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，这是数学物理学目前的主要问题之一。PI将使用几何分析和非线性偏微分方程的技术来研究复杂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平坦的Calabi-Yau流形的极限。这与镜像对称理论密切相关，镜像对称理论的灵感来自于物理考虑。第二个项目关注的是紧致Kahler流形上的Ricci流的长时间行为，在最困难的情况下，崩溃发生在无限时间。Ricci流被广泛地用于证明三维流形的Poincare定理和几何化定理，而理解它在高维流形上的行为是该领域的一个中心问题。第三个项目是集中在唐纳森的计划，以延长丘的解决方案的卡拉比猜想在卡勒几何辛四流形。这将提供一种新的分析工具，以构造辛形式四维流形作为高度非线性偏微分方程的解，并将在辛拓扑学中有惊人的应用。该奖项反映了NSF的法定使命，并已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的支持。

项目成果

Gaps in the Support of Canonical Currents on Projective K3 Surfaces

投影 K3 表面上规范电流的支持差距

• DOI: 10.1007/s12220-023-01526-0
• 发表时间: 2024
• 期刊: The Journal of Geometric Analysis
• 作者: Filip, Simion;Tosatti, Valentino
• 通讯作者: Tosatti, Valentino

Canonical currents and heights for K3 surfaces

• DOI: 10.4310/cjm.2023.v11.n3.a2
• 发表时间: 2021-03
• 期刊: Cambridge Journal of Mathematics
• 影响因子: 1.6
• 作者: Simion Filip;Valentino Tosatti

Leafwise flat forms on Inoue-Bombieri surfaces

• DOI: 10.1016/j.jfa.2023.110015
• 发表时间: 2021-06
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Daniele Angella;Valentino Tosatti