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Differential Equations and the Geometry of Manifolds

微分方程和流形几何

基本信息

批准号：
2105478
负责人：
Jeff Viaclovsky
金额：
$ 49.84万
依托单位：
University of California-Irvine
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105478&HistoricalAwards=false
关键词：
Differential Equations Geometry Manifolds

项目摘要

The focus of this project is to better understand the relationship between the geometry and the topology of a space. The latter, topology, is the study of properties of a space that are invariant under continuous stretching or bending of a space. Geometry involves understanding distances. For example, if the surface of our planet is viewed as a sphere one can measure distances on it by computing arclengths of great circles. One can imagine deforming the Earth by pushing in or pulling on small or large regions to warp the geometry. There are many ways to make this notion precise in terms of minimizing a total energy measurement. This idea can be generalized to higher dimensional objects called manifolds, which are generalized versions of the surface of our planet. For example, the space that we live in is three-dimensional, and if one includes time, we are in a four-dimensional universe. In order to understand these types of higher-dimensional objects, one attempts to find the best way to measure distances on them that use the least amount of energy, and maximize the symmetries of the space. These projects will define appropriate energies on such spaces, and seek out the important optimal geometries that minimize the total energy. The PI will participate in mentoring, outreach and organization of conferences in the mathematics community.In more technical terms, this research will use solutions of partial differential equations, which are geometric in origin to study properties of differentiable manifolds. The main areas of concentration of the PI's research are the study of gravitational instantons in dimension four (both compact and complete non-compact), the study of collapsing sequences Ricci-flat metrics on K3 surfaces, the construction of a global moduli space of scalar-flat Kahler ALE metrics, and the study of the orbifold Yamabe problem. In ongoing work with Hein, Sun, and Zhang, the PI has constructed new examples of Ricci-flat metrics on K3 surfaces, which collapse to an interval, with Heisenberg nilmanifolds occurring as fibers in the regular collapsing regions. In this case, gravitational instantons of type ALH-star bubble off, and it is of interest to have a better understanding of this class of instantons. In joint work with Chen and Zhang, the PI has constructed examples of collapsing sequences of Ricci-flat metrics on the K3 surface that has both ALG and ALG-star bubbles, and it is also of interest to have a better understanding of these types of instantons, which have quadratic volume growth. In joint work with Han, the PI will conduct further study of the moduli space of scalar-flat Kahler ALE metrics for certain groups at infinity and finding new examples of such metrics on non-Artin components of deformations of isolated quotient singularities. In joint work with Ju, the PI is studying compactness and existence results for the orbifold Yamabe problem, which differs substantially from the smooth case due to the failure of the positive mass theorem for ALE metrics. Finally, the PI is committed to integrating research and education and cultivating intellectual development on many levels.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.

这个项目的重点是更好地理解空间几何和拓扑之间的关系。后者，拓扑学，是研究空间在连续拉伸或弯曲下不变的性质。几何学涉及到理解距离。例如，如果我们把地球表面看作一个球体，就可以通过计算大圆的弧长来测量其上的距离。人们可以想象通过推入或拉动小区域或大区域来扭曲几何图形来使地球变形。有许多方法可以使这一概念在最小化总能量测量方面变得精确。这个想法可以推广到称为流形的更高维的物体上，流形是我们星球表面的广义版本。例如，我们生活的空间是三维的，如果包括时间，我们就处于一个四维的宇宙中。为了理解这些类型的高维物体，人们试图找到最好的方法来测量它们上的距离，使用最少的能量，并最大化空间的对称性。这些项目将在这样的空间上定义适当的能量，并寻找使总能量最小的重要的最佳几何形状。PI将参与数学界的指导、推广和会议的组织。在更专业的术语中，这项研究将使用偏微分方程解，它的起源是几何的，以研究可微流形的性质。PI的主要研究领域是四维(紧致和完全非紧致)引力瞬子的研究，K3曲面上折叠序列Ricci平坦度规的研究，标量平坦Kahler Ale度规的整体模空间的构造，以及Orbilold Yamabe问题的研究。在与Hein，Sun和Zhang正在进行的工作中，PI在K3曲面上构造了Ricci平坦度量的新例子，K3曲面收缩成区间，Heisenberg零流形作为纤维出现在规则的收缩区域中。在这种情况下，ALH-STAR类型的引力瞬子起泡，对这类瞬子有更好的了解是有意义的。在与Chen和Zhang的合作中，PI已经在同时具有ALG和ALG-STAR气泡的K3表面上构造了Ricci平坦度规的坍塌序列的例子，而且更好地理解这些类型的瞬子也是有趣的，它们具有二次体积增长。在与韩的合作中，PI将对某些群在无穷远处的标量平坦Kahler ale度量的模空间进行进一步的研究，并在孤立商奇点的变形的非Artin分量上寻找此类度量的新例子。在与Ju的联合工作中，PI正在研究Orbilold Yamabe问题的紧性和存在性结果，由于ALE度量的正质量定理的失败，该问题与光滑情况有本质的不同。最后，PI致力于将研究和教育相结合，并在多个层面上培养智力发展。这一奖项反映了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。