Differential Equations and the Geometry of Manifolds

这个项目的重点是更好地理解空间的几何和拓扑之间的关系。后者，拓扑学，是研究在空间的连续拉伸或弯曲下不变的空间的性质。几何学涉及理解距离。例如，如果我们的星球表面被视为一个球体，我们可以通过计算大圆的弧长来测量它的距离。你可以想象通过向内推或拉小的或大的区域来扭曲几何形状，从而使地球变形。有许多方法可以使这个概念在最小化总能量测量方面精确。这个想法可以推广到更高维的物体，称为流形，这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，我们就在一个四维的宇宙中。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的距离，使用最少的能量，并最大限度地提高空间的对称性。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化的重要的最佳几何形状。PI将参与数学界的指导，推广和会议组织。在更技术性的术语中，这项研究将使用偏微分方程的解，这是几何起源，以研究可微流形的性质。PI研究的主要集中领域是四维（紧致和完全非紧致）引力瞬子的研究，K3表面上的Ricci平坦度量的崩溃序列的研究，标量平坦Kahler ALE度量的全局模空间的构建，以及轨道Yamabe问题的研究。在与Hein，Sun和Zhang的持续工作中，PI已经在K3曲面上构建了Ricci平坦度量的新例子，这些曲面坍缩到一个区间，海森堡流形作为纤维出现在规则的坍缩区域中。在这种情况下，ALH型恒星的引力瞬子会冒泡，对这类瞬子有更好的理解是很有意义的。在与Chen和Zhang的联合工作中，PI已经在K3表面上构建了Ricci平坦度量的塌陷序列的例子，该表面具有ALG和ALG星泡，并且更好地理解这些类型的瞬子也很有趣，它们具有二次体积增长。在与Han的合作中，PI将进一步研究无穷远某些群的标量平坦Kahler ALE度量的模空间，并在孤立商奇点的变形的非Artin分量上找到这种度量的新例子。在与Ju的联合工作中，PI正在研究轨道Yamabe问题的紧致性和存在性结果，由于ALE度量的正质量定理的失败，该问题与光滑情况有很大不同。最后，PI致力于整合研究和教育，并在多个层面上培养智力发展。该奖项反映了NSF的法定使命，并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，被认为值得支持。

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.