Differential Equations and the Geometry of Manifolds

研究PI的一个重要动机是了解空间的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，而前者，几何学，涉及到距离的理解，更严格。例如，我们星球的表面是一个球体，人们通过计算大圆的弧长来测量其上的距离(地球实际上是一个扁平的球体，但它非常接近完美的球体)。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。这样的形变不如人们熟悉的圆形地球那么有吸引力，有很多方法可以让这个概念变得非常精确，使某种总能量测量最小化。这与物理原理直接相关，物理原理认为，物理系统的状态将趋向于使总能量最小的最终构型。这个想法可以推广到被称为流形的高维物体，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。在这个方案中的项目是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优几何。更专业地说，PI的研究是广义地使用起源于几何的偏微分方程解来研究可微流形的性质。PI的主要研究领域是推广爱因斯坦条件的临界度量的存在性，黎曼流形上二次曲率泛函的研究，临界ALE度量和轨道流形的研究，以及临界度量的模空间的性质。在与Matt Gursky的合作中，PI证明了各种四维流形上临界度量的存在性。本文的目的是进一步研究这类解的模空间的性质。这与Tian-Viaclovsky以前研究过的Ordiold紧性定理有关。PI已经证明了某些紧致流形上的Yamabe问题是不可解的，这表明在光滑流形的情况下，Orbilold Yamabe问题比光滑流形上的问题更加微妙。PI建议进一步探索这一现象，以及与ALE空间质量概念的联系。关于紧致流形上的反自对偶度量的存在性已经有了大量的研究，它们已经被证明是大量存在的。因此，该建议的另一个目的是在某些情况下理解模空间的全局性质；特别是对于orbillold-锥反自对偶度量。最后，PI致力于将研究和教育相结合，从多个层面培养智力发展。国际数学联合会一直积极参与数学界的外联和会议组织工作。

An important motivation for the research of the PI is to understand the relationship between the geometry and the topology of a space. The latter, topology, is the study of properties of a space which are invariant under continuous stretching or bendings of a space, while the former, geometry, involves understanding distances and is more rigid. For example, the surface of our planet is a sphere, and one measures distances on it by computing arclengths of great circles (the Earth is actually an oblate spheroid, but it is very close to being perfectly spherical). One can imagine deforming the Earth by pushing in or pulling on small or large regions to warp the geometry. Such a deformation is less appealing than the familiar round Earth, and there are many ways to make this notion very precise in terms of minimizing some sort of total energy measurement. This is directly related to physical principles which say that the state of a physical system will tend towards a final configuration which minimizes the total energy. 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 which use the least amount of energy, and maximize the symmetries of the space. The projects in this proposal are to define appropriate energies on such spaces, and to seek out the important optimal geometries which minimize the total energy.In more technical terms, the research of the PI is, broadly speaking, to 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 existence of critical metrics generalizing the Einstein condition, the study of quadratic curvature functionals on Riemannian manifolds, the study of critical ALE metrics and orbifolds, and properties of moduli spaces of critical metrics. In joint work with Matt Gursky, the PI has proved existence of critical metrics on various four-manifolds. The proposed research is to further study the properties of the moduli space of such solutions. This is related to orbifold compactness theorems previously studied by Tian-Viaclovsky. The PI has previously demonstrated non-solvability of the Yamabe problem on certain compact orbifolds, which showed that the orbifold Yamabe problem is more subtle than in the case of smooth manifolds. The PI proposes further exploration of this phenomenon, and of connections with the notion of mass of ALE spaces. There has been a considerable amount of research on the existence of anti-self-dual metrics on compact manifolds; they have been shown to exist in abundance. Another goal of the proposal is therefore to understand global properties of the moduli space in certain cases; especially for orbifold-cone anti-self-dual metrics. Finally, the PI is committed to integrating research and education and cultivating intellectual development on many levels. The PI has been active in outreach and organization of conferences in the mathematics community.