Differential Equations and the Geometry of Manifolds

研究PI的一个重要动机是理解空间的几何和拓扑之间的关系。后者，拓扑学，是对空间在连续拉伸或弯曲下不变的性质的研究，而前者，几何学，涉及到对距离的理解，并且更加刚性。例如，我们星球的表面是一个球体，人们通过计算大圆的弧来测量它上的距离（地球实际上是一个扁球体，但它非常接近完美的球体）。你可以想象通过推或拉小或大的区域来扭曲地球的几何形状。这种变形不像我们熟悉的地球那样吸引人，而且有很多方法可以使这个概念非常精确，以最小化某种总能量测量。这与物理原理直接相关，即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，我们就在一个四维的宇宙中。为了理解这些高维物体，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。该方案中的项目是在这样的空间上定义适当的能量，并寻找重要的最佳几何形状，使总能量最小化。用更专业的术语来说，PI的研究，从广义上讲，是使用几何起源的偏微分方程的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度量和轨道的研究，以及临界度量的模空间的性质。在与Matt Gursky的联合工作中，PI证明了各种四流形上的临界度量的存在。本文拟进一步研究此类解的模空间性质。这与Tian-Viaclovsky先前研究的轨道紧性定理有关。PI先前已经证明了在某些紧形轨道上的Yamabe问题的不可解性，这表明在光滑流形的情况下，轨道的Yamabe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。关于紧流形上反自对偶度量的存在性已经有了相当多的研究；它们已经被证明大量存在。因此，该建议的另一个目标是在某些情况下理解模空间的整体性质；特别是对于轨锥反自对偶指标。最后，PI致力于整合研究和教育，培养多层次的智力发展。PI一直积极地在数学界外展和组织会议。

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.