Variational Problems and Nonlinear Equations in Geometry

几何与分析的相互作用至少可以追溯到世纪，至今仍是数学研究中一个重要而活跃的领域。几何学的经典主题源于我们对理解物理世界的某些属性的渴望，例如角度，距离和某些形状的属性。微分几何的发展是为了使用微积分的工具来理解弯曲空间的几何形状-例如，广义相对论预测的物质空间的曲率，或者肥皂泡的性质（结果与描述黑洞的方程有关）。在同样的方式，笛卡尔意识到，平面几何可以研究使用代数，所以微分几何可以研究使用技术分析，特别是微分方程。 该项目的研究涉及几何和数学物理的不同问题，但通过数学分析在其研究中所发挥的作用而统一起来。 椭圆算子的泛函行列式是一个起源于谱理论和数学物理的问题，PI认为对特定问题的分析是一个导致四阶椭圆方程的变分问题。相应的拉格朗日量是无界的，解的存在性及其定性性质是高度非平凡的。类似的方程被用来模拟薄膜的性质。PI研究的另一个问题涉及黎曼度量的模空间，这些度量是由二次曲率量的积分给出的泛函的临界点。在PI与J. Viaclovsky的工作中，他们构造了临界点的新例子，但这种构造自然会导致关于解的模空间的各种各样的问题。二次曲率泛函的一个重要例子是Weyl泛函。临界点被称为巴赫平坦度量，包括一些重要的例子，如自对偶度量。PI研究的一个模空间问题是Singer关于正数量曲率自对偶流形的线性化问题。

The interaction of geometry and analysis dates back to at least the eighteenth century, and yet continues to be an important and highly active field of mathematical research. The classical subject of geometry grew out of our desire to understand certain properties of the physical world such as angles, distances and properties of certain shapes. Differential geometry was developed to use the tools of calculus to understand the geometry of curved spaces--for example, the curvature of space by matter as predicted by general relativity, or the properties of soap bubbles (which turn out to be related to the equations describing black holes). In the same way that Descartes realized that plane geometry can be studied using algebra, so differential geometry can be studied using techniques from analysis, especially differential equations. The research in this project involves disparate problems from geometry and mathematical physics but are united by the role played by mathematical analysis in their study. The functional determinant of an elliptic operator is a problem originating in spectral theory and mathematical physics, and the analysis of the particular problem the PI considers is a variational problem leading to a fourth order elliptic equation. The associated Lagrangian is unbounded, and the existence of solutions and their qualitative properties is highly nontrivial. Similar equations are used to model the properties of thin films. Another problem the PI studies concerns the moduli space of Riemannian metrics that are critical points of functionals given by the integral of quadratic curvature quantities. In work of the PI with J. Viaclovsky, they constructed new examples of critical points, but this construction naturally leads to various conjectures about the moduli space of solutions. An important example of a quadratic curvature functional is the Weyl functional. Critical points are called Bach-flat metrics, and include important examples such as self-dual metrics. One moduli space problem the PI studies is a question of Singer about the linearized problem for self-dual manifolds of positive scalar curvature.