Variational Problems and Nonlinear Equations in Geometry

几何和分析的相互作用至少可以追溯到18世纪，但仍然是数学研究的一个重要和高度活跃的领域。几何这门经典学科源于我们想要了解物理世界的某些属性，如角度、距离和某些形状的属性。发展微分几何是为了使用微积分的工具来理解弯曲空间的几何--例如，广义相对论预测的空间由物质所产生的曲率，或者肥皂泡的性质(事实证明，这与描述黑洞的方程有关)。就像笛卡尔意识到平面几何可以用代数来研究一样，所以微分几何也可以用分析中的技巧来研究，特别是微分方程。这个项目的研究涉及与几何和数学物理完全不同的问题，但由于数学分析在他们的研究中所起的作用而联系在一起。椭圆算子的函数行列式是一个起源于谱理论和数学物理的问题，而PI所考虑的特定问题的分析是一个导致四阶椭圆方程的变分问题。相关的拉格朗日是无界的，解的存在及其定性性质是非常不平凡的。类似的方程也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积分给出的泛函的临界点。在与J.Viaclovsky合作的PI中，他们构造了临界点的新例子，但这种构造自然导致了关于解的模空间的各种猜想。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指标，包括自对偶指标等重要例子。PI研究的一个模空间问题是Singer关于正标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.