Mathematical Sciences: Laplacians on Reimannian Manifolds

主要研究人员将通过比较定义在流形上的不同偏微分算子来分析流形的几何和拓扑。众所周知，偏微分算子的某些不变量与定义该算子的空间的几何之间存在着一定的关系。主要研究人员将使用热核方法来近似与微分算子相关的不变量。他将使用这种方法试图找到拉普拉斯算子的指数定理，该算子最初是为了研究物理学中的某些问题而引入的。指数定理是用于将空间的拓扑和几何与空间上定义的微分方程式联系起来的主要工具。因此，如果调查者能够为这个特定的拉普拉斯算子找到一个指数，他就建立了一种方法，可以应用于空间几何的研究。理解曲率在这种指数理论中的作用将是很重要的。流形是将我们对形状、形状和距离的理解形式化的数学对象；曲面是二维流形的一个例子。偏微分方程是用来描述大多数物理现象的方程。研究这些方程的解本身就是数学的一个主要分支。首席研究员将研究涉及这两个主题的问题，以及偏微分方程形和解之间的关系。//

The principal investigator will analyze the geometry and topology of manifolds by comparing different partial differential operators defined on the manifold. It is known that there is a relationship between certain invariants of a partial differential operator and the geometry of the space on which the operator is defined. The principal investigator will use heat kernel methods to approximate the invariants associated to the differential operators. He will use this approach to try to find an index theorem for a Laplace operator which was originally introduced to study certain problems in physics. The index theorem is the main tool used to link the topology and geometry of a space to the differential equations defined on the space. Hence if the investigator can find an index for this particular Laplace operator he will have established a method which can be applied to the study of the geometry of the space. It will be important to understand the role of curvature in such an index theory. A manifold is the mathematical object which formalizes our understanding of form, shape, and distance; a curved surface is an example of a two dimensional manifold. Partial differential equations are the equations used to describe most physical phenomena. The study of the solutions of these equations is a major branch of mathematics in itself. The principal investigator will study problems which involve both of these topics and the relationship between shape and solutions of partial differential equations. //