Inverse Problems For Partial Differential Equations - A Geometric Analysis Perspective

本项目将研究各种医学成像技术的数学模型。这些问题被表述为偏微分方程（PDE）中的逆问题，其中人们希望从微分方程的解的数据中获得有关微分方程的信息。这个问题在几何环境中还没有得到很好的理解，其中PDE发生在流形上，本研究的目标是在这个方向上推进该领域。本项目将从几何学和拓扑学中引入新颖和创新的思想来克服这些困难。这个项目将通过提供不同领域之间的联系来丰富数学。此外，它将使成像技术应用于更广泛的几何设置，以提供更有效和准确的非侵入性检测技术。

This project will study mathematical models of various medical imaging techniques. These problems are formulated as inverse problems in partial differential equations (PDE) where one wishes to obtain information about a differential equation from data about its solutions. This problem is not well understood in the geometric setting where the PDE is taking place on a manifold and the goal of this research is to advance the field in this direction. This project will introduce novel and innovative ideas from geometry and topology to overcome some of these difficulties. This project will enrich mathematics by providing links between different fields. Furthermore, it will enable the application of imaging techniques in a broader geometric setting to provide more efficient and accurate non-invasive detection techniques.