Inverse Problems For Partial Differential Equations - A Geometric Analysis Perspective

这个项目将研究各种医学成像技术的数学模型。这些问题被表示为偏微分方程(PDE)中的反问题，其中人们希望从关于其解的数据中获得关于该方程的信息。这个问题在发生在流形上的偏微分方程组的几何背景下并没有被很好地理解，本研究的目标是在这个方向上推动该领域的发展。这个项目将从几何学和拓扑学中引入新颖和创新的想法，以克服其中的一些困难。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，它还将使成像技术在更广泛的几何环境中应用，以提供更高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.