Inverse problems in partial differential equations and medical imaging

当科学家和工程师试图在不穿透甚至不接触物体的情况下确定物体内部（例如人体、原子核、飞机机翼、地球或太阳）的特性时，他们会分析其对声波、弹性波、电磁波或量子机械波的影响。在数学上，波的传播是由偏微分方程来模拟的，在提议的研究中考虑的一类问题包括从解（波）的可用测量中确定这种方程的系数（它编码未知的介质属性）。在对这类问题的分析理解方面的几项突破揭示了一个美丽的潜在数学结构，并导致了该领域活动的爆炸式增长。取得了很大进展，但一些根本性和难点问题还没有解决。我的研究将试图理解其中一些分析问题。我还将与工程方面的研究人员合作，开发下一代医疗诊断仪器，用于成像组织的电学特性。

When scientists and engineers seek to determine properties of the interior of an object (e.g. the human body, the nucleus of an atom, an airplane wing, the earth, or the sun) without penetrating or even accessing the object, the analyze its effect on acoustic, elastic, electromagnetic or quantum mechanical waves. Mathematically, wave propagation is modeled by Partial Differential Equations and the class of problems considered in the proposed research consists in determining the coefficients of such and equation (which encode the unknown medium properties), from available measurements of the solutions (the waves). Several breakthroughs in the analytical understanding of such problems have uncovered a beautiful underlying mathematical structure and have led to an explosion of activity in the field. There has been considerable progress, yet some fundamental and difficult problems remain open. My research will attempt to understand some of these analytic problems. I will also work in collaboration with researchers in engineering towards the development of next generation medical diagnostic instrumentation for imaging electric properties of tissue.