This project is concerned with applications of semiclassical analysis in studying the behavior of the solutions of partial differential equations in terms of the geometry of their underlying spaces. Semiclassical analysis studies the transition between quantum and classical mechanics. A unifying idea in this area is to study the behavior of quantum mechanical objects such as eigenvalues (energy levels) and eigenfunctions (quantum states) in terms of classical objects like the geodesic flow. The Principal Investigator studies Kac's well-known inverse spectral problem for analytic domains by introducing new tools. In particular, the project will investigate explicit formulas for the complicated wave invariants in order to improve existing results on this subject. The Principal Investigator will also pursue applications of semiclassical analysis to another celebrated problem, Calderon's inverse problem with partial data.
The Principal Investigator works in the field of partial differential equations. His project focuses on two very important inverse problems in mathematics and physics, namely, Kac's inverse problem and Calderon's inverse problem. Kac's inverse problem asks roughly whether one can determine the shapes of objects (such as stars) from the wave frequencies that they emit. The Principal Investigator approaches this problem by developing new computational tools that lead to an improved understanding of the relationship between the waves and the shapes of the objects. Such inverse spectral problems, as some engineers have recently suggested, may also have interesting applications in shape-matching, copyright, and medical imaging analysis. The Calderon inverse problem (or, in its technical formulation, the problem of electric impedance tomography) asks whether one can determine the conductivity of a body on the basis of current and voltage measurements taken at its surface. This inverse method has been proposed as a valuable diagnostic tool in medicine for the early detection of breast cancer. The past ten years have seen considerable developments in this area, but the problem with partial data (meaning for measurements taken only on pieces of the surface) remains an open problem. The project investigates the partial data problem with the aid of new analytical techniques.
