Mathematical Sciences: Partial Differential Equations and Complex Variables

点击翻译按钮获取中文摘要

DMS-9501046 PI: Treves Treves will continue his investigation of analysis on manifolds equipped with locally integrable structures which allow holomorphic extension in appropriate complexifications (hypo-analytic structures). He will look at global aspects, especially for elliptic structures (which generalize both the Dolbeault and the De Rham structures), as well as microlocal aspects based on variants of the Fourier and Radon transforms. He also intends to investigate boundary values of holomorphic objects (functions, cohomology, classes) and their interpretation as hyperfunctions. In particular he will try to expand the use of the FBI transform and the wave front set to characterize the extensions. Another question which Treves will continue to investigate is the possibility or impossibility of solving the associated systems of partial differential equations. Treves investigates the possibility of solving differential equations when there are more equations than unkowns. Some of these equations occur naturally in certain geometric and physical processes. One fundamental question to answer is whether some of these solutions can be reconstructed from their values along rays, like in X-ray tomography or magnetic resonance imaging. This is quite unclear when dealing with highly sophisticated types of solutions, and some deep understanding of what is going on is needed, an sought.