Mathematical Sciences: Partial Differential Equations and Complex Variables

DMS-9501046 PI: Treves Treves将继续研究局部可积结构流形的分析，这些结构允许在适当的复化（次解析结构）下全纯扩展。他将着眼于全局方面，特别是椭圆结构（它概括了Dolbeault和De Rham结构），以及基于傅里叶变换和Radon变换变体的微局部方面。他还打算研究全纯对象（函数、上同调、类）的边值及其作为超函数的解释。特别是，他将尝试扩展使用FBI变换和波前集来表征扩展。特里夫斯将继续研究的另一个问题是解相关的偏微分方程组的可能性或不可能性。特里夫斯研究了当方程多于未知量时解微分方程的可能性。其中一些方程自然地出现在某些几何和物理过程中。要回答的一个基本问题是，这些解中的一些是否可以从它们沿着射线的值重建，比如在x射线断层扫描或磁共振成像中。在处理高度复杂的解决方案类型时，这是相当不清楚的，并且需要对正在发生的事情有一些深刻的理解。

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.