Contact Geometry, Complex Analysis and Imaging

《接触几何、复数分析与成像》DMS-0203705DR摘要。爱泼斯坦的研究建议包含四个不同领域的项目：(1)接触结构的不变量，(2)CR几何问题，(3)多参数微扰理论，(4)磁共振成像。(1)中提出的工作很可能是与Richard Melrose合作完成的，他们打算利用他们之前在Heisenberg演算上的工作来寻找接触流形的不变量。一个特别有趣的问题是利用Toeplitz算子的谱流不变量来区分具有同伦超平面场的接触结构。(2)中的工作是Dr。Epstein正在进行的关于CR-流形可嵌入性的项目中，他建议考虑维度大于3的CR-结构，以理解为什么一些CR-函数是不稳定的，以及为什么一些CR-结构在底层CR-结构的变形下是稳定的。在(3)中描述的项目涉及解析地参数化“自伴”解析族的特征值和特征空间的问题。从广义上讲，爱泼斯坦博士想看看现代多元复变函数理论对多参数微扰问题的看法。初步调查表明，现在有可能在这一领域取得真正进展。他的主要目标是找到合理的简单和显式的解析空间，在这些空间上自伴的族的特征值和特征空间被解析地参数化。最后，(4)是爱泼斯坦博士研究的一个全新方向。他计划研究使用非均匀背景场进行磁共振成像的可行性。爱泼斯坦的工作是将分析和代数应用于计量学问题。他的主要研究重点是接触流形的几何和分析。从几个复变量到微分几何，从拓扑学到偏微分方程组，这些都是产生的背景。利用与理查德·梅尔罗斯合作开发的工具，爱泼斯坦博士希望阐明这些无处不在的空间的分析和几何形状。爱普斯坦博士还计划研究使用非均匀背景场进行磁共振成像的可行性。这在很大程度上超出了当今技术的能力。如果它被证明是可行的，它将允许磁共振成像的许多新应用，也应该导致更便宜的设备。这是一个多方面的问题，具有重要的数学和工程方面。

Abstract for "Contact geometry, complex analysis and imaging"DMS- 0203705Dr. Epstein's research proposal contains projects in four distinct areas: (1)Invariants of contact structures, (2) Problems in CR-geometry, (3)Multi-parameter perturbation theory, (4) Magnetic resonance imaging. Theproposed work in (1) will likely be done in collaboration with Richard Melrose.They intend to use their prior work on the Heisenberg calculus to findinvariants of contact manifolds. A problem of particular interest is to usespectral flow invariants of Toeplitz operators to distinguish contactstructures with homotopic hyperplane fields. The work in (2) is part ofDr. Epstein's ongoing project on embeddability of CR-manifolds, he proposes toconsider CR-structures in dimensions greater than 3 to understand why someCR-functions are unstable and why some are stable under deformation of theunderlying CR-structure. The project described in (3) concerns the problem ofanalytically parameterizing the eigenvalues and eigenspaces of a "self adjoint"analytic family of matrices. In broad terms Dr. Epstein would like to see whatthe modern theory of functions of several complex variables has to say aboutmulti-parameter perturbation problems. Preliminary investigations show thatreal progress is now possible in this field. His main goal is to findreasonably simple and explicit analytic spaces on which the eigenvalues andeigenspaces of a self adjoint family are analytically parametrized. Finally(4) is an entirely new direction for Dr. Epstein's research. He plans to studythe feasibility of magnetic resonance imaging using inhomogeneous backgroundfields.Dr. Epstein's work is in the application of analysis and algebra to problems ingeometry. A principal focus of his research is the geometry and analysis ofcontact manifolds. These arise is many contexts from several complex variablesto differential geometry to topology to partial differential equations. Usingthe tools developed in collaboration with Richard Melrose, Dr. Epstein hopes toelucidate the analysis and geometry of these ubiquitous spaces. Dr. Epsteinalso plans to investigate the feasibility of magnetic resonance imaging usinginhomogeneous background fields. This is largely beyond the capabilities ofpresent day technology. If it proves feasible it would allow many newapplications of MR imaging and should also lead to less expensive equipment.This is a multifaceted problem with significant mathematical andengineering aspects.