Contact Geometry, Complex Analysis and Imaging

摘要：DMS- 0203705Dr。Epstein的研究计划包括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)多参数摄动理论，(4)磁共振成像。(1)中提议的工作可能会与Richard Melrose合作完成。他们打算利用他们之前在海森堡微积分上的工作来找到接触流形的不变量。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。在(2)的工作是博士的一部分。在Epstein正在进行的cr -流形可嵌入性项目中，他建议考虑大于3维的cr -结构，以理解为什么有些cr -函数在潜在cr -结构变形下不稳定，而有些则稳定。在(3)中描述的项目涉及“自伴随”解析矩阵族的特征值和特征空间的解析参数化问题。从广义上讲，爱泼斯坦博士想看看现代复数变量函数理论对多参数摄动问题是怎么说的。初步调查表明，在这一领域有可能取得真正的进展。他的主要目标是找到合理的简单和显式的解析空间，在这些解析空间上，自伴随族的特征值和特征空间被解析参数化。最后(4)是爱泼斯坦博士研究的一个全新方向。他计划研究利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几何问题。他的主要研究重点是接触流形的几何和分析。从几个复杂变量到微分几何，从拓扑学到偏微分方程，这些都出现在许多情况下。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。爱泼斯坦博士还计划研究利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，它将允许许多磁共振成像的新应用，也将导致更便宜的设备。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.