Contact Geometry, Complex Analysis and Imaging

摘要“接触几何，复杂的分析和成像“DMS-0203705 Dr.爱泼斯坦的研究计划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）多参数扰动理论，（4）磁共振成像。在（1）中提出的工作可能会与Richard Melrose合作完成。他们打算利用他们以前在海森堡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使用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-流形的可嵌入性的项目，他建议考虑大于3维的CR-结构，以理解为什么一些CR-函数是不稳定的，为什么一些在底层CR-结构的变形下是稳定的。 在（3）中描述的项目涉及的问题解析参数化的特征值和特征空间的“自伴“分析家庭的矩阵。 从广义上讲，爱泼斯坦博士想看看现代多复变函数理论对多参数摄动问题有什么看法。初步调查表明，这一领域现在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明确的解析空间上的特征值andeigenspaces的自伴家庭解析参数化。 最后（4）是爱泼斯坦博士研究的一个全新方向。 他计划研究使用非均匀背景场的磁共振成像的可行性。爱泼斯坦博士的工作是将分析和代数应用于测量问题。他的主要研究重点是接触流形的几何和分析。从多复变函数到微分几何、拓扑学、偏微分方程，这些问题都是在许多背景下出现的.爱泼斯坦博士希望利用与理查德·梅尔罗斯合作开发的工具来阐明这些无处不在的空间的分析和几何学。爱泼斯坦博士还计划研究利用不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，它将允许许多新的应用磁共振成像，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.