Indices and Relative Indices in Contact and CR-Geometry

命题：DMS-9970487主要研究者：Charles Epsteinlet(X，H)表示具有紧结构H的紧致奇维流形。基本例子是(1)紧流形的余旋丛，(2)复流形的严格伪凸边界。任何接触流形都有一个特殊的伪微分算子代数，称为Heisenberg代数。爱泼斯坦和梅尔罗斯在蒙特维尔和吉列曼工作的基础上，确定了海森堡代数中的一族零阶投影算子，推广了定义在复流形边界上的Szego投影子。他们证明了如果S、S是两个这样的投影子，那么S定义的从S的像到S的像的投影就是Fredholm映射。设我(S、S)表示这一限制的指标。我们称其为相对指数。通过考虑作用于向量丛上的Heisenberg算子，许多经典指标问题可以转化为广义Szego投影子对的相对指标问题。这项研究的主要目的是得到更经典的几何不变量的相对指数项的公式，例如Dirac算子的指数和全纯欧拉特征。这种类型的广义相对指数公式将允许Epstein和Melrose完全解决Atiyah-Weinstein问题的一般情况，得到由两个球丛之间的接触微分同态所定义的FIO的指数的几何公式。除此之外，研究人员计划寻找接触结构的解析不变量。利用Lempert提出的方法和Heisenberg代数的推广，他希望得到I(S，S‘)的第二个公式，该公式对于研究严格伪凸、紧CR-流形上的CR-结构的小的可嵌入变形是有用的。部分微分方程是物理、化学、经济等领域问题的主要数学描述工具，而决定一个方程有多少解(如果有解)是一个基本问题。一般来说，这个问题太微妙了，不能准确地解决，但有一个数字叫做弗雷德霍尔姆指数，可以估计解的数量。本课题的主要结果是“Atiyah-SingerIndex定理”。爱泼斯坦提出的研究致力于将这些想法扩展到更一般的方程类，这些方程出现在“接触流形”的背景下，“接触流形”是力学和控制理论中问题的自然领域。正在开发的分析工具应该会有许多应用，而不仅仅是弗雷德霍尔姆指数的公式。

AbstractProposal: DMS-9970487Principal Investigator: Charles EpsteinLet (X,H) denote a compact, odd dimensional manifold with acontact structure, H. The basic examples are (1) The co-spherebundle of a compact manifold, (2) The strictly pseudoconvexboundary of a complex manifold. Any contact manifold has adistinguished algebra of pseudo-differential operators called theHeisenberg algebra. Epstein and Melrose, building on work ofBoutet de Monvel and Guillemin, identified a family of order zeroprojection operators in the Heisenberg algebra, generalizing theSzego projector defined on the boundary of a complex manifold.They established that if S, S' are two such projectors then theprojection defined by S from the image of S' to the image of S isa Fredholm map. Let i(S,S') denote the index of thisrestriction. We call this the relative index. By consideringHeisenberg operators which act on vector bundles, many classicalindex problems can be re-phrased as relative index problems forpairs of generalized Szego projectors. The main thrust of theproposed research is to obtain formulae for the relative index interms of more classical geometric invariants, such as indices ofDirac operators and holomorphic Euler characteristics. A generalrelative index formula of this type will allow Epstein andMelrose to completely settle the general case of theAtiyah-Weinstein problem, asking for a geometric formula for theindex of an FIO defined by a contact diffeomorphism between twoco-sphere bundles. Beyond this the investigator plans to searchfor analytic invariants of contact structures. Using an approachsuggested by Lempert and an extension of the Heisenberg algebrahe hopes to obtain a second formula for i(S,S'), which should beuseful in studying the small embeddable deformations of theCR-structure on a strictly pseudoconvex, compact CR-manifold.Partial differential equations are the principal tool formathematical descriptions of problems in physics, chemistry,economics, etc., and it is a fundamental problem to decide howmany solutions, if any, such an equation has. In general thisproblem is too delicate to solve exactly, but there is a numbercalled the Fredholm index which can estimate the number ofsolutions. The main result in this subject is the "Atiyah-SingerIndex Theorem." The research proposed by Epstein is concernedwith extending these ideas to more general classes of equationswhich arise in the context of "contact manifolds," a naturalarena for problems in mechanics and control theory. Analyticaltools are being developed which should have many applicationsbeyond the formula for the Fredholm index.