Indices and Relative Indices in Contact and CR-Geometry

设（X，H）表示具有接触结构的紧致奇维流形H.基本的例子有：（1）紧致流形的余球丛，（2）复流形的严格伪凸边界。 任何接触流形都有一个著名的伪微分算子代数，称为海森堡代数。 Epstein和Melrose在Boutet de Monvel和Guillemin工作的基础上，在海森堡代数中确定了一族零阶投影算子，推广了定义在复流形边界上的Szego投影仪。他们建立了如果S、S'是两个这样的投影仪，那么由S定义的从S'的像到S的像的投影伊萨Fredholm映射。 设i（S，S '）表示该约束的指数. 我们称之为相对指数。 通过引入作用在向量丛上的Heisenberg算子，许多经典指标问题可以转化为广义Szego投影对的相对指标问题。 本文的主要研究方向是利用更经典的几何不变量，如Dirac算子和全纯Euler特征线的指数，得到相对指数的计算公式。 这种类型的一般相对指数公式将允许爱泼斯坦和梅尔罗斯完全解决一般情况下的阿蒂亚-温斯坦问题，要求一个几何公式的指数的FIO定义的两个co-sphere束之间的接触的同构。 除此之外，研究者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，这对于研究严格伪凸紧致CR流形上CR结构的小嵌入变形是有用的。偏微分方程是物理，化学，经济等问题的数学描述的主要工具，而决定这样一个方程有多少解（如果有的话）是一个基本问题。 一般来说，这个问题太微妙，无法精确解决，但有一个称为Fredholm指数的数字可以估计解决方案的数量。 本课题的主要结果是“Atiyah-Singer指标定理”。爱泼斯坦提出的研究是将这些思想扩展到更一般的方程类，这些方程类出现在“接触流形”的背景下，这是力学和控制理论问题的自然舞台。 正在开发的分析工具应该有许多应用程序以外的公式为Fredholm指数。

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.