Indices and Relative Indices in Contact and CR-Geometry

摘要（X，H）表示具有接触结构的紧奇维流形H，其基本例子有：(1)紧流形的共球束，(2)复流形的严格拟凸边界。任何接触流形都有一个被称为海森堡代数的伪微分算子。Epstein和Melrose在boutet de Monvel和Guillemin的工作基础上，确定了海森堡代数中的一组阶零投影算子，推广了在复杂流形边界上定义的zego投影算子。他们建立了，如果S， S‘是两个这样的投影，那么由S定义的从S’的图像到S的图像的投影就是弗雷德霍尔姆地图。设i（S，S'）表示这个约束的索引。我们称之为相对指数。通过考虑作用于向量束的heisenberg算子，许多经典的指标问题可以转化为广义Szego投影对的相对指标问题。本研究的主要目的是获得更经典的几何不变量的相对指标项的公式，如狄拉克算子的指标和全纯欧拉特征。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，要求一个由两个共球束之间的接触微分同构定义的前io指数的几何公式。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代数的扩展，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ccr结构的小可嵌入变形是有用的。偏微分方程是对物理、化学、经济学等领域的问题进行数学描述的主要工具，如果一个方程有解，那么确定它有多少个解是一个基本问题。一般来说，这个问题太微妙了，无法精确地解决，但是有一个叫做弗雷德霍尔姆指数的数字可以估计出解决方案的数量。这门课的主要结果是“Atiyah-SingerIndex定理”。爱泼斯坦提出的研究是将这些想法扩展到更一般的方程类别，这些方程出现在“接触流形”的背景下，这是力学和控制理论问题的自然舞台。分析工具正在开发中，除了弗雷德霍尔姆指数的公式之外，还应该有许多应用。

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.