Three Problems in Complex Analysis

复分析中的三个问题

• 批准号: 9971935
• 负责人: David Catlin
• 金额: $ 16.71万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971935&HistoricalAwards=false
• 关键词: Three; Problems; Complex; Analysis

AbstractAward: DMS-9971935Principal Investigator: David CatlinThe proposal is concerned with three separate problems inalgebraic geometry, differential geometry and in partialdifferential equations.The first problem is concerned with acomplex variables analog of the classical theorem of E. Artin,which states that a non-negative polynomial in n real variablescan be written as a sum of squares of rational functions. Theproposal seeds to determine exactly when a given nonnegativepolynomial in n complex variables can be written as a quotient oftwo functions, each of which is the sum of squares of holomorphicpolynomials. A conjecture is stated, which as a special casewould likely lead to a constructive version of Artin's theorem.The second problem is concerned with the Kuranishi problem, whichseeks geometric conditions involving the Levi form that implythat a given local neighborhood of a CR manifold can be embeddedinto complex Euclidean space. The proposal describes a possibleapproach to the problem of finding such an embedding when theassociated CR structure is minimally smooth. The third problemis concerned with finding sufficient conditions for eithercompactness or hypoellipticity of the complex Neumann problem inthe case when the boundary of the domain is of infinite type.It is well-known fact that not all results in mathematics can beimmediately applied to so-called real world problems. Sometimesthe proof of an important statement depends on abstract methodswhich cannot be easily applied in more concrete situations whichcall for an explicit construction. For the first problem, theproposer is examining a well-known theorem in algebra which usesabstract methods to explain why certain polynomials are alwaysnonnegative. In order for this theorem to be used in the realworld, it would be very important to find a constructive proof ofthis theorem. The second part of the proposal has to do with aproblem in geometry in which a a surface in effect is not smoothand has sharp corners or twists built into it. Such problems arealso important for applications because in problems having to dowith the shapes of objects, it is rarely the case that theobjects are completely smooth. Although the problem that isstudied is somewhat theoretical, it often happens that techniquesdeveloped in one problem can often be used in related areas aswell.

摘要奖项：DMS-9971935主要研究员：David Catlin该提案涉及代数几何、微分几何和偏微分方程中的三个独立问题。第一个问题涉及 E. Artin 经典定理的复变量模拟，该定理指出 n 个实变量的非负多项式可以写成有理数的平方和 功能。准确确定 n 个复数变量中的给定非负多项式何时可以写成两个函数的商的提案种子，每个函数都是全纯多项式的平方和。 提出了一个猜想，作为一种特殊情况，它可能会导致 Artin 定理的构造性版本。第二个问题与 Kuranishi 问题有关，该问题寻求涉及 Levi 形式的几何条件，这意味着 CR 流形的给定局部邻域可以嵌入到复杂的欧几里得空间中。该提案描述了一种可能的方法，用于解决当相关 CR 结构最低限度平滑时找到这种嵌入的问题。 第三个问题涉及在域边界为无限类型的情况下，找到复杂诺依曼问题的紧性或亚椭圆性的充分条件。众所周知，并非所有数学结果都可以立即应用于所谓的现实世界问题。有时，一个重要命题的证明依赖于抽象方法，而这些方法不能轻易应用于需要显式构造的更具体情况。对于第一个问题，提出者正在研究代数中的一个著名定理，该定理使用抽象方法来解释为什么某些多项式总是非负的。为了将该定理应用于现实世界，找到该定理的建设性证明非常重要。 该提案的第二部分与几何问题有关，其中表面实际上并不光滑，并且内置有尖角或扭曲。 此类问题对于应用程序也很重要，因为在与物体形状有关的问题中，物体很少是完全光滑的。 尽管所研究的问题在某种程度上是理论性的，但经常发生的是，在一个问题中开发的技术也可以在相关领域中使用。