Three Problems in Complex Analysis

AbstractAward：DMS-9971935主要研究者：大卫卡特林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，它指出，一个非负多项式在n个真实的变量可以写为一个平方和的合理的功能。该建议种子，以确定当一个给定的非负多项式在n个复变量可以写为两个函数的商，其中每个是平方和的holomorphicpolynomials。 第二个问题是Kuranishi问题，它寻求包含Levi形式的几何条件，该条件意味着CR流形的给定局部邻域可以嵌入复欧氏空间。该建议描述了一种可能的方法来找到这样的嵌入时，相关联的CR结构是最小光滑的问题。 第三个问题是当区域的边界是无限型时，求复Neumann问题的紧性或亚椭圆性的充分条件，众所周知，并非所有的数学结果都能直接应用于所谓的真实的世界问题。有时，一个重要命题的证明依赖于抽象的方法，而这些方法在需要明确构造的更具体的情况下不容易应用.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。为了使这个定理在现实世界中得到应用，找到一个构造性的证明是非常重要的。 第二部分的建议涉及到几何问题，其中一个表面实际上是不光滑的，有尖锐的角落或扭曲。这样的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，很少有物体是完全光滑的。 尽管所研究的问题是理论性的，但在一个问题中发展起来的技术往往也可以用于相关领域。

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.