Three Problems in Complex Analysis

摘要奖：DMS-9971935首席研究员：David Catlin该建议涉及代数几何、微分几何和偏微分方程式中的三个独立问题。第一个问题涉及一个复变量，类似于E.Artin的经典定理，该定理指出，n个实变量中的非负多项式可以写成有理函数的平方和。我们的建议是准确地确定给定的n个复变数中的非负多项式何时可以写成两个函数的商，每个函数都是全纯多项式的平方和。第二个问题是关于Kuranishi问题的，它寻找包含Levi形式的几何条件，这些条件意味着CR流形的给定局部邻域可以嵌入到复欧氏空间中。该方案描述了当关联CR结构是最小光滑时找到这样一个嵌入的问题的一种可能的方法。第三个问题是在区域边界是无穷大的情况下，找到复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立即应用于所谓的现实世界问题。有时，重要陈述的证明依赖于抽象的方法，而这些方法不容易应用于需要显式构造的更具体的情况。对于第一个问题，提出者正在研究代数中的一个著名定理，该定理使用抽象的方法来解释为什么某些多项式总是非负的。为了将该定理用于现实世界，找到该定理的构造性证明将是非常重要的。提案的第二部分与几何学中的一个问题有关，在这个问题中，一个表面实际上并不光滑，而且里面有尖锐的拐角或扭曲。这类问题对于应用程序也很重要，因为在与对象形状有关的问题中，对象很少是完全光滑的。虽然所研究的问题有些理论性，但在一个问题中发展的技术往往也可以用于相关领域，这是经常发生的。

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.