Geometric Function Theory in Several Complex Variables

P.I.提议的主要部分是继续他在几乎复杂的流形上的工作(与S·伊瓦什科维奇教授合作)。几乎复流形不是复流形的无动机推广。在格罗莫夫的工作之后，它们的相关性变得毋庸置疑，在几何学，特别是辛几何中有着深刻的应用。一方面，它们提供了复杂流形所缺乏的非凡的灵活性，另一方面，它们有时提供了一种更清晰的方法来处理几个复杂变量中的问题。最近在该地区开展的工作，尤其是伊瓦什科维奇和P.I.以及内夫顿·巴利的工作表明，真正基本的问题仍然需要调查。P.I.打算致力于分析问题，例如小林-罗伊登度规或用Poletsky的方法构造多重次调和函数。几乎复流形上多重次调和函数的几何作用最近在Gaussier-Sukhov和Ivashkovich以及P.I.的工作中得到了很好的说明，与此相关的是P.I.打算研究的几个问题，特别是具有一致界的全纯矩阵的Cartan引理。到目前为止，这个问题只在最近才被P.I.和B.Berndtsson教授用精炼的最大值原理在复数维1中解决。这项建议是在复杂分析方面，但在几个活跃的数学领域的汇合点。其中涉及到一些几何问题，微分不等式起着重要的作用。P.I.的工作领域主要是几个复变量理论。许多“真正的问题”通过使用复杂的方法得到了最好的解释，即使最初的问题似乎不需要使用复数。它可以追溯到求解代数方程的起源，但它继续存在于例如傅里叶积分理论(在工程学课程中教授)，并在最近的发展中继续下去。复杂分析与常微分方程或偏微分方程组理论有着深刻而丰富的联系，后者允许研究物理现象(在宏观层面上)。它适用于动力系统的研究。复杂分析还与几何学有各种联系，后者更多地指向对自然现象的定性(而不是定量)描述。正是在这种有意义的背景下，私家侦探提出了工作建议。

The main part of the proposal by the P.I.is to continue his work (in collaboration with Prof. S. Ivashkovich) on almost complex manifolds. Almost complex manifolds are not an un-motivated generalization of complex manifolds. After the work of Gromov, their relevance became unquestionable, with deep applications to geometry and in particular to symplectic geometry. On one hand they allow a remarkable flexibility that complex manifolds lack, on the other hand they sometimes provide a much clearer approach to problems in several complex variables. Recent work in the area, in particular by Ivashkovich and the P.I., and by Nefton Pali, shows that truly basic questions still need investigation. The P.I. intends to work on analytic questions, e.g. the Kobayashi-Royden metric or the construction of pluri-subharmonic functions by the method of Poletsky. The geometric role of pluri-subharmonic functions on almost complex manifolds has been well illustrated recently in the works of Gaussier-Sukhov and of Ivashkovich and the P.I.Related to the above there are several questions that the P.I. intends to work on, especially the Cartan Lemma for holomorphic matrices with uniform bounds. The problem was so far solved only recently and only in complex dimension 1 by the P.I. and Prof B. Berndtsson, using a refined maximum principle. This proposal is on the Complex Analysis side but at the confluence of several active areas of Mathematics. Some geometry is involved, differential inequalities play an important role. The field in which the P.I. works is mainly the theory of several complex variables. Many ``real problems'' receive their best explanation by using complex methods, even if the initial problem does not seem to call for using complex numbers. It goes back to the origin, for solving algebraic equations, but it continued for example in the theory of Fourier integrals (that are taught in engineering classes), and it continues in the most recent developments. Complex analysis has deep and rich connections to the theory of ordinary or partial differential equations, which allows the study of physical phenomena (at the macroscopic level). It applies to the study of dynamical systems. Complex Analysis has also various links to Geometry which is more directed to a qualitative (rather than quantitative) description of natural phenomena. It is in this meaningful context that the P.I. proposes to work.