Geometric Function Theory in Several Complex Variables

p.i.提议的主要部分是继续他的工作（与S. Ivashkovich教授合作），研究几乎复杂的流形。几乎复流形并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，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，它们有时为几个复杂变量的问题提供了更清晰的方法。最近在该领域的研究，特别是伊瓦什科维奇和私家侦探，以及内夫顿·帕里的研究表明，真正的基本问题仍需要调查。P.I.打算研究解析问题，例如Kobayashi-Royden度规或用Poletsky的方法构造多次谐波函数。最近在gausier - sukhov和Ivashkovich以及pi的著作中已经很好地说明了复数次调和函数在几乎复流形上的几何作用。与上述有关，pi打算研究几个问题，特别是具有一致界的全纯矩阵的Cartan引理。到目前为止，这个问题直到最近才被P.I.和B. Berndtsson教授用一种改进的极大值原理在复杂的1维空间中解决。这个建议是在复杂分析方面，但在几个活跃的数学领域的汇合。涉及到一些几何问题，微分不等式起着重要作用。pi的工作领域主要是几个复杂变量的理论。许多“实际问题”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。它可以追溯到原点，用于解代数方程，但它在例如傅立叶积分理论中继续（在工程课上教的），它在最近的发展中继续。复杂分析与常微分方程或偏微分方程理论有着深刻而丰富的联系，它允许研究物理现象（在宏观层面上）。它适用于动力系统的研究。复分析与几何学也有不同的联系，后者更倾向于对自然现象进行定性（而不是定量）的描述。正是在这种有意义的背景下，私家侦探计划才会发挥作用。

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.