Mathematical Sciences: Envelopes of Holomorphy and Holomorphic Motions

斯洛德科夫斯基教授将研究全纯包络的结构，他特别感兴趣的是与它们的构造和单叶性有关的问题。还将讨论全纯包络中解析圆盘和簇的存在性以及Levi-平坦超曲面的唯一性等问题。找到这些问题的答案将非常有助于解决涉及全纯运动的可扩性的第二组问题以及与选择的全纯公理相关的问题。这里研究的领域涉及到多个复变量的函数理论。通过要求多个复变量的函数是可微的(如在初等微积分中)所强加的结构是非常丰富的。特别地，给定其中定义了这样一个函数的集合，重要的是知道所有这样的函数在保持它们的可微性的情况下可以扩展到的最大集合。这种“全纯包络”还没有被很好地理解，斯洛德科夫斯基教授的项目之一就是开发找到它的方法。

Professor Slodkowski will investigate the structure of envelopes of holomorphy where he is especially interested in problems related to their construction and univalence. Some questions of existence of analytic discs and varieties in envelopes of holomorphy and of the uniqueness of Levi-flat hypersurfaces will also be addressed. Finding answers to these questions should be very helpful in solving a second set of problems involving the extendibility of holomorphic motions and related questions concerning the holomorphic axiom of choice. The areas under investigation here involve the theory of functions of several complex variables. The structure imposed by requiring that a function of several complex variables be differentiable (as in elementary calculus) is a very rich one. In particular, given a set in which such a function is defined, it is important to know the largest set to which all such functions can be extended maintaining their differentiability. This "envelope of holomorphy" is not well understood and one of Professor Slodkowski's projects is to develop methods to find it.