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，他对包络的构造和一元性问题尤其感兴趣。讨论了全纯包络中解析盘和变异的存在性问题以及列维平面超曲面的唯一性问题。找到这些问题的答案对于解决涉及全纯运动的可扩展性和全纯选择公理的相关问题是非常有帮助的。这里研究的领域涉及到复数变量的函数理论。要求几个复杂变量的函数是可微的（如在初等微积分中）所强加的结构是一个非常丰富的结构。特别地，给定一个定义了这样一个函数的集合，重要的是要知道所有这样的函数可以扩展到的最大集合，并保持它们的可微性。这个“全纯包络”还没有被很好地理解，斯洛德科夫斯基教授的一个项目就是寻找找到它的方法。

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.