Mathematical Sciences: Geometric and Analytic Theory of Holomorphic Functions of Several Complex Variables

工作重点在数学理论的几个复杂的 变数将集中在四个主要方向。 第一 关于一阶d杆边界的可去奇异性 运营商 这些微分算子模拟共轭 域的边界上的微分，约化全纯 边界值为零。 这些想法可以追溯到 黎曼的著名发现， 一元有界全纯函数。 一些尖锐 结果已获得的情况下，球在两个复杂的 尺寸，但没有更高的。 此外，工作将 在比较可移动集的有界解决方案的d- 针对所有解决方案的可移除集的栏操作符。 第二个重点将寻求扩大早期的工作， 从全真实的子流形到简单的多项式凸性 真实的流形 开始的地方将是真实的流形 穿过原点。 这项研究的目的是 提供保证多项式的几何准则 任意连续函数的逼近 一般来说，真实的解析上的真实的全纯函数 流形并不扩展为复解析的。 如果一个减少 解析性对次谐性或偶性要求 多谐性，扩展是可能的。 这项工作将寻求 从以下方面来描述扩展的障碍： 流形的几何形状 最后，将努力获得边界唯一性 结果的分析品种的尺寸至少为2。 的 这条调查线的来源是一个简单的观察： 二维（复）有界球中的两个解析圆 通过边界位于球表面的曲线， 重合或边界相交在非常薄的集合中。 的 对高等品种的这一结果的适当说明是 难以捉摸，虽然最近的工作点的概念，峰集 作为边界交叉点的适当背景。

Work focusing on the mathematical theory of several complex variables will concentrate on four major directions. The first concerns removable singularities for first order d-bar boundary operators. These differential operators emulate conjugate differentiation on the boundary of domains, reducing holomorphic boundary values to zero. The ideas may be traced back to Riemann's celebrated discovery for the removable singularities of bounded holomorphic functions of one variable. Some sharp results have been obtained for the case of a ball in two complex dimensions, but nothing for higher. In addition, work will be done in comparing removable sets for bounded solutions of the d- bar operator against sets removable for all solutions. A second thrust will seek to expand earlier work on polynomial convexity from totally real submanifolds to simply real manifolds. The place to begin will be real manifolds passing through the origin. The goal of such research is to provide geometric criteria which guarantee polynomial approximation of arbitrary continuous functions. In general, real holomorphic functions on real analytic manifolds do not extend to be complex analytic. If one reduces the requirement of analyticity to that of subharmonicity or even pluriharmonicity, extensions are possible. This work will seek to describe the obstructions to extension in terms of the geometry of the manifold. Finally, efforts will be made to obtain boundary uniqueness results for analytic varieties of dimension at least two. The source of this line of investigation is a simple observation: two analytic discs in a two (complex) dimensional ball, bounded by curves whose boundaries lie on the surface of the ball either coincide or the boundaries intersect in very thin sets. The proper statement of this result for higher varieties has been elusive, although recent work points to the concept of peak sets as the proper context for the boundary intersections.