Mathematical Sciences: Applications of Nonstandard Analysis to Measure Theory, Potential Theory, and Related Topics

勒布教授是世界上最著名的 基本概念的应用， 数学称为非标准分析。 这种结构，首先 由逻辑学家引入，是一个巧妙的严格发展， 无穷大和无穷小的经典概念 号码 因此，非标准分析是一种数学分析。 莱布尼茨的哲学观点在他的发明中的辩护 的微积分。 勒布教授是领先的实践者， 使用非标准方法来证明“标准”会导致 广泛的数学分析和应用。 勒布的 研究非标准测度理论和非标准测度理论 而非标准概率论导致了 对概率论、数学物理和 数理经济学 Loeb测度的概念， 由首席研究员，已成为一个标准的工具， 概率论、随机分析和势理论。 Loeb教授目前的建议有两个部分。 首先他 提出将联合收割机的非标准整合理论与 非标准凸包。 应用程序设想到巴拿赫 空间和Banach空间值随机过程。 第二、 他建议继续他关于理想概念的工作（勒布） 边界，为非常一般的潜在理论而发展。 这 研究承诺对经典的 潜力理论

Professor Loeb is one of the world's leading experts in the applications of the fundamental foundational concept in mathematics called nonstandard analysis. This construct, first introduced by logicians, is an ingenious rigorous development of the classical concepts of infinitesimally large and small numbers. Nonstandard analysis, therefore, is a mathematical vindication of Leibnitz' philosophical viewpoint in his invention of the calculus. Professor Loeb is the leading practitioner in the use of nonstandard methods to prove "standard" results in a broad range of mathematical analysis and applications. Loeb's work in nonstandard measure theory and nonstandard measure theory and nonstandard probability theory has led to important contributions to probability theory, mathematical physics, and mathematical economics. The notion of Loeb measure, introduced by the principal investigator, has become a standard tool in probability theory, stochastic analysis, and potential theory. The current proposal by Professor Loeb has two parts. First, he proposes to combine his nonstandard integration theory with nonstandard convex hulls. Applications are envisaged to Banach spaces and Banach space valued stochastic processes. Secondly, he proposes to continue his work on the concept of ideal (Loeb) boundary, developed for very general potential theories. This research promises to have important implications for classical potential theory.