Operator theory methods in pure and applied mathematics

该建议在一个共同的算子理论框架内统一了当前对数学分析界感兴趣的几个研究。研究的主题包括：通过pleemelj对称研究多元Beurling变换和势理论中的庞加莱变分原理；非连通平面域上Bergman正交多项式的锐估计的推导，是Carleman和Widom经典工作的延续；基于精确代数原理的自由星型代数构造性决策方法的阐述详细研究了hele - shaw型演化规律下运动边界的定性特征。这个项目的完成将涉及算子理论技术和实际代数的结合，以及潜在的理论和复杂的分析方法。这个项目的新颖之处部分在于需要改进和升级目前存在的用于研究这些问题的工具。该项目解决了几个当前感兴趣的问题，因为它们与广泛的非数学学科相关。这些学科包括但不限于：经典场论（静电或引力）；在统计学、工程学和经济学问题中出现的最优化理论；流体力学的经典和量子解释。首席研究员将继续在物理学家、计算机科学家和经济学家之间传播数学思想方面发挥积极作用。这将通过他计划在不久的将来组织的专门出版物和几次科学会议来完成。该项目的教学部分将侧重于研究生的监督，博士后的指导（不限于专门研究数学的个人），以及年轻数学人才的识别，特别是在首席研究员所在机构的本科生中。

The proposal unifies within a common operator theory framework several investigations of current interest to the community of mathematical analysts. Among the topics under consideration are the following: a study of the multivariate Beurling transform via Plemelj symmetrization and the Poincare variational principle in potential theory; the derivation of sharp estimates for Bergman orthogonal polynomials on disconnected planar domains, which represents a continuation of classical work of Carleman and of Widom; the elaboration of constructive decision methods (based on precise algebraic principles) in a free star-algebra; a detailed study of the qualitative features of moving boundaries under Hele-Shaw-type evolution laws. The completion of this project will involve a combination of operator theory techniques and real algebra, along with potential theoretic and complex analytical methods. The novelty of the project lies partly in the need to refine and upgrade the tools that currently exist for studying such problems.The project addresses several questions that are of current interest because of their relevance to a wide range of nonmathematical disciplines. Such disciplines include, but are not limited to, the following: classical field theory (electrostatic or gravitational); optimization theory as it arises in statistics, engineering, and economics problems; classical and quantum interpretations of fluid mechanics. The Principal Investigator will continue to play an active role in disseminating mathematical ideas among physicists, computer scientists, and economists. This will be accomplished through specialized publications and several scientific meetings that he plans to organize in the near future. The pedagogical component of the project will focus on the supervision of graduate students, the mentoring of postdocs (not confined to individuals specializing in mathematics), and the identification of young mathematical talent, especially among the undergraduates at the Principal Investigator's home institution.