Comprehensive Study of Functional and Real Analysis

泛函分析和实分析的综合研究

基本信息

批准号：
07304014
负责人：
KOMATSU Hikosaburo
金额：
$ 13.18万
依托单位：
Science University of Tokyo (1996)The University of Tokyo (1995)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1996
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07304014/
关键词：
functional analysis real analysis theory of real variables function spaces operator theory operator algebras represetation theory and harmonic analysis partial differential equations

项目摘要

This is a comprehensive research project involving all active mathematicians in the fields of Functional and Real Analysis. The aim is to have a proper perspective of recent progress in these fields in Japan as well as in the world so that coordinated research activities are possible for the future. For this purpose we made five groups according to the subjects.Each group conducted standing seminars and a nationwide meeting each year. The Investigators met three times every year in order to maintain close correspondence between the groups, and organized a joint symposium each year.In view of this character of the research project it is difficult to summarize our results in a few words but the following could be said. Until recently both Functional Analysis and Real Analysis are so much specialized that many results are profound but interest only few specialists of particular fields. However, we are having now a new trend in which those profound results are applied to an unexpected problem to make a breakthrough of a field which is stagnated for a long time. Our research project has promoted this desirable tendency of researches.For example, some of the function spaces appearing in Real Analysis have been proved to play a essential role in the theory of partial differential equations, although they were introduced only by theoretical necessity in Real Analysis. Toeplitz operators, which were introduced as examples of operators not equivalent to multiplications, are recognized as the same as pseudodifferential operators, and have found applications to the representation theory of Lie groups. Many approaches have been made for non-linear problems. A new lights was shed on the classical WKB method of perturbation.

这是一个综合性研究项目，涉及泛函分析和实分析领域的所有活跃数学家。目的是对日本以及世界范围内这些领域的最新进展有一个正确的认识，以便将来可以协调研究活动。为此，我们按照主题分为五个小组，每个小组每年举办一次常设研讨会和一次全国性会议。为了保持各小组之间的密切联系，研究人员每年举行三次会议，并每年组织一次联合研讨会。鉴于该研究项目的这一特点，很难用几句话来概括我们的结果，但可以如下：说。直到最近，泛函分析和实分析都非常专业化，以至于许多结果都很深刻，但只有少数特定领域的专家感兴趣。然而，我们现在出现了一种新的趋势，即这些深刻的成果被应用于意想不到的问题，以突破长期停滞的领域。我们的研究项目推动了这一理想的研究趋势。例如，实分析中出现的一些函数空间已被证明在偏微分方程理论中发挥着重要作用，尽管它们只是出于理论需要而在实分析中引入的。 Toeplitz 算子是作为不等价于乘法的算子示例引入的，被认为与伪微分算子相同，并且已应用于李群表示论。针对非线性问题已经提出了许多方法。对经典 WKB 微扰方法有了新的认识。