Applied Mathematical Logic

应用数理逻辑

• 批准号: 0456653
• 负责人: Kenneth Kunen
• 金额: --
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-05-15 至 2009-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0456653&HistoricalAwards=false
• 关键词: Applied; Mathematical; Logic

The investigator's research combines a number of diverse areas inmathematics: logic, set theory, algebra, topology, and analysis,as well as some automated reasoning techniques from computer science.In topology, the investigator focuses on properties of scattered spaces,compact homogeneous spaces, and Bohr topologies. Topology and analysisare integrated in this research, especially in the area of Bohr topologies,since the subject involves the standard cardinal functions of generaltopology (such as weight, character, etc.), but is studied via thetheory of group representations, which is part of harmonic analysis.Also, compact homogeneous spaces are often constructed with the aidof measures on the spaces. Logic and set theory are relevant becausestatements about topology and measure theory are frequently independentof the usual axioms of set theory; when a result is proved independent,the methods used are those of formal logic. For example, the notion ofthe Cantor-Bendixson sequence and scattered spaces is 100 years old,but there are still questions about the cardinals which can arisein the sequence of Cantor-Bendixson derivatives; part of theinvestigator's research studies how this sequence varies in differentmodels of set theory. In algebra, the investigator works on algebraicsystems such as quasigroups and loops. Automated reasoning toolsare very useful here. These algebraic systems are described byfairly simple axioms, and a computer search can often revealinteresting new consequences of these axioms. However, theinvestigator combines the computer use with classical argumentsinvolving combinatorics and group theory.The investigator's research studies a number of topics in puremathematics which arose naturally in an attempt to generalizeproperties of the physical universe. For example, topology arisesnaturally in an attempt to generalize the geometry of physical space.Measure theory is a natural extension of the notion of probability.The research also studies algebraic properties of loops, which naturallygeneralize the concept of groups, which arise in the study of symmetry.There is also a computational component to this research, especiallyinvolving loops. Frequently in this area, one wants to knowwhether one equation implies another. A proof of such an implicationinvolves symbolic manipulation which can be performed by a computer program.In recent years, the hardware and software have become powerful enoughto discover new implications and to solve problems which had beenintractable without computer assistance. The investigator'sresearch here is of interest both for the mathematics itself,and for the advancement of the computer tools used.

研究者的研究结合了许多不同领域的插入物质：逻辑，固定理论，代数，拓扑和分析，以及来自计算机科学的某些自动推理技术。在拓扑中，研究人员专注于散射空间的特性，紧凑的同质空间和Bohr拓扑。 整合在这项研究中的拓扑和分析，尤其是在BOHR拓扑领域，因为该主题涉及通用学的标准基本功能（例如重量，角色等），但通过群体表示的一部分进行了研究，这是和谐分析的一部分。同样，紧凑的同质空间通常与该空间的辅助辅助构建。 逻辑和集合理论是相关的，因为关于拓扑结构和测量理论通常独立于通常的理论公理。当结果被证明是独立的时，使用的方法是形式逻辑的方法。 例如，Cantor-Bendixson序列和散落空间的概念已有100年的历史，但是关于红衣主教的问题仍然存在，可能会引起Cantor-Bendixson衍生物的序列。 Investigator的一部分研究研究了该序列如何在集合理论的不同模型中变化。 在代数中，研究人员在代数系统（如准元素和循环）上工作。 自动推理工具在这里非常有用。 这些代数系统是通过简单的公理来描述的，并且计算机搜索通常可以揭示这些公理的新后果。 然而，研究人员将计算机的使用与经典参数互动的组合和群体理论相结合。 例如，拓扑结构是为了概括物理空间的几何形状。量学理论是概率概念的自然扩展。该研究还研究循环的代数特性，该循环的代数特性自然化了群体的概念，这在对称性研究中也是该研究的计算组成部分。 通常在这个领域，一个人想知道一个方程意味着另一个方程。 可以通过计算机程序执行的象征性操纵的这种含义的证明。近年来，硬件和软件已经变得足够强大，以发现新的含义并解决没有计算机援助的情况下可以提取的问题。 对于数学本身以及使用的计算机工具的发展，这里的研究人员的研究既令人感兴趣。