Applied Mathematical Logic

研究者的研究整合了数学中的许多不同领域：逻辑、集合论、代数、拓扑和分析，以及计算机科学中的一些自动推理技术。在拓扑学中，研究者专注于Stone空间、紧齐性空间和Bohr拓扑的性质。测度论被用来构造具有有趣性质的拓扑空间，而空间的拓扑性质被用来证明关于空间上可能存在的测度的定理。这涉及到给任意抽象群或其他结构一个拓扑;这一主题起源于20世纪30年代的调和分析;这一领域的现代问题涉及到一般拓扑、傅立叶级数和泛函分析。逻辑和集合论是相关的，因为拓扑和测度论的结果经常独立于集合论的通常公理;当证明结果是独立的时，所用的方法是形式逻辑的方法。在代数中，研究者研究代数系统，如拟群和循环。 自动推理工具在这里非常有用，主要是在非联想系统的研究中。 这些系统是由相当简单的公理描述的，计算机搜索经常可以揭示这些公理的有趣的新结果。然而，调查员结合了计算机的使用与涉及组合学和群论的经典论点。 这项研究有两条不同但相关的线索，第一条线索涉及传统纯数学知识的扩展。 这里没有具体的实际应用，尽管拓扑学是自然产生的，试图概括物理空间的几何性质，而测度论是概率概念的自然延伸。第二个线索涉及自动推理（AR）工具。 AR允许计算机从给定的知识中得出逻辑结论。 这门学科从20世纪60年代就已经存在了，但直到最近几年，硬件和软件才变得足够强大，能够发现没有计算机辅助就无法发现的结论。第二条线索是研究人员改进AR工具并使用这些工具创建新的结果数学的工作的延续。 这不仅对数学本身有意义，而且因为它展示了工具的力量，这些工具可以应用于其他科学和工程领域的推理任务，以及机器人代理的自主决策。

The investigator's research integrates 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 ofStone spaces, compact homogeneous spaces, and Bohr topologies.Topology and analysis are integrated in this research; measure theory isused to construct topological spaces with interesting properties,while the topological properties of a space are used to prove theoremsabout the possible measures which can exist on the space.The investigator studies Bohr topologies, which involve giving a topologyto arbitrary abstract groups or other structures; this subject has itsroots in the harmonic analysis of the 1930s; modern questions in thisarea relate to general topology, Fourier series, and functional analysis.Logic and set theory are relevant because results in topology and measuretheory are frequently independent of the usual axioms of set theory; when aresult is proved independent, the methods used are those of formal logic. In algebra, the investigator works on algebraic systems such asquasigroups and loops. Automated reasoning tools are very usefulhere, primarily in the study of non-associative systems. Thesesystems are described by fairly simple axioms, and a computer searchcan often reveal interesting new consequences of these axioms. However, the investigator combines the computer use withclassical arguments involving combinatorics and group theory. There are two distinct, but related, threads to this research.The first thread involves the expansion of our knowledge of traditionalpure mathematics. There is no specific practical application in mindhere, although topology arises naturally in an attempt to generalizeproperties of the geometry of physical space, and measuretheory is a natural extension of the notion of probability.The second thread involves automated reasoning (AR) tools. AR allows thecomputer to derive logical conclusions from given knowledge. This subjecthas been in existence since the 1960s, but it is only in recent years thatthe hardware and software have become powerful enough to discover conclusionswhich could not have been discovered without computer assistance.This second thread is a continuation of the investigator's work inimproving the AR tools and using these tools to create new resultsin mathematics. This is of interest not only for themathematics itself, but because it demonstrates the power of the tools,which can then be applied to reasoning tasks in other areas of scienceand engineering, as well as to autonomous decision making by robotic agents.