Applied Mathematical Logic

研究者的研究结合了许多不同的数学领域：逻辑，集合论，代数，拓扑学和分析，以及计算机科学中的一些自动推理技术。在拓扑学中，研究者专注于分散空间，紧致齐次空间和玻尔拓扑的性质。 本研究将拓扑学和分析学结合起来，特别是在玻尔拓扑学领域，因为本课题涉及到一般拓扑学的标准基数函数（如权、特征等），但它是通过调和分析的一部分群表示理论来研究的，紧齐性空间也常常借助于空间上的测度来构造。 逻辑和集合论是相关的，因为关于拓扑和测度论的陈述经常独立于集合论的通常公理;当一个结果被证明是独立的时，所使用的方法是形式逻辑的方法。 例如，Cantor-Bendixson序列和分散空间的概念已经有100年的历史了，但是关于Cantor-Bendixson导数序列中可能出现的基数仍然存在问题;研究者的研究部分研究了这个序列在集合论的不同模型中如何变化。 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。 这些代数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经典的论点，涉及组合学和群论。调查员的研究研究了一些纯粹数学的主题，这些主题自然地出现在试图概括物理宇宙的性质。 例如，拓扑学很自然地试图概括物理空间的几何学;测度论是概率概念的自然延伸;该研究还研究了环的代数性质，它自然地概括了在对称性研究中出现的群的概念;该研究也有一个计算部分，特别是涉及环。 在这一领域，人们常常想知道一个方程是否意味着另一个方程。 证明这样一个蕴涵涉及到可以由计算机程序执行的符号操作。近年来，硬件和软件已经变得强大到足以发现新的蕴涵，并解决没有计算机辅助就难以解决的问题。 调查员的研究在这里是感兴趣的数学本身，并为先进的计算机工具使用。

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.