Applied Mathematical Logic

研究人员的研究结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，研究人员重点研究了分散空间、紧齐性空间和玻尔拓扑的性质。在这项研究中，拓扑学和分析是结合在一起的，特别是在Bohr拓扑学领域，因为这门学科涉及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群表示理论来研究的，而且紧致齐性空间通常是借助于空间上的测度来构造的。逻辑和集合论是相关的，因为关于拓扑学和测度论的命题往往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，但仍然存在关于Cantor-Bendixson导数序列中可能出现的基数的问题；研究人员的部分研究研究了这个序列如何在集合论的不同模型中变化。在代数中，研究人员研究的是代数系统，如拟群和环。自动推理工具在这里非常有用。这些代数系统用相当简单的公理来描述，计算机搜索通常可以揭示这些公理的有趣的新结果。然而，研究人员将计算机的使用与涉及组合学和群论的经典论证相结合。研究人员的研究研究了纯数学中的一些主题，这些主题是自然产生的，试图概括物理宇宙的属性。例如，拓扑学是为了推广物理空间的几何而自然出现的。测度论是概率概念的自然扩展。研究还研究了环的代数性质，它自然地推广了群的概念，这是在对称性研究中出现的。这项研究也有计算部分，特别是涉及环。在这一领域，人们经常想知道一个等式是否意味着另一个等式。这种含义的证明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足够强大，足以发现新的含义并解决以前不需要计算机帮助就能解决的问题。研究人员在这里的研究既对数学本身感兴趣，也对所使用的计算机工具的进步感兴趣。

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.