Applied Mathematical Logic

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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.