The investigator's research combines a number of diverse areas in mathematics: 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 analysis are integrated in this research, especially in the area of Bohr topologies, since the subject involves the standard cardinal functions of general topology (such as weight, character, etc.), but is studied via the theory of group representations, which is part of harmonic analysis. Also, compact homogeneous spaces are often constructed with the aid of measures on the spaces. Logic and set theory are relevant because statements about topology and measure theory are frequently independent of the usual axioms of set theory; when a result is proved independent, the methods used are those of formal logic. For example, the notion of the Cantor-Bendixson sequence and scattered spaces is 100 years old, but there are still questions about the cardinals which can arise in the sequence of Cantor-Bendixson derivatives; part of the investigator's research studies how this sequence varies in different models of set theory. In algebra, the investigator works on algebraic systems such as quasigroups and loops. Automated reasoning tools are very useful here. These algebraic systems are described by fairly simple axioms, and a computer search can often reveal interesting new consequences of these axioms. However, the investigator combines the computer use with classical arguments involving combinatorics and group theory.
The investigator's research studies a number of topics in pure mathematics which arose naturally in an attempt to generalize properties of the physical universe. For example, topology arises naturally 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 naturally generalize the concept of groups, which arise in the study of symmetry. There is also a computational component to this research, especially involving loops. Frequently in this area, one wants to know whether one equation implies another. A proof of such an implication involves symbolic manipulation which can be performed by a computer program. In recent years, the hardware and software have become powerful enough to discover new implications and to solve problems which had been intractable without computer assistance. The investigator's research here is of interest both for the mathematics itself, and for the advancement of the computer tools used.
