Applied Mathematical Logic

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

The investigator's research integrates 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 Stone spaces, compact homogeneous spaces, and Bohr topologies. Topology and analysis are integrated in this research; measure theory is used to construct topological spaces with interesting properties, while the topological properties of a space are used to prove theorems about the possible measures which can exist on the space. The investigator studies Bohr topologies, which involve giving a topology to arbitrary abstract groups or other structures; this subject has its roots in the harmonic analysis of the 1930s; modern questions in this area relate to general topology, Fourier series, and functional analysis. Logic and set theory are relevant because results in 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. In algebra, the investigator works on algebraic systems such as quasigroups and loops. Automated reasoning tools are very useful here, primarily in the study of non-associative systems. These 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. There are two distinct, but related, threads to this research. The first thread involves the expansion of our knowledge of traditional pure mathematics. There is no specific practical application in mind here, although topology arises naturally in an attempt to generalize properties of the geometry of physical space, and measure theory is a natural extension of the notion of probability. The second thread involves automated reasoning (AR) tools. AR allows the computer to derive logical conclusions from given knowledge. This subject has been in existence since the 1960s, but it is only in recent years that the hardware and software have become powerful enough to discover conclusions which could not have been discovered without computer assistance. This second thread is a continuation of the investigator's work in improving the AR tools and using these tools to create new results in mathematics. This is of interest not only for the mathematics itself, but because it demonstrates the power of the tools, which can then be applied to reasoning tasks in other areas of science and engineering, as well as to autonomous decision making by robotic agents.