Applied Mathematical Logic

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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.