Richard Neal Ball

My interests lie in ordered algebra, and my primary expertise lies in lattice-ordered groups (hereafter l-groups). My training and dissertation, written under the direction of W. C. Holland, were in l-permutation groups. For the first eight years after getting my Ph.D., I intensively investigated completions of l-groups. There are two or three main themes to this work. The first theme was essentially topological: convergence and Cauchy structures. It develops that l-groups are richly endowed with intrinsic notions of convergence, most of which lead to interesting completions. The second theme is lattice theoretical, and culminated in a demonstration of the existence and uniqueness of the distinguished completion. This completion is a sublattice of the essential hull of a given l-group in the category of distributive lattices; it contains virtually all known completions of interest. This work is summarized in the survey article "Completions of l-Groups," in Lattice Ordered Groups, Advances and Techniques,A.M.W. Glass and W.C. Holland, (eds.), Kluwer Academic Publishers, 1989. Since my visits to Wesleyan in 1983 and 1985, most of my work has been centered on C(X), the set of continuous extended-real valued functions on a space X. This very classical topiccan be approached from a number of directions (Banach spaces, C^* -algebras, f-rings, etc.), but I prefer to think of C(X) as simply an archimedean l-group with weak unit. (The category of such things is called W). Almost all my work on C(X) has been in collaboration with my close friend A. W. Hager, from whom I have learned most of what I know about C(X), and this collaboration continues. The decisive penetration which led to our stuff was the characterization of the epimorphisms in W. The ideas lead directly to locales, via SpFi, the category of spaces with filters. My interests in locales has resulted in a long paper on C(L), the real-valued maps on the locale L, and a collaboration underway with J. Walters-Wayland. Since 1992, my good friend and colleague J. N. Hagler and I have been working in topological dynamics. Our results can be described as developing the general topology of flows. We are currently finishing a long article on projectives Boolean flows. Finally, in a very recent collaboration with J. N. Hagler and Y. Sternfeld, I have written a paper about lattice-theoretical aspects of hereditarily indecomposable continua. This work also continues.

Academic Experience:University of Denver Distinguished Teacher, 1994;Boise State University College of Arts and Sciences Research Award, 1987