Extending the Scope of Geometrical Model Theory

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Buechler's proposed research is to extend the scope ofgeometrical stability theory outside of the context of afirst-order theory. His test project is an analysis of thegeometries induced by bounded linear operators on a Hilbert spaceand more complex structures like von Neumann algebras. With hisstudent Berenstein Buechler has shown that many self-adjointoperators induce dependence relations satisfying all of theconditions of the dividing dependence relation. Buechler hopesthat the dividing dependence relation will give insight into thestructure of von Neumann algebras. In another project Buechlerwill investigate "dividing relative to a closure operator". Whilethis study is analogous to the study of p-simple types in asuperstable theory it presents dramatically different resultswhen the original theory is not simple and the closure operatoris chosen creatively. Applications to metric spaces and Vaught'sconjecture are expected. Buechler will also study a class ofscale-free networks with model-theoretic methods. Scale-freenetworks are ubiquitous in nature and technology. Random graphtheorists have discovered some of their properties, such as thedegree functions. However, techniques for building models arelimited. Model theorists have developed techniques for buildinggraphs that are random relative to some constraints. Thesemethods may lend themselves to building scale-free graphs withspecified parameters.Frequently a significant advancement in science occurs when aproblem arising in one area is viewed from the perspective ofanother discipline. For example, problems in genetics haveyielded to techniques from graph theory. In mathematics algebrahas lead to great insight into geometry and knot theory.Buechler's specialty is model theory, a subfield of mathematicallogic. Recently, Hrushovski applied model theory to solveproblems in number theory. Buechler is adapting these samemodel-theoretic methods with an eye to problems in analysis andnetwork theory. In analysis Buechler is looking at themodel-theoretic content of operator theory, which has connectionsto mathematical physics. The networks Buechler will study are atthe heart of such disparate systems as the metabolic pathways ina cell and the World Wide Web. Buechler will attempt to adaptmodel-theoretic techniques for constructing graphs of interest inpure mathematics to building models of these networks arising innature and technology. Graduate students will be involved in allof these projects. The cross-disciplinary nature of the work willrequire the students to learn science that their normalcurriculum would not expose them to, and to learn the value ofviewing research in a broader context.