Buechler's proposed research is to extend the scope of geometrical stability theory outside of the context of a first-order theory. His test project is an analysis of the geometries induced by bounded linear operators on a Hilbert space and more complex structures like von Neumann algebras. With his student Berenstein Buechler has shown that many self-adjoint operators induce dependence relations satisfying all of the conditions of the dividing dependence relation. Buechler hopes that the dividing dependence relation will give insight into the structure of von Neumann algebras. In another project Buechler will investigate "dividing relative to a closure operator". While this study is analogous to the study of p-simple types in a superstable theory it presents dramatically different results when the original theory is not simple and the closure operator is chosen creatively. Applications to metric spaces and Vaught's conjecture are expected. Buechler will also study a class of scale-free networks with model-theoretic methods. Scale-free networks are ubiquitous in nature and technology. Random graph theorists have discovered some of their properties, such as the degree functions. However, techniques for building models are limited. Model theorists have developed techniques for building graphs that are random relative to some constraints. These methods may lend themselves to building scale-free graphs with specified parameters.
Frequently a significant advancement in science occurs when a problem arising in one area is viewed from the perspective of another discipline. For example, problems in genetics have yielded to techniques from graph theory. In mathematics algebra has lead to great insight into geometry and knot theory. Buechler's specialty is model theory, a subfield of mathematical logic. Recently, Hrushovski applied model theory to solve problems in number theory. Buechler is adapting these same model-theoretic methods with an eye to problems in analysis and network theory. In analysis Buechler is looking at the model-theoretic content of operator theory, which has connections to mathematical physics. The networks Buechler will study are at the heart of such disparate systems as the metabolic pathways in a cell and the World Wide Web. Buechler will attempt to adapt model-theoretic techniques for constructing graphs of interest in pure mathematics to building models of these networks arising in nature and technology. Graduate students will be involved in all of these projects. The cross-disciplinary nature of the work will require the students to learn science that their normal curriculum would not expose them to, and to learn the value of viewing research in a broader context.
