Extending the Scope of Geometrical Model Theory

Buechler提出的研究是将几何稳定性理论的范围扩展到一阶理论之外。 他的测试项目是分析由希尔伯特空间上的有界线性算子和更复杂的结构（如冯诺依曼代数）诱导的几何。Buechler和他的学生Berenstein一起证明了许多自伴算子导出的依赖关系满足可除依赖关系的所有条件。Buechler希望，划分依赖关系将使深入了解冯诺依曼代数的结构。在另一个项目中，Buechler将研究“相对于闭包运算符的除法”。虽然这项研究是类似于研究的p-简单类型在一个稳定的理论，它提出了显着不同的结果时，原来的理论是不简单的，封闭算子的选择是创造性的.并对度量空间和Vaught猜想的应用作了展望。Buechler还将用模型论方法研究一类无标度网络。无尺度网络在自然界和技术中无处不在。 随机图论学家已经发现了它们的一些性质，比如度函数。 然而，建立模型的技术是有限的。 模型理论家们已经开发出了一些技术来构建相对于某些约束来说是随机的图。 这些方法可以用来构造具有特定参数的无标度图。当一个领域出现的问题从另一个学科的角度来看待时，往往会出现科学上的重大进步。 例如，遗传学的问题已经屈服于图论的技术。在数学中，代数学使人们对几何和纽结理论有了深刻的认识。比希勒的专长是模型论，这是逻辑学的一个分支。最近，Hrushovski应用模型理论来解决数论中的问题。Buechler正在调整这些相同的模型理论方法，着眼于分析和网络理论中的问题。在分析Buechler是看在模型理论的内容运营商理论，这有连接到数学物理。 Buechler将要研究的网络是诸如细胞内代谢途径和万维网等不同系统的核心。Buechler将尝试调整模型理论技术，用于构建纯数学中感兴趣的图，以构建自然和技术中出现的这些网络的模型。 研究生将参与所有这些项目。这项工作的跨学科性质将要求学生学习他们的正常课程不会让他们接触的科学，并学习在更广泛的背景下看待研究的价值。

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.