Extending the Scope of Geometrical Model Theory

扩展几何模型理论的范围

• 批准号: 0140062
• 负责人: Steven Buechler
• 金额: $ 11.48万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0140062&HistoricalAwards=false
• 关键词: Extending; Scope; Geometrical; Model; Theory

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.

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希尔伯特空间上由有界线性算子和更复杂的结构（如冯·诺伊曼代数）引起的几何。比赫勒和他的学生证明了许多自伴随算子诱导出的依赖关系满足划分依赖关系的所有条件。Buechler希望这种分依赖关系能使我们对冯·诺伊曼代数的结构有更深入的了解。在另一个项目中，buechler将研究“相对于闭包运算符的除法”。虽然本研究类似于超稳定理论中p-简单类型的研究，但当原始理论不简单并且创造性地选择闭包算子时，结果却截然不同。期望将其应用于度量空间和沃特猜想。Buechler还将用模型理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。随机图理论家已经发现了它们的一些性质，比如度函数。然而，构建模型的技术是有限的。模型理论家已经开发了一些技术来构建相对于某些约束的随机图。这些方法可用于构建带有指定参数的无标度图。通常，当从另一学科的角度看待一个领域出现的问题时，科学就会取得重大进展。例如，遗传学的问题已经让位于图论的技术。在数学中，代数使我们对几何和结理论有了深刻的认识。Buechler的专长是模型理论，这是数学的一个分支。近年来，赫鲁晓夫斯基将模型理论应用于数论问题的解决。Buechler将这些相同模型理论的方法用于分析和网络理论中的问题。在分析中，Buechler着眼于算子理论的模型理论内容，它与数学物理有联系。Buechler将研究的网络是细胞内代谢途径和万维网等不同系统的核心。Buechler将尝试将模型理论技术用于构造纯数学中感兴趣的图，以构建自然和技术产生的这些网络的模型。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到的科学，并学习在更广泛的背景下观察研究的价值。