Structural Invariants for Higher-Rank Graphs

C*-代数理论是在20世纪30年代和40年代发明的，作为量子力学的数学模型。除了与物理学的联系外，C*-代数还为研究各种数学对象提供了一个有用的框架。C*-代数在物理学之外的应用也开始出现；例如，人们可以使用C*-代数来研究量子信息论和有向图或网络。相反，与有向图相关的C*-代数已被证明是增强了我们对C*-代数这类整体的理解的关键例子。这个项目专注于有向图的推广--高秩图--及其相关的C*-代数。更好地理解与高阶图相关的C*-代数将加强我们对C*-代数的整体理解，以及它们对其他数学领域的适用性。除了让研究生和本科生参与关于高阶图C*-代数的研究外，这个项目还将通过两种方式增加蒙大拿大学和附近大学数学家的智力机会。首先，它启动了蒙大拿大学研究型研究生的交流计划，使他们能够在一所提供高级研究生课程或研讨会的大学学习一学期，以补充学生的研究兴趣。第二，首席研究员将在蒙大拿大学接待国内和国际研究学者，并将促进这些学者在蒙大拿州逗留期间访问其他地区机构。2000年，Kumkin和Pask.引入了高秩图，希望他们的相关C*-代数，如图C*-代数，将提供更广泛的关于C*-代数的重要见解。然而，高秩图C*-代数的结构比图C*-代数复杂得多，这种复杂性限制了高秩图C*-代数在其他数学领域的应用。为了扭转这一趋势，这个项目将(1)提高我们在其他环境中识别高秩图C*-代数的能力；(2)开发新的、更强大的工具来分析高秩图C*-代数。对于第一个目标，主要研究人员将澄清高秩图C*-代数与它们的扭曲对应之间的关系，并研究高秩图C*-代数与其他数学分支(如李代数、格、小波和多分辨分析)之间的联系。第二个目标将通过研究高秩图C*-代数的不变量来实现，如K-理论、群胚结构、KMS状态和Cartan子代数。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The theory of C*-algebras was invented in the 1930s and 1940s as a mathematical model for quantum mechanics. In addition to their connections with physics, C*-algebras provide a useful framework for studying a variety of mathematical objects. Applications of C*-algebras beyond physics are also beginning to arise; for example, one can use C*-algebras to study both quantum information theory, and directed graphs, or networks. Conversely, the C*-algebras associated to directed graphs have proved to be key examples which have enhanced our understanding of the class of C*-algebras as a whole. This project focuses on a generalization of directed graphs -- higher-rank graphs -- and their associated C*-algebras. A better understanding of the C*-algebras associated to higher-rank graphs will strengthen our understanding of C*-algebras as a whole, and also their applicability to other areas of mathematics. In addition to involving graduate and undergraduate students in research about higher-rank graph C*-algebras, this project will also enhance the intellectual opportunities available to mathematicians at the University of Montana and at nearby universities, in two ways. First, it initiates an exchange program for research-level graduate students at the University of Montana, to enable them to spend a semester studying at a university which is offering an advanced graduate course or seminar that complements the student's research interests. Second, the principal investigator will host national and international research scholars at the University of Montana, and will facilitate these scholars' visits to other regional institutions during their stay in Montana.Higher-rank graphs were introduced by Kumjian and Pask in 2000, in the hopes that their associated C*-algebras, like graph C*-algebras, would provide important insights about C*-algebras more generally. However, the structure of higher-rank graph C*-algebras is much more intricate than that of graph C*-algebras, and this complexity has limited the applicability of higher-rank graph C*-algebras to other areas of mathematics up to now. To reverse this trend, this project will (1) improve our ability to recognize higher-rank graph C*-algebras in other contexts; and (2) develop new, stronger tools for analyzing higher-rank graph C*-algebras. Towards the first goal, the principal investigator will clarify the relationship between higher-rank graph C*-algebras and their twisted counterparts, as well as investigate connections between higher-rank graph C*-algebras and other branches of mathematics (such as Lie algebras and lattices, and wavelets and multiresolution analyses). The second goal will be achieved by studying invariants of higher-rank graph C*-algebras such as their K-theory, groupoid structure, KMS states, and Cartan subalgebras.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.