Structural Invariants for Higher-Rank Graphs
高阶图的结构不变量
基本信息
- 批准号:1800749
- 负责人:
- 金额:$ 27万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2018
- 资助国家:美国
- 起止时间:2018-09-01 至 2024-08-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
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.
C*代数理论是在20世纪30年代和40年代作为量子力学的数学模型而发明的。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,可以使用C*代数来研究量子信息理论和有向图或网络。相反,与有向图相关的C*-代数已被证明是增强我们对C*-代数整体理解的关键例子。这个项目的重点是有向图-高阶图-及其相关的C*-代数的推广。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,以及它们在其他数学领域的适用性。除了让研究生和本科生参与高阶图C*-代数的研究外,该项目还将以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,它为蒙大拿大学的研究级研究生启动了一个交换项目,使他们能够在提供高级研究生课程或研讨会的大学学习一个学期,以补充学生的研究兴趣。其次,首席研究员将在蒙大拿大学接待国内和国际研究学者,并在蒙大拿期间为这些学者访问其他地区机构提供便利。高阶图是由Kumjian和Pask在2000年引入的,他们希望他们的相关C*-代数,就像图C*-代数一样,能够提供关于C*-代数更普遍的重要见解。然而,高阶图C*-代数的结构比图C*-代数复杂得多,这种复杂性至今限制了高阶图C*-代数在其他数学领域的应用。为了扭转这一趋势,该项目将(1)提高我们在其他情况下识别高阶图C*-代数的能力;(2)开发新的,更强大的工具来分析高阶图C*-代数。为了实现第一个目标,首席研究员将澄清高阶图C*-代数和它们的扭曲对应物之间的关系,以及研究高阶图C*-代数和其他数学分支(如李代数和格,小波和多分辨率分析)之间的联系。第二个目标将通过研究高阶图C*-代数的不变量来实现,如它们的k理论、群样结构、KMS状态和Cartan子代数。该奖项反映了美国国家科学基金会的法定使命,并通过使用基金会的知识价值和更广泛的影响审查标准进行评估,被认为值得支持。
项目成果
期刊论文数量(8)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Monic representations of finite higher-rank graphs
有限高阶图的模态表示
- DOI:10.1017/etds.2018.79
- 发表时间:2020
- 期刊:
- 影响因子:0.9
- 作者:FARSI, CARLA;GILLASPY, ELIZABETH;JORGENSEN, PALLE;KANG, SOORAN;PACKER, JUDITH
- 通讯作者:PACKER, JUDITH
Analyzing the Weyl Construction for Dynamical Cartan Subalgebras
分析动态嘉当子代数的 Weyl 构造
- DOI:10.1093/imrn/rnab114
- 发表时间:2021
- 期刊:
- 影响因子:1
- 作者:Duwenig, Anna;Gillaspy, Elizabeth;Norton, Rachael
- 通讯作者:Norton, Rachael
Moves on k-graphs preserving Morita equivalence
- DOI:10.4153/s0008414x21000055
- 发表时间:2020-06
- 期刊:
- 影响因子:0
- 作者:C. Eckhardt;Kit Fieldhouse;D. Gent;E. Gillaspy;Ian Gonzales;D. Pask
- 通讯作者:C. Eckhardt;Kit Fieldhouse;D. Gent;E. Gillaspy;Ian Gonzales;D. Pask
K-theory for real k-graph C∗-algebras
实 k 图 C 代数的 K 理论
- DOI:10.2140/akt.2022.7.395
- 发表时间:2022
- 期刊:
- 影响因子:0.6
- 作者:Boersema, Jeffrey L.;Gillaspy, Elizabeth
- 通讯作者:Gillaspy, Elizabeth
Spectral triples and wavelets for higher-rank graphs
高阶图的谱三元组和小波
- DOI:10.1016/j.jmaa.2019.123572
- 发表时间:2020
- 期刊:
- 影响因子:1.3
- 作者:Farsi, Carla;Gillaspy, Elizabeth;Julien, Antoine;Kang, Sooran;Packer, Judith
- 通讯作者:Packer, Judith
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Elizabeth Gillaspy其他文献
Purely Atomic Representations of Higher-Rank Graph $$\varvec{C}^{\varvec{*}}$$ -Algebras
- DOI:
10.1007/s00020-018-2493-z - 发表时间:
2018-09-29 - 期刊:
- 影响因子:0.900
- 作者:
Carla Farsi;Elizabeth Gillaspy;Palle Jorgensen;Sooran Kang;Judith Packer - 通讯作者:
Judith Packer
Elizabeth Gillaspy的其他文献
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{{ truncateString('Elizabeth Gillaspy', 18)}}的其他基金
Conference: Groundwork for Operator Algebras Lecture Series (GOALS) 2022
会议:算子代数基础讲座系列 (GOALS) 2022
- 批准号:
2154574 - 财政年份:2022
- 资助金额:
$ 27万 - 项目类别:
Standard Grant
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