Noncommutative Geometry Models in Physics

物理学中的非交换几何模型

• 批准号: 1205440
• 负责人: Matilde Marcolli
• 金额: $ 18.75万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-15 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1205440&HistoricalAwards=false
• 关键词: Noncommutative; Geometry; Models; Physics

This grant is for research in applications of Noncommutative Geometry (NCG) to models in theoretical physics. One part focuses on applications to quantum Hall systems and quasicrystals. The goal is to extend the NCG approach to fractional Hall effects. For quasicrystal structures, the aim is to combine algebro-geometric techniques and NCG methods. A second part of the research deals with the use of NCG techniques in spin foams models, as developed in recent work of the PI, with the aim of combining this method with the almost commutative geometries used in the NCG approach to particle physics models. A third part of the proposal deals with extending models of particle physics and cosmology based on NCG, and to connect these models of low energy limits of String Theory. The last part proposes a new point of view, based on the correspondence between BPS states counting and statistical physics model of crystal melting recently developed in String Theory. Broader impact: The PI was invited speaker in the Mathematical Physics session of the International Congress of Mathematicians, Hyderabad 2010, and plenary speaker in the European Congress of Mathematics, Amsterdam 2008. The PI graduated twelve PhD students over the past five years (eight belonging to ethnic minorities) and is currently supervising seven other graduate students. The PI has also supervised eleven undergraduates at Caltech (four belonging to underrepresented groups) on summer research projects. In the past two years, seven papers were coauthored by the PI with Caltech undergraduates. The PI is also mentoring two postdocs (one from an underrepresented group). For outreach the PI has authored four research monographs and has edited eight volumes arising from conferences, workshops and special programs that she co-organized over the past few years. In addition to these publications and to frequent lectures at workshops and conferences, the PI also delivered five general audience lectures on her research over the past three years.

该补助金用于研究非交换几何（NCG）在理论物理模型中的应用。一部分集中在量子霍尔系统和准晶的应用。目标是将NCG方法扩展到分数霍尔效应。对于准晶结构，目标是将联合收割机代数几何技术和NCG方法结合起来。研究的第二部分涉及使用NCG技术的自旋泡沫模型，在最近的工作中开发的PI，与这种方法相结合的目的几乎可交换的几何形状中使用的NCG方法粒子物理模型。建议的第三部分涉及基于NCG的粒子物理学和宇宙学的扩展模型，并将这些模型连接到弦理论的低能极限。最后一部分基于弦论中BPS态计数与最近发展起来的晶体熔化统计物理模型之间的对应关系，提出了一个新的观点。更广泛的影响：PI受邀在2010年海得拉巴国际数学家大会的数学物理会议上发言，并在2008年阿姆斯特丹欧洲数学大会上发言。在过去的五年里，PI毕业了12名博士生（8名属于少数民族），目前正在指导其他7名研究生。PI还监督了加州理工学院的11名本科生（4名属于代表性不足的群体）的夏季研究项目。在过去的两年里，PI与加州理工学院的本科生共同撰写了七篇论文。PI还指导了两名博士后（其中一名来自代表性不足的群体）。在外联方面，PI撰写了四部研究专著，并编辑了八卷来自她在过去几年中共同组织的会议，研讨会和特别计划的书籍。除了这些出版物和经常在研讨会和会议上演讲，PI还在过去三年中就她的研究发表了五次普通听众演讲。