Noncommutative Geometry Models in Physics

该基金用于研究非交换几何在理论物理模型中的应用。其中一部分着重于量子霍尔系统和准晶体的应用。目标是将NCG方法扩展到分数霍尔效应。对于准晶体结构，目标是将代数几何技术与NCG方法相结合。研究的第二部分涉及在自旋泡沫模型中使用NCG技术，这是PI最近的工作中发展起来的，目的是将这种方法与NCG方法中使用的几乎交换几何相结合，用于粒子物理模型。提案的第三部分涉及基于NCG的粒子物理和宇宙学的扩展模型，并将这些弦理论的低能量极限模型连接起来。最后，根据弦理论中BPS态计数与晶体熔化统计物理模型的对应关系，提出了一种新的观点。更广泛的影响：PI在2010年海德拉巴国际数学家大会数学物理会议上受邀发言，并在2008年阿姆斯特丹欧洲数学大会上受邀发言。在过去的五年里，PI培养了12名博士生（其中8名是少数民族），目前正在指导另外7名研究生。PI还监督了加州理工学院的11名本科生（其中4名属于代表性不足的群体）的暑期研究项目。在过去的两年中，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（其中一名来自一个代表性不足的群体）。在外联方面，PI撰写了四本研究专著，并编辑了八卷，这些内容来自她在过去几年中参与组织的会议、研讨会和特别项目。除了这些出版物和经常在讲习班和会议上发表演讲外，PI在过去三年中还就她的研究进行了五次普通听众讲座。

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.