CAREER: Physical Properties of New Materials via Mathematics -- Methods and Applications

本研究项目的目标是运用现代数学技术，促进对新型纳米材料物理性质的理解。PI有新的研究项目，这些项目将受到a)解析几何方法和b)非交换几何方法的攻击。这将导致新的通用技术的发展，这些技术也可以应用于其他领域。该材料源于三周期恒定平均曲率（CMC）表面及其补面，这些补面形成线状网络。本文给出了CMC表面上狄拉克方程的有效理论，计算了导线网络图上拉普拉斯算子的特征值谱，探索了先前发现的非交换/交换对偶，研究了导线网络的非交换拓扑不变量和量子霍尔效应的模拟。PI将使用的方法和工具范围从非交换几何、代数和表示理论、奇点理论和微分几何到凝聚态物理。上述项目将从不同的角度为新型纳米材料的数学和物理性质提供重要的见解。这种跨学科的努力汇集了数学和物理学的工具，并促进了两个学科之间的对话。该结果可能对光伏发电有用，从而提高太阳能电池的效率。这个项目更广泛的影响是其教育方面的固有影响，以及涉及物理学几个领域的拟议研究的主题，以及物理科学和数学科学之间的跨学科性质。同时担任数学系和物理系教授的考夫曼通过与数学系的同事讨论研究项目，促进了数学系和物理系之间的对话。她正在为本科生开发一门数学和物理相结合的课程，一个数学和物理的联合系列研讨会，一个跨学科的本科生研究项目，以及一个关于数学和物理问题的互动博客新网站。这是从她开发跨学科课程作为远程学习课程的经验演变而来的。该提案有很强的外联成分。PI计划为小学生举办一次实地考察日，为印第安纳州的教师举办一次专题讨论会，并为公众举办半年一次的系列讲座。考夫曼计划加入普渡大学K-12科学拓展项目。考夫曼积极地为数学和物理领域的女性树立了榜样，她参与了“投身科学的女性”和“投身物理的女性”活动，并为投身物理领域的女性提供了服务。她还指导一名少数族裔研究生与普渡大学化学系进行联合研究项目。

The goal of this research project is to advance the understanding of physical properties of new nano-materials by applying modern mathematical techniques. The PI has new research projects, which will be attacked by using a) analytical and b) noncommutative geometric methods. This will lead to the development of new general techniques that can be applied to other fields as well. The materials stem from triply periodic constant mean curvature (CMC) surfaces and their complements which form wire networks. This gives an effective theory for the Dirac equation on the CMC surface, the computation of the eigenvalue spectrum of the Laplace operator on the graph of the wire network, the probing of a previously found noncommutative/commutative duality and the study of noncommutative topological invariants and the analogue of the quantum Hall effect for the wire networks. The methods and tools that the PI will employ range from methods in noncommutative geometry, algebra and representation theory, singularity theory and differential geometry to condensed matter physics. The above projects will provide important insights from different perspectives into the mathematical and physical properties of new nanomaterials. This interdisciplinary effort brings together tools from mathematics and physics and fosters the dialogue between the two disciplines. The outcome may be useful in photovoltaics, leading to increased effectiveness of solar cells. The broader impact aspects of this project are inherent in its educational aspects and the subject matter of the proposed research which involves several fields of physics, as well as the interdisciplinary nature between the physical and mathematical sciences. Kaufmann, holding a joint appointment in mathematics and physics, is fostering the dialogue between these two departments by discussing research projects with colleagues in both departments. She is undertaking the development of a combined math-physics course for undergraduates, a joint seminar series in math and physics, undergraduate research projects with an interdisciplinary subject and a new website as an interactive blog about questions from math and physics. This evolves from her experience in developing an interdisciplinary course as a distance learning class. The proposal has a strong outreach component. The PI plans a field day for elementary school children, a symposium for Indiana teachers as well as a semiannual lecture series for the general public. Kaufmann plans to join the Science K-12 Outreach Program at Purdue University. Kaufmann actively serves as a role model for women in mathematics and physics, and by her involvement in Women in Science and Women in Physics initiatives and outreach for women in physics. She is also supervising a minority graduate student working on a joint research project with the chemistry department at Purdue.