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

本研究项目的目的是通过应用现代数学技术来促进对新的纳米材料物理性质的理解。PI有了新的研究项目，将使用a)解析和b)非交换几何方法来攻击这些项目。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。这种材料来源于三周期的常平均曲率(CMC)表面及其互补部分，它们形成了金属丝网。这为CMC表面上的狄拉克方程、导线网络图上拉普拉斯算符的本征值谱的计算、对已发现的非对易/对易对偶的探索、对非对易拓扑不变量的研究以及对导线网络量子霍尔效应的模拟提供了有效的理论依据。PI将使用的方法和工具范围从非对易几何、代数和表示论、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的工具，并促进了这两个学科之间的对话。这一成果可能在光伏领域有用，从而提高太阳能电池的效率。该项目具有更广泛的影响，其教育方面和拟议研究的主题涉及多个物理领域，以及物理科学和数学科学之间的跨学科性质。考夫曼是数学和物理系的联合教授，他正在通过与两个系的同事讨论研究项目来促进这两个系之间的对话。她正在为本科生开发一门数学和物理相结合的课程，一系列关于数学和物理的联合研讨会，一项跨学科主题的本科生研究项目，以及一个关于数学和物理问题的互动博客的新网站。这源于她开发一门跨学科课程作为远程学习课程的经验。该提案有很强的外展成分。PI计划为小学生举办一次野外活动，为印第安纳州的教师举办研讨会，并为普通公众举办半年一次的系列讲座。考夫曼计划加入普渡大学的科学K-12拓展计划。考夫曼积极为数学和物理领域的妇女树立榜样，参与妇女参与科学和物理活动，并在物理学领域为妇女开展宣传活动。她还在指导一名少数族裔研究生与普渡大学化学系的一个联合研究项目。

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.