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

Purdue University, West Lafayette IN

Investigators

Abstract

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.