Mathematics

The collection of frequencies at which a geometric structure resonates is called spectrum of that structure. Encoded in the spectrum is a great deal of information about geometric form, which is difficult to extract. One might ask: How does the sound of a bell determine its shape, or vice versa? A new approach to the problem of relating geometry to the spectrum, based on a concept called the hypoelliptic Laplacian, has shown great promise. The purpose of this project is to build a new theoretical foundation for the hypoelliptic Laplacian, and then develop its applications in harmonic analysis and elsewhere. Expected outcomes will include a clearer and deeper overall understanding of the hypoelliptic Laplacian, and a broadening of the range of applications to which it may be applied. It also provides significant training and mentoring opportunities for graduate students and postdoctoral fellows in geometric and harmonic analysis, distributed across the three sites involved in the project.

This project is a collaboration between Nigel Higson of The Pennsylvania State University (USA), Universidad Nacional de Colombia (Colombia), Universidad de Los Andes (Colombia), Pontificia Universidad Javeriana (Colombia), Universidad de las Sabana (Colombia) with funding from the US National Science Foundation.

To learn more, contact project lead Dr. Nigel Higson: ndh2@psu.edu.