The goal of this project is to study a new geometric concept, polynomial inclusions, and their applications. It was first discovered by Newton that an interior point in a hollow ellipsoid feels no gravitational force, and that for a filled ellipsoid the gravity inside is a quadratic function of the position. Polynomial inclusions are a generalization of ellipsoids in terms of Newtonian potential. An interior point in a hollow polynomial inclusion feels a gravitational force that is a polynomial of the position. Polynomial inclusions have proven to be useful for many modeling and design problems. Specifically, the following applications pertaining to polynomial inclusions will be investigated: (i) predictive models for design of synthetic complex materials, (ii) optimal structures for fusion reactors and minimum field concentration, and (iii) inverse source problem for medical and geological imaging. This research project will be integrated with educational and outreach activities such as one-to-one mentoring, educational module development, training graduate students, and fostering interdisciplinary collaborations.
The Newtonian potential induced by a polynomial inclusion of degree k is precisely a polynomial of degree k inside the body. The interest in polynomial inclusions arises from that partial differential equations admit closed-form simple solutions for polynomial inclusions as for ellipsoids. The following fundamental technical problems concerning polynomial inclusions will be addressed in the project: (i) proving the existence and uniqueness of polynomial inclusions, (ii) numerically computing polynomial inclusions, (iii) explicit parametrization of polynomial inclusions and (iv) employing polynomial inclusions to solve aforementioned engineering problems. The outcomes of this project may have an impact on the classic Hilbert's sixteenth problem as well as industrial problems ranging from structural engineering, fusion designs and to imaging. The project will also strengthen the connection between the Department of Mathematics and the Department of Mechanical Aerospace Engineering at Rutgers University by means of recruiting students of diverse backgrounds into the research team.
