Polynomial inclusions: open problems and potential applications

这个项目的目标是研究一个新的几何概念，多项式包含及其应用。牛顿首先发现，中空椭球体的内点没有重力，而对于填充的椭球体，内点的重力是位置的二次函数。多项式包含是椭球的牛顿势的推广。中空多项式夹杂中的内点感受到的引力是该位置的多项式。多项式包含已被证明对许多建模和设计问题很有用。具体地说，将研究与多项式夹杂有关的以下应用：(I)合成复杂材料设计的预测模型，(Ii)聚变反应堆和最小场浓度的最佳结构，以及(Iii)医学和地质成像的逆源问题。这一研究项目将与教育和外联活动相结合，如一对一指导、教育模块开发、培训研究生和促进跨学科合作。由k次多项式包含在体内诱导的牛顿势正好是k次多项式。对多项式包含的兴趣源于偏微分方程允许多项式包含的闭合形式的简单解，就像对于椭球一样。多项式包含的基本技术问题包括：(1)多项式包含的存在唯一性证明；(2)多项式包含的数值计算；(3)多项式包含的显式参数化；(4)利用多项式包含来解决上述工程问题。这个项目的结果可能会对经典的希尔伯特第十六问题以及从结构工程、核聚变设计到成像的工业问题产生影响。该项目还将通过招募不同背景的学生加入研究小组，加强罗格斯大学数学系和机械航空航天工程系之间的联系。

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.