Matrix and Operator Pencils Network

矩阵铅笔和算符铅笔是许多物理科学中出现的数学问题，包括非常古老和经典的问题，如弹性力学中的Cosserat问题，以及现代技术产生的问题，如如何通过反馈控制系统使空气动力学不稳定的喷气式战斗机能够飞行，或者如何在粒子加速器中保持等离子体稳定。这些系统可以被认为是多项式方程，其中的系数不仅是实数：相反，它们是矩阵(通常太大，以至于仅仅将它们存储在计算机上就会有问题)或更一般的数学对象，被称为“算子”。多项式方程必须在某种意义上“解”，以确定系统稳定或不稳定、可控或不可控的物理参数的值。这些方程的“解”集合被称为“谱”。至少从20世纪60年代初开始，数学家、物理学家和工程师就已经对这些问题进行了独立的研究，但这三个群体还没有充分地接触到彼此。数学家往往没有意识到应用中的最新挑战，而应用科学家还没有能够从算子铅笔光谱和逼近它们的新方法的研究方面的最新进展中受益。这笔赠款的目的是将三个社区结合在一起，在现有的一些小联络点和三个社区对跨学科研究的新热情的基础上，举行会议，解决悬而未决的问题，通过网站在更广泛的科学界传播他们的工作，并建立新的研究联盟，这些联盟应该比网络更持久。网络将增加数学家接触更现实和更具挑战性的应用问题的机会，将最新的新数学技术应用于工程问题，并使这三个社区都能够从设计具有有趣和多样化结构约束的矩阵和算子铅笔的数值方法和软件的巨大挑战中受益并做出贡献。

Matrix and operator pencils are mathematical problems which arise in many physical sciences, including very old and classical problems like the Cosserat problem in elasticity, as well as problems arising from modern technology, such as how to make an aerodynamically unstable jet fighter flyable by means of a feedback control system or how to keep a plasma stable in a particle accelerator.These systems may be thought of as polynomial equations in which the coefficients are not just real numbers: rather, they are matrices (often so large that just to store them on a computer would be problematic) or more general mathematical objects called `operators'. The polynomial equations have to be `solved' in some sense to determine values of a physical parameter for which a system is stable or unstable, controllable or uncontrollable. The sets of `solutions' of these equations are called `spectra'.There has been independent research on these problems by mathematicians, physicists and engineers at least since the early 1960s, but the three communities have not engaged with each other sufficiently. Mathematicians are often unaware of the newest challenges in applications, while applied scientists have not yet been able to benefit from the recent advances in the study of the spectra of operator pencils and new methods to approximate them. The purpose of this grant is to bring the three communities together, building on some small existing points of contact and a new enthusiasm in the three communities for interdisciplinary research, to hold meetings, to solve outstanding problems, to disseminate their work in the wider scientific community through a website and to establish new research alliances which should outlast the network.The network will enhance the exposure of mathematicians to more realistic and challenging application problems, bring recent new mathematical technologies to bear on engineering problems, and allow all three communities to benefit from and contribute to the great challenge of devising numerical methods and software for matrix and operator pencils with interesting and diverse structural constraints.