Matrix and Operator Pencils Network

矩阵和算子束是出现在许多物理科学中的数学问题，包括非常古老和经典的问题，如弹性中的Cosserat问题，以及现代技术中出现的问题，例如，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。其中系数不仅仅是真实的数：相反，它们是矩阵（通常如此之大，以至于仅仅将它们存储在计算机上将是有问题的）或称为“算子”的更一般的数学对象。多项式方程必须在某种意义上“求解”，以确定系统稳定或不稳定、可控或不可控的物理参数值。这些方程组的“解”集被称为“谱”，至少从1960年代初开始，数学家、物理学家和工程师就对这些问题进行了独立研究，但这三个群体相互之间没有充分接触。数学家通常不知道应用中的最新挑战，而应用科学家还没有能够从算子束光谱研究的最新进展和近似它们的新方法中受益。这笔赠款的目的是将三个社区聚集在一起，建立在一些现有的小接触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，通过一个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，这些联盟应比该网络更持久。该网络将增加数学家对以下方面的接触：更现实和具有挑战性的应用问题，使最近的新的数学技术承担工程问题，并允许所有三个社区受益，并有助于设计数值方法和软件的矩阵和算子铅笔有趣的和不同的结构约束的巨大挑战。

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.