Perron-Frobenius theory and max-algebraic combinatorics of nonnegative matrices

极大代数是数学中一个发展迅速的分支，它具有解决数学、科学和工程中一类非线性问题的潜力，当算术加法被极大值运算所代替，算术乘法被加法所代替时，这些非线性问题可以被赋予线性问题的形式。除了处理非线性问题的优势，如果他们是线性的，极大代数的技术使我们在许多情况下，有效地描述了所有解决方案的集合，从而选择一个最好的一个关于一个指定的标准。它还提供了一个代数编码的一类组合或组合优化问题。虽然基础的最大代数创建的第一个开创性的文件产生的心脏英格兰50年前，它主要是2000年后，我们看到了显着的扩张，这一领域的一些研究中心在世界各地（例如巴黎，伯克利，圣地亚哥，德尔夫特，麦迪逊和莫斯科）。如今，它渗透到一系列数学领域，从代数拓扑，泛函分析，线性代数和几何，到非线性，离散和随机优化以及数学生物学。部分或全部致力于极大代数的会议、小型研讨会、讲习班和其他活动的数量正在增加。出版了一些研究专著，其中三本是2005年以来出版的。应用程序是理论（例如在离散事件动态系统，控制理论和优化）和实际（荷兰铁路网的分析）。随着极大代数的迅速发展和最新的研究成果，利用极大代数的组合技巧来加强极大代数与传统线性代数之间的相互作用似乎是一个很好的时机。这意味着，例如，发展Perron Frobenius理论的半环，发展理论的最大代数张量，解决平均支付游戏和最大代数矩阵方程。这将对理解矩阵的一系列性质产生直接影响，这些性质在数学的其他领域，物理学，计算机科学，工程学，生物学和其他地方以类似于传统线性代数的方式找到应用。例如，它将使研究人员能够解决最大代数方程组，通过使用最大代数模型而不是传统模型来帮助分析复杂的信息技术系统，在具有最大线性动力学的系统中找到稳定状态，建模和解决固态物理中出现的问题，或某些类型的调度问题。PI将把这项研究与现有的英国热带数学工作组的工作联系起来，该工作组由他担任主席的伦敦数学学会资助。该小组的研究会议每年在沃里克、曼彻斯特和伯明翰举行三次，来自英国多所大学的30多名同事参加。PI将与一些国际领先的极大代数中心建立合作网络和战略伙伴关系，以进一步推动该领域的发展。预计，作为该项目的结果，PI将获得支持，于2014年或2015年在伯明翰举办一次热带数学国际会议，并在未来创建一个热带数学中心（CTM），该中心将有几个资助的研究项目。CTM将组织国际研究研讨会和会议，为工业合作伙伴提供专业知识，并提供专业的本科和研究生课程。它将与曼彻斯特大学的CICADA研究小组和巴黎理工学院现有的类似中心密切合作。

Max-algebra is a rapidly evolving branch of mathematics with potential to solve a class of non-linear problems in mathematics, science and engineering that can be given the form of linear problems, when arithmetical addition is replaced by the operation of maximum and arithmetical multiplication is replaced by addition. Besides the advantage of dealing with non-linear problems as if they were linear, the techniques of max-algebra enable us in many cases to efficiently describe the set of all solutions and thus to choose a best one with respect to a specified criterion. It also provides an algebraic encoding of a class of combinatorial or combinatorial optimisation problems.Although the foundations of max-algebra were created in the first pioneering papers produced in the heart of England 50 years ago, it is mainly after 2000 that we see a remarkable expansion of this area in a number of research centres worldwide (e.g. Paris, Berkeley, San Diego, Delft, Madison and Moscow). Nowadays it penetrates a range of areas of mathematics from algebraic topology, functional analysis, linear algebra and geometry, to non-linear, discrete and stochastic optimisation and mathematical biology. The number of conferences, mini-symposia, workshops and other events devoted partly or wholly to max-algebra is increasing. A number of research monographs have been published, three of them since 2005. Applications are both theoretical (for instance in discrete-event dynamic systems, control theory and optimisation) and practical (analysis of the Dutch railway network). Following the recent remarkable expansion of max-algebra and latest research findings, it seems to be the right time to use the recently developed powerful combinatorial techniques of max-algebra to strengthen the interplay between max-algebra and conventional linear algebra. This means for instance to develop the Perron-Frobenius theory in semirings, develop the theory of max-algebraic tensors, solve the mean-payoff games and max-algebraic matrix equations. This will have an immediate impact on the understanding of a range of properties of matrices which find applications in other areas of mathematics, in physics, computer science, engineering, biology and elsewhere in a way similar to that of conventional linear algebra. For instance it will enable researchers to solve systems of max-algebraic equations, help to analyse complex systems of information technology by using a max-algebraic rather than traditional model, find a steady regime in systems with max-linear dynamics, model and solve problems arising in solid state physics, or in certain types of scheduling problems.To feed into this project and also to help to address the challenges, the PI will link this research with the work of the existing UK working group in tropical mathematics funded by the London Mathematical Society, which he chairs. Research meetings of this group are organised three times a year in Warwick, Manchester and Birmingham and are attended by more than 30 colleagues from a number of UK universities. The PI will form collaborative networks and strategic partnership with a number of internationally leading centres in max-algebra to further advance the field. It is expected that as a consequence of this project the PI will obtain support to organise an international conference on tropical mathematics at Birmingham in 2014 or 2015 and in the future to create a centre for tropical mathematics (CTM), which will have several funded research projects. CTM will organise international research workshops and conferences, provide expertise for industrial partners and for specialised undergraduate and postgraduate courses. It will closely cooperate with the research group CICADA at the University of Manchester and with the existing similar centre at the Ecole Polytechnique in Paris.