Finite Geometries and their Applications

有限几何及其应用

• 批准号: 8726-2013
• 负责人: Bruen, Aiden
• 金额: $ 0.8万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635197
• 关键词: Finite; Geometries; their; Applications

My aim is to continue an established program of research in discrete and applied mathematics, specializing in finite geometries and designs with applications to communication theory. Modern technology deals with the storage and transmission of digital information.The groundwork was laid in the nineteen forties by Claude E Shannon, a mathematician and engineer. In particular he showed how to digitize [band-limited] analog signals in a practical way. Many practical problems in the communications area remain open. They give rise to basic mathematical problems which still remain unsolved despite the efforts of notable mathematicians and Fields medalists. I continue to work on these problems, such as the following:1.the existence problem for finite projective planes and block designs.2.the longest linear MDS codes problem I have contributed some results [e.g. for the order 10 case for planes] towards a partial solution of 1 and I have co-authored a major result for problem 2. For 1, the main tool is a collection of results in blocking sets which I have developed. The work in 2, published in the Inventiones, was inspired by Segre's brilliant use of the Hasse-Weil theorem for curves. The main practical examples of these MDS codes are the widely-used and ubiquitous Reed-Solomon codes which first came to prominence for error control with the invention of the compact disc player. The broad agenda of my ongoing research program continues to be my work in finite geometries and designs and related areas of discrete mathematics such as graph theory, but always with an eye to applications.I have written two patent applications in the communications area in the past and am working on two others.I am eager to vigorously pursue this ongoing research programme.

我的目标是继续在离散数学和应用数学方面的既定研究计划，专门研究有限几何和设计及其在通信理论中的应用。现代技术涉及数字信息的存储和传输。20世纪40年代，数学家和工程师克劳德·E·香农奠定了这一基础。特别是，他展示了如何以一种实用的方式将[带限]模拟信号数字化。通信领域的许多实际问题仍然悬而未决。它们产生了基本的数学问题，尽管著名的数学家和菲尔兹奖牌获得者做出了努力，这些问题仍然没有得到解决。我继续研究这些问题，例如：1.有限射影平面和区块设计的存在性问题。2.最长线性MDS码问题我已经为问题1的部分解贡献了一些结果[例如，对于平面的阶数为10的情况]，并且我与人合著了问题2的一个主要结果。对于1，主要工具是我开发的块集合中的结果的集合。发表在《发明》杂志上的2中的工作灵感来自于Segre对曲线的Hasse-Weil定理的巧妙运用。这些MDS码的主要实际例子是广泛使用且无处不在的里德-所罗门码，它随着光盘播放器的发明而首次因差错控制而变得突出。我正在进行的研究计划的主要议程仍然是我在有限几何和设计以及离散数学(如图论)的相关领域的工作，但总是着眼于应用。我过去曾在通信领域撰写过两项专利申请，目前正在进行另外两项申请。我渴望积极推进这一正在进行的研究计划。