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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568612
• 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和应用数学研究的既定计划，专门从事有限几何和设计与应用通信理论。 现代技术涉及数字信息的存储和传输，数学家和工程师克劳德·香农在20世纪40年代奠定了基础。特别是他展示了如何以实用的方式对[带限]模拟信号进行滤波。通信领域的许多实际问题仍然悬而未决。它们产生了一些基本的数学问题，尽管著名的数学家和菲尔兹奖获得者做出了努力，这些问题仍然没有得到解决。我继续致力于解决这些问题，例如： 1.有限射影平面和区组设计的存在性问题。 2.最长线性MDS码问题 我已经贡献了一些结果[例如，为10阶情况下的飞机]对部分解决方案1，我合著了一个主要结果的问题2。对于1，主要工具是我开发的阻塞集合中的结果集合。这项工作在2，发表在发明，灵感来自塞格雷的辉煌使用的哈塞-韦尔定理的曲线。这些MDS码的主要实际例子是广泛使用和无处不在的里德-所罗门码，随着光盘播放器的发明，里德-所罗门码首先在差错控制方面变得突出。 我正在进行的研究项目的广泛议程仍然是我在有限几何和设计以及离散数学相关领域的工作，如图论，但始终着眼于应用。我已经写了两个专利申请在通信领域在过去，并正在对其他两个工作。我渴望积极追求这个正在进行的研究计划。