Finite Geometries and their Applications

我的目标是继续在离散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与通信理论的应用。现代技术涉及数字信息的存储和传输。基础是在20世纪40年代由数学家和工程师克劳德·E·香农奠定的。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未解决。它们引起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，但这些问题仍然没有得到解决。我继续研究这些问题，例如：有限射影平面和块设计的存在性问题。对于最长的线性MDS代码问题，我已经为1的部分解决方案贡献了一些结果[例如，对于平面的10阶情况]，并且我已经共同撰写了问题2的主要结果。对于1，主要工具是我开发的块集结果集合。发表在《发明》（Inventiones）杂志上的这项工作，灵感来自于Segre对曲线的哈斯-韦尔定理（Hasse-Weil theorem）的出色运用。这些MDS代码的主要实际示例是广泛使用和无处不在的里德-所罗门代码，该代码首先因激光唱机的发明而突出错误控制。我正在进行的研究计划的广泛议程继续是我在有限几何和设计以及离散数学相关领域（如图论）的工作，但总是着眼于应用。我过去在通信领域写过两份专利申请，目前正在处理另外两份。我渴望积极地进行这个正在进行的研究项目。

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.