Finite Geometries and their Applications

我的目标是继续开展离散和应用数学的既定研究计划，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。数学家和工程师克劳德·E·香农 (Claude E Shannon) 在十九世纪四十年代奠定了基础。他特别展示了如何以实用的方式将[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。尽管著名数学家和菲尔兹奖获得者付出了努力，但它们引发了一些基本的数学问题，但这些问题仍未得到解决。我继续研究这些问题，例如：1.有限射影平面和分块设计的存在性问题。2.最长线性MDS代码问题我已经贡献了一些结果[例如对于飞机的 10 阶情况] 朝向问题 1 的部分解决方案，并且我与人共同创作了问题 2 的主要结果。对于问题 1，主要工具是我开发的分块集中的结果集合。 2 中的作品发表在 Inventiones 上，其灵感来自于 Segre 对曲线 Hasse-Weil 定理的出色运用。这些 MDS 代码的主要实际例子是广泛使用且无处不在的 Reed-Solomon 代码，随着光盘播放器的发明，该代码首次在错误控制方面脱颖而出。我正在进行的研究计划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（例如图论）的工作，但始终着眼于应用。我过去在通信领域撰写了两项专利申请，并且正在研究另外两项专利申请。我渴望大力开展这项正在进行的研究计划。

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.