Extremal Combinatorics

极值组合学

基本信息

批准号：
EP/G056730/1
负责人：
Peter Keevash
金额：
$ 41.9万
依托单位：
Queen Mary University of London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2010
资助国家：
英国
起止时间：
2010 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FG056730%2F1
关键词：
Extremal Combinatorics

项目摘要

Combinatorics forms a challenging and fundamental part of mathematics, but is in the happy position of being relatively accessible to a wider audience. Its theorems find a wide number of direct applications both to other areas of mathematics and other academic disciplines, and thus it makes its influence felt indirectly as these disciplines, in turn, apply the theoretical power of combinatorics in more practical settings. The three main objectives of this project are both fundamental combinatorial problems and also have potential applications to additive number theory, and topics in computer science, including the theories of complexity, codes, and communication. Looking more broadly beyond the immediate applications, there are many other more practical fields to which ideas from Combinatorics make contributions, including physics, electrical engineering, bioinformatics, economics, and internet modelling. One of its most exciting and rapidly developing branches is Extremal Combinatorics, which deals with finding the extremal values of a function defined on some class of combinatorial objects. In fact, most combinatorial problems can be thought of as having such a characterisation, but even just focussing on those which are naturally expressed in such terms leads to some of the most elegant and deep questions in discrete mathematics. The basic concepts needed to understand the problems posed are remarkably simple, particularly in constrast with many other areas of mathematics, but this belies the ingenious ideas and techniques which mathematicians have been led to develop over the years in solving these problems.

Combinatorics构成了数学的具有挑战性和基本的一部分，但在更广泛的受众中相对易于访问。它的定理在其他数学和其他学科的其他领域都发现了广泛的直接应用，因此，由于这些学科在更实用的环境中应用理论上的能力，因此它使其受到间接的影响。该项目的三个主要目标既是基本的组合问题，又在添加数理论和计算机科学中的主题（包括复杂性，代码和沟通理论）中具有潜在的应用。除了直接应用外，还可以更广泛地展现，还有许多其他更实用的领域，其中组合的想法做出了贡献，包括物理学，电气工程，生物信息学，经济学和互联网建模。极端组合学是其最令人兴奋的和快速发展的分支机构之一，它涉及在某些组合对象上定义的函数的极值。实际上，大多数组合问题都可以被认为是具有这样的特征，但即使只是以这种术语表达的人，也会导致一些在离散数学中最优雅，最深切的问题。了解所带来的问题所需的基本概念非常简单，尤其是在许多其他数学领域的约束中，但是这掩盖了多年来数年来解决这些问题的巧妙思想和技术。