Combinatorial designs and their generalizations

点击翻译按钮获取中文摘要

This research aims to improve our understanding of combinatorial designs and some of their generalizations. In the most basic case, a design on some ground set is a collection of subsets which covers any two distinct elements equally often. This is a familiar idea in many geometric models, since two different points uniquely determine a line. Designs underlie well-known puzzles, such as Sudoku and the “prisoner's hat problem”. More importantly, by their nature designs are useful in information theory (through their connection with error-correcting codes), computer science (software testing, network design) and statistics (experimental design). These applications have placed the focus on finite designs, and the beautiful geometries arising from finite fields (binary sequences, for instance) form a natural starting point.******It is possible to extend the basic definition above in a number of ways. In a t-design, any t distinct points are to be contained in the same number of blocks. In a graph decomposition, the edges or connections between pairs are to be partitioned into copies of a given small structure. A further possible extension includes the use of edge colours to model two or more simultaneous relationships. As one toy example, a bridge tournament might require every pair of players to be partners exactly once and opponents exactly twice.******Although there are many special constructions known, the general existence question for designs and graph decompositions is notoriously hard. This research is mainly concerned with the challenging and most general cases, and especially in situations where standard methods don't apply.******A recent celebrated theorem on existence of designs has brought this topic to the forefront of combinatorial mathematics. Even as this theory is complete for extremely large designs, there is still considerable work to be done. In particular, the reasons why specific designs fail to exist leads to wonderful connections to other areas of mathematics, including algebra, analysis and geometry.******In comparison with applied mathematics and other sciences, this research is admittedly a step removed from direct industrial applications. However, offsetting this, it is sufficiently general to have end-use in a wide variety of applications. Moreover, the additional structure that information-based problems impose on designs, codes, or arrays is quite often mathematically natural. In this way, the research is guided closely by its applications.