Algebraic Methods in Combinatorics and Finite Geometry

该项目的重点是组合学和有限几何中的代数方法。组合学是数学中一个快速发展的领域。它拥有丰富的计算、科学和工程应用，从算法分析到人类基因组测序再到手机技术。在组合学中，人们考虑离散的(相对于连续的)结构，如图、超图、设计、拟阵和有限几何。本研究项目的中心任务之一是证明关于这些结构的存在性、枚举性和构造性结果。本研究项目集中在组合学和有限几何中产生的各种矩阵的代数不变量。所考虑的典型矩阵是设计和有限几何的关联矩阵，集合系统的高阶包含矩阵，以及图的邻接或拉普拉斯矩阵。所研究的不变量包括模秩、谱和Smith范式。这位调查者打算追求几个研究方向。一个方向是应用这些不变量来解决设计理论、极值组合学和有限几何中的存在问题。另一个方向是使用代数不变量来区分具有相同参数的非同构组合结构。

The project's focus is on algebraic methods in combinatorics and finite geometry. Combinatorics is a fast-growing area of mathematics. It has a wealth of computational, scientific, and engineering applications, ranging from algorithm analysis to human genome sequencing to cellular phone technology. In combinatorics one considers discrete (as opposed to continuous) structures such as graphs, hypergraphs, designs, matroids, and finite geometries. One central task of this research project is to prove existential, enumerative, and constructive results concerning these structures.This research project is centered on algebraic invariants of various matrices arising in combinatorics and finite geometry. Typical matrices considered are incidence matrices of designs and finite geometries, higher inclusion matrices of set systems, and adjacency or Laplacian matrices of graphs. The invariants under investigation include modular ranks, spectrum, and Smith normal forms. The investigator intends to pursue several research directions. One direction is to apply these invariants to solve existential problems in design theory, extremal combinatorics, and finite geometry. Another direction is to use algebraic invariants for the purpose of distinguishing non-isomorphic combinatorial structures with the same parameters.