Combinatorics is a loosely-organized area of mathematics having to do with counting and optimizing discrete mathematical structures. Topology is an area of mathematics having to do with deforming and distinguishing continuous mathematical structures. Topological combinatorics refers to the application of ideas from topology to problems in combinatorics.
This thesis consists of an introduction, followed by five papers on various topics in topological combinatorics. Each paper is given in its original form, as its own chapter, with co-authors and current publication status indicated at the beginning of the chapter. Here we give definitions and summarize the results of the later chapters, including directions for further research.