From the prefaceThe London Taught Course Centre (LTCC) for PhD students in the MathematicalSciences has the objective of introducing research students to abroad range of advanced topics. For some students, these topics mightinclude one or two in areas directly related to their PhD projects, but therelevance of most will be much less clear or even apparently non-existent.However, all of us involved in mathematical research have experienced thatextraordinary moment when the penny drops and some tiny gem of informationfrom outside one’s immediate research field turns out to be the keyto unravelling a seemingly insoluble problem, or to opening up a new vistaof mathematical structure. By offering our students advanced introductionsto a range of different areas of mathematics, we hope to open their eyes tonew possibilities that they might not otherwise encounter.Each volume in this series consists of chapters on a group of relatedthemes, based on modules taught at the LTCC by their authors. Thesemodules were already short (five two-hour lectures) and in most cases thelecture notes here are even shorter, covering perhaps three-quarters of thecontent of the original LTCC course. This brevity was quite deliberate onthe part of the editors: we asked contributors to keep their chapters shortin order to allow as many topics as possible to be included in each volume,whilst keeping the volumes digestible. The chapters are “advanced introductions”,and readers who wish to learn more are encouraged to continueelsewhere. There has been no attempt to make the coverage of topics comprehensive.That would be impossible in any case — any book or series ofbooks which included all that a PhD student in mathematics might needto know would be so large as to be totally unreadable. Instead, what wepresent in this series is a cross-section of some of the topics, both classicaland new, that have appeared in LTCC modules in the nine years since itwas founded.The present volume covers the general area of algebra, logic andcombinatorics. The main readers are likely to be graduate students andmore experienced researchers in the mathematical sciences, looking forintroductions to areas with which they are unfamiliar. The mathematicspresented is intended to be accessible to first year PhD students, whatevertheir specialised areas of research, though we appreciate that how “elementary”or “advanced” any particular chapter appears to be will differwidely from reader to reader. Whatever your mathematical background,we encourage you to dive in, and we hope that you will enjoy reading theseconcise introductory accounts written by experts at the forefront of currentresearch.