Topics in Hyperplane Arrangements

A hyperplane is an affine (not necessarily through the origin) subspace of a (finite-dimensional) vector space of dimension one less than the ambient dimension (hence hyperplane, as in ambient dimension three they are just planes). A (finite) collection of such objects is called a hyperplane arrangement.

A very basic example would be the coordinate (hyper)planes in Euclidean space. A more intriguing one is given by all hyperplanes defined by \(x_i=x_j\) for \(i < j\), which even in dimension three is clearly quite different; this is called the braid arrangement.

There is a voluminous literature on hyperplane arrangements, and nearly every instance has parenthetical limits placed on it too. One of the most common one is (complex), but (essential) is popular too, describing the case where all the hyperplanes intersect in one point. This weighty research monograph focuses nearly exclusively on (real) (central) arrangements, where the ambient space is real and all hyperplanes go through the origin. (Though not necessarily just the origin; the braid arrangement above intersect mutually in a line, and so is not essential.)

If you want to know more about exactly what topics in hyperplane arrangements this book covers, please be directed to the terse but highly informative introduction. The main focus is on combinatorics and algebra coming from the intersections of both the hyperplanes themselves and the half-spaces they define. As an example, there are several quite advanced algebraic reformulations of the famous Zaslavsky formula for the number of regions the hyperplanes divide space into.

The authors have a knack for naming things (Lie and Zie elements, Janus monoid) but also regrettably put nearly all references into extremely telegraphic endnotes for each chapter. As a result, it’s hard to tell what is original, even when it comes to names, though a MathSciNet search and reading the endnotes suggests nearly everything is new. A related issue is that the extremely thorough notation index has no page numbers, nor is necessarily keyed well to the index; since there is a lot of notation, this is worth keeping in mind.

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Karl-Dieter Crisman teaches mathematics at Gordon College in Massachusetts, where he also gets to work on open source software, the mathematics of voting, and examining connections between all of these and issues of belief and faith.