Abstract

What is geometric combinatorics? This question is a bit controversial, but at least in part, it is the study of geometric objects and their combinatorial structure. Rather than trying to define this precisely at the outset, in this lecture we’ll mainly give examples that appear in the 2004 PCMI graduate courses. 1. Polytopes A popular class of examples are the convex polytopes, that is, convex hulls of finite point sets in R d. These form the main topic of the graduate course by Ziegler, but also play prominent roles in the undergraduate courses by Swartz and Thomas, and in the undergraduate faculty course by Su (as well as making cameo appearances in the graduate courses by Barvinok, Fomin, Forman, MacPherson, and Wachs!). In R 2, convex polytopes are polygons such as triangles, quadrilaterals, pentagons, hexagons, etc. In R 3 they can be more interesting, such as the triangular prism depicted in Figure 1(a). a b

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