Some Chapters from “combinatorial Commutative Algebra” (ezra Miller and Bernd Sturmfels)

Of the 18 chapters in the book, we will only cover the 5 listed below and thus allow for a more leisurely pace. There is also some supplementary material in the bibliography to expand upon if needed. The selected chapters are thematically divided into two groups. The first one is by far the larger and is concerned with the interaction between ideas from (simplicial) geometry and topology on the one hand, and from basic commutative or homological algebra (resolutions, Hilbert series, etc.) on the other. The second “group” consists only of Chapter 12 on Ehrhart polynomials; nevertheless, it seems appropriate to include this material because it is very intuitive and easy to explain, lies at the crossroads of algebra and geometry, and contains some quite surprising results (the relation of the Ehrhart polynomial to Todd classes and Grothendieck–Riemann–Roch, Brion’s formula, and Barvinok’s Theorem). Chapter 1 contains standard introductory material from commutative algebra, but it is probably a good idea to review it anyway, making good use of the combinatorial/pictorial setup of monomial ideals provided there. Highlights of Chapter 3 are the very suggestive pictures of 3-dimensional staircases on the one hand, and the close connection between monomial ideals in 3 variables to the representation of planar graphs on the other. This latter material is well known to (and in some cases actively studied by) people in the local combinatorics community, and is so of interest to both the algebraically and the combinatorially inclined. Chapter 4 reveals what was “really” going on in Chapter 3. It unifies concepts from geometry, topology and algebra, and introduces one of the central algebraic/polyhedral objects in the book: the hull complex, due to Bayer and Sturmfels. It contains a wealth of examples that make the connection between algebra and polyhedral geometry very explicit. One of the both visually and mathematically most suggestive constructions — Alexander duality — is introduced in Chapter 5. This chapter revolves around the interaction of three different kinds of duality: polyhedral duality resp. duality of planar graphs, generators vs. irreducible decompositions of monomial ideals, and Alexander duality. All this comes packaged with pretty pictures of staircases and dual graphs. The last chapter proposed for this seminar, Chapter 12 on Ehrhart polynomials, is on a different theme. Ehrhart polynomials build yet another bridge from the combinatorics and