For your first reading assignment, your goal will be understand of the idea of a simplicial complex—or one of its many generalizations, if you’re already familiar with this idea (see below). Simplicial complexes are the basis of many of the algorithms we’ll study in this class—and for a vast array of algorithms used in mesh processing, scientific computing, and Hollywood. They are also one common way to discretize topological spaces, which are the starting point for our study of geometry in the smooth setting.
We will discuss this concept further in class, so don’t worry if you’re utterly confused! Also, please feel free to discuss and ask questions here, by posting in the comments.
Listed below are some suggested readings (from “beginner” to “expert”). For this first reading assignment, you are also free to search for references that you find easier or more interesting:
- Simplicial Complex (up through & including “Closure, Star, and Link”) – Perhaps it seems lazy to send a link to the Wikipedia (which is the first result on Google), but then again, I wrote much of this page many years ago—and made the figures. It would be a useful exercise to contrast with Abstract Simplicial Complex.
- Simplicial Complexes and Simplicial Complexes – two brief introductions, both by Herbert Edelsbrunner.
- Cell Complexes – more advanced (and nicely illustrated) introduction by Jeff Erickson.
- Algebraic Topology (pp. 102–104) – very nice (and free!) book by Allen Hatcher, who here discusses $\Delta$-complexes, a close cousin of simplicial complexes. Those of you who want to go really deep might also take a look at pp. 5–10 on cell (CW) complexes, as well as an additional supplement on simplicial CW structures.
- A Survey of Efficient Structures for Digital Geometry Processing, Section 2.2 – I include this reference not because it is particularly good introduction to simplicial complexes—in fact, there are some errors!—but because it gives a rough sense of the kind of document you will write for your final project (albeit a bit different in focus).
Submission: Please send an email to email@example.com and firstname.lastname@example.org no later than 10:00 AM on Tuesday, January 19; please include the string DDGSpring2016 in your subject line, so that we can be sure we catch your submission. Your email for readings should always include:
- a short (2-3 sentence) summary of what you read, and
- at least one question about something you found confusing / incomplete / not addressed.
For this particular reading, your summary should include a clear, concise definition for a simplicial complex, i.e., one that could be used to quickly communicate the idea to one of your peers without ambiguity. For those of you already familiar with simplicial complexes, try to summarize the taxonomy of different complex types (simplicial complex, oriented simplicial complex, $\Delta$-complex, CW-complex, simplicial set, …).