--------------------------------------------- PAPER EDITS section changes: - intro should be shorter (Sasha pruned the intro) - do we need the overview paragraphs? maybe put them at the intro of each section instead (Sasha moved two to intros, scrapped others) - change section 1 to "constructors" or "making simplicial complexes" or "working with simplicial complexes" (?) and change emphasis (Ben began making this change, Sasha contributed) - pull a bunch of section 2 stuff into this (Ben began making this change, Sasha contributed) - combine the examples in intro + section 2, i.e. example 0.0 should be merged with 2.3 (Ben began making this change, Sasha contributed) - start with Stanley Reisner ideal definition etc. give the constructor example - then give examples of the database (example 1.3) (Sasha: atm, the structure is different based on how Ben added things, but I think it works) - end with the maps (example 1.2) - edit example 1.2 to start with a sentence (no longer have example environments) - change section 2 to "homological algebra," or "___ combinatorics" (__ could be algebraic, polyhedral, etc.?) - give the link, Hochster's formula, shellability and CM stuff - section 3 renamed to "commutative algebra" (too broad?), or "free resolutions" remove: - bounds on h vector don't seem necessary (Sasha commented them out) other things: - tighten up exposition everywhere (Sasha did a passthrough of his own writing) - we don't define an abstract simplicial complex which is a mistake (Sasha added this to intro, together with the basic definitions) - point out that our package isn't well suited for huge numbers of vertices, i.e. bad for topological data analysis (Sasha added a sentence in opening paragraph of section 1) - need a better title: "simplicial complexes in Macaulay2" or "Some simplicial complexes in Macaulay2" (Sasha changed this, still open to feedback) - homogenize pictures, i.e. the placement of vertices is done in an absolute way as opposed to, e.g. "above right, above left," etc. - we reuse k for a field and index which is slightly taboo. pick another letter in some way--maybe use A for comm. ring? (Sasha changed index to \ell. added a sentence in section 2 to assume k is a field) - use \dotsc or \dotsb instead of ... or \cdots - sub/superscript hell in some places, e.g. Alexander dual - we could just take out the example environments entirely, any references to examples can be finessed--e.g. we can refer to to refer to an example (and more generally we can label everything uniquely) (Ben began making this change) - Section 2 or 3 needs to have some relabelling of the ring S. --------------------------------------------- FIRST WEEK PLANNING - find examples from other places in math that we can reproduce in macaulay2 - showcase functionality, specifically on simplicial maps which are new - do some relative homology, try to find some interesting topological results - browse through some papers in jsag journal to get ideas for structure - showcase database functionality - basic fact that we represent complexes by Stanley-Reisner ideals - showcase Lybenziunk complexes and general homogenization feature - database could be used to give examples of relative homology --------------------------------------------- MORE DETAILED SECTION PLANNING - Section 2 (combinatorics) - f-vectors, h-vectors, Cohen-Macaulay + shellability are the themes for this section. - Should give some of the motivation in identifying Cohen-Macaulay rings - Prop 3.1 ch 3 of Stanley gives a condition for checking CM - Show examples of shellable + non-shellable - Rudin ball/Ziegler ball is Cohen-Macaulay but not shellable - Certain "chess-board" complexes are pure but not Cohen-Macaulay - Use Reisner's criterion (corollary 5.3.9) in Bruns-Herzog for an example + non-example - Another way to check CM is to look at the minimal free resolution to compute projective dimension and compare with dimension (may be worth mentioning) - Lemma 5.1.8 relates h and f-vectors - Use Thm 5.1.15 in Bruns-Herzog on an example to illustrate connection with h-vector - Ex 1.2 in Stanley for h-vector relating to Poincaré series of P1 x P1 - References: Stanley, Bruns-Herzog, ... - Section 3 (resolutions) - Give examples of all(?) of our complex constructions - Scarf resolution - Nearly scarf ideals: pick something in database, construct the nearly scarf ideal, then show that scarf complex is the same - If you have a resolution supported on a complex, one way to determine if its a resolution is to do all induced subcomplexes of the lcm sublattice of the ring - Reuse ex 1.2 above to compute minimal free resolution of irrelevant ideal of P1 x P1 - Taylor resolution that's not minimal - References: Miller-Sturmfels, Peeva, ... - Section 4 (topology) - Simplicial maps, relative homology, database of small 2/3-manifolds - Maybe take a classical map (Hopf fibration), then approximate by simplicial complexes and illustrate the example - To this end, find classical examples in topology. e.g. in Hatcher/Munkres - References: Munkres, Hatcher, Tight polyhedral submanifolds and tight triangulations, introduction to piecewise linear topology