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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 <insert Ben's silly ascii symbol here> 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.

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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

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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 mentoning)
- 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 poincare 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 (toplogy)
- 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 toplogy. e.g. in Hatcher/Munkres
- References: Munkres, Hatcher, Tight polyhedral submanifolds and tight triangulations, introduction to piecewise linear topology
