Damon and Affleck originally intended Will to be a physics prodigy, and their first draft used that idea. They were told by people who read their script that it would work better if Will was a math prodigy, so later drafts used that. They also planned to work in a subplot about Will being recruited by the U.S. government to do cryptology and about Will being chased by government agents. They were told by other people who read the script that this subplot didn’t work well, so they eliminated it. Damon and Affleck wrote every draft of the film, although they changed it several times in response to suggestions by people who read the various drafts. (Actually, Damon wrote the first draft of the film and later asked Affleck to re-write it and become the co-writer of the film.)
In any case, none of this has anything to do with who was given the Oscar for writing the film. When a number of writers work on a film, there is (always? sometimes? I’m not sure) a hearing before the Screenwriter’s Guild. The Guild looks at all the drafts of the film and decides who has made major contributions to the actual film. Those writers chosen by the Guild are the ones whose names appear on screen as the writers of the film. Those writers, as a team, are the ones who are eligible for an Oscar. Who gets how much money for the writing of the various drafts of the screenplay depends on the producers of the film. When they buy a script to produce it, they pay the writers right then (and possibly also promise them a cut of profits). When they get another writer or writers to do a new draft with some revisions in it, they pay those new writers for the new draft at that point.
The Snake Lemma is probably the basic result in homological algebra. It’s used to get the “long-exact sequence in homology” to work out right. Originally it arose in the simplicial homology of topological spaces. Then it was generalized to a result in R-modules. Nowadays it’s been extended to a result in any abelian category.
The standard proof is at the second level, and relies on the well-worn technique of “diagram chasing”, which is why it works so much better visually than written out in formulas. The general result can either be done directly (but confusingly to less categorically-adept students) or by invoking the (high-powered) theorem that any abelian category can be faithfully embedded in an apropriate category of R-modules, and then the diagram-chasing proof suffices.
So, the statement (assuming I get the coding right). Consider the following diagram with exact rows:
A_1--->A_2--->A_3--->0
| | |
|f_1 |f_2 |f_3
V V V
0--->B_1--->B_2--->B_3
Then there is a map s:ker(f[sub]3[/sub])->coker(f[sub]1[/sub]) such that the long sequence
Basically, the proof goes by picking an element in A[sub]3[/sub] that gets killed by f[sub]3[/sub], then “chasing” it around to an element in B_1. This defines the map s, but some student – me, when I first saw this – will raise the objection that certain non-canonical choices were made along the way, and so s isn’t well-defined.
A student in the movie indeed raises the point and Jill Clayburgh gives the correct diagram chase to show that any two possible images differ from each other by the image under f[sub]1[/sub] of an element of A[sub]1[/sub]. Thus, s is well-defined as a map into coker(f[sub]1[/sub]).
> A student in the movie indeed raises the point . . .
Daniel Stern plays the student, incidentally. The interplay between Stern and Clayburgh is even more interesting than the fact that the theorem is given. Writing the correct diagram on the board and having Clayburgh give the correct explanation (as far as it goes, but the scene starts well into her explanation, so we only hear the end of the statement of the proof) is interesting in itself, but the fact that we can tell that Sterm is following the proof and asking exactly the sort of questions an intelligent student would ask makes it better.
Now that I think about it, it seems to me that the one unrealistic thing about the interaction between Stern and Clayburgh in the movie is the exchange at the end of the film. Stern is supposed to be a grad student. Now, the Snake Lemma is the sort of proof that one would learn in one’s first year in grad school. A really good student might even learn it in their senior year of college. There’s a discussion between Stern and Clayburgh at the end of the film (which is only supposed to be a couple of weeks later than the beginning of the film) in which Stern is talking about a theorem he claims he can solve. The theorem is (it sounds to me like) the complete classification of finite simple groups. This problem was indeed solved several years after the film. The problem is that it wouldn’t be several weeks between the time that a student learned the Snake Lemma and began work on trying to solve the complete classification problem but more like several years. (Incidentally, I am aware that the complete classification problem was not a single theorem but a huge mathematical subject worked on by dozens of mathematicians in hundreds of articles over decades of research.)
NUMB3RS was slightly different than most. Most of the equations were right, but it was about how a subject would be approached or how mathematicians do things. Nominally, the point of the show is how mathematics (and, thus, mathematicians) solves real-world problems. If you don’t have a single character who solves a problem like a mathematician, you’ve lost half the point right there.
I wonder if they really sent the episodes to you to get your honest feedback, or they sent them to you on the off chance you’ll encourage your students to watch? (Or, possibly even more cynical, they were hoping to get at least one comment of ‘this is great!’ so they could use it for advertising, and ignore all the ‘this is stupid’ comments).
Well, yes. The cynical comment is what I meant by “feedback”. My point is not that they asked advice afterwards, but that the expense of coordinating such marketing efforts is at least what it would cost to get a grad student on staff in the first damn place.