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ping Herman Rubin, second attempt; other reactions also welcome
Posted:
Oct 2, 2013 7:59 AM


Dear Professor Rubin,
In an old sci.math thread (1994), you once wrote:
> (...) Unless one goes into such things as cohomology > of groups, algebraic topology is NOT fundamental. (...)
Do you happen to remember that remark, and would you care to elaborate a bit?
I would have thought that at least, say, the idea of an homotopy is 'fundamental'. But let's agree that it's not (for example because we view sets, functions, relations, proofs, etc. as more fundamental). Then the question becomes what /is/ fundamental about cohomology of groups in that sense?
BTW, if anyone else has some idea on what Prof. Rubin could have meant, I'd be very interested, too.
For convenience, here's the original 1994 post, as found on Google's archive:
/// [Begin quote from Google groups archive]
In article <1994May13.112540.806@rhodes> mili...@vax.rhodes.edu writes: > In article <CppI6...@ennews.eas.asu.edu>, wol...@enuxsa.eas.asu.edu (Hans Juergen Wolters) writes: >> In article <27...@alice.att.com>, Peter Shor <sh...@alice.att.com> wrote: >>> >>> Something nobody has mentioned, and which I think is absolutely essential >>> for math majors, is a complex analysis course (one semester should be >>> enough). >> You must have missed my post or you consider me a nobody, >> but I said that complex analysis is taught at most German >> universities usually in the third or fourth semester.
> I've noticed that in America we tend to ignore things like geometry, > advanced calculus and complex variables at the undergraduate level. I think > this is a mistake, Although I do think the general courses that are required > should be required (like 2 semesters of abstract algebra and 2 semesters > of real analysis). This is what I would like an undergrad math major to see:
> Intro to Higher Mathematics (basically a course on proof and axioms > which contains (possibly) a construction of the Real number > system. This could contain some advanced calculus material > or lead to such a course)
This, of course, should come very early, it should be mandatory no later than high school. Nobody can UNDERSTAND calculus without at least having this at an intuitive level.
> Linear Algebra > > Multilinear Algebra
These are overblown. With a good understanding of mathematical concepts, there is not that much material here. Some of linear algebra is of applied importance, but not much of theoretical importance, and, except in a few areas, multilinear algebra, beyond the bare minimum, does not seem to me to be of much importance.
> Abstract Algebra (2 semesters) > > Real Analysis (2 semesters)
I would take the customary material, and teach all of it in much less time. I see no real problem with that.
> Topology (1 semester  it would be nice if we could cut away some of the point > set topology in order to do some algebraic topology  for example, > see the book by Gamelin and Greene)
My own preference would be otherwise. General point set topology is very poorly taught these days; the "horrible counterexamples" are what are needed to understand the concepts. Unless one goes into such things as cohomology of groups, algebraic topology is NOT fundamental.
> > Differential Geometry (1 semester  perhaps something on the order of > the first 4 chapters in the book by Do Carmo)
Good, but not fundamental.
> Differntial Equations  Applied Mathematics (1 or 2 semesters  the emphasis > should be on the Laplace and Fourier Transforms and the classical > work on Partial Differential Equations. Some of the elementary techniques > of solving ODE's should be done in elementary calculus (together > with the applications of these techniques))
UGH. The actual obtaining of solutions to differential equations should be relegated to a computational methods course. As for Laplace transforms, I know of uses for them, and while I often have need to solve a differential equation, I have yet to find a case where the linear equation is more easily solvable by Laplace transforms than directly. Similar arguments hold for the quite important Fourier transform. Physics is not mathematics.
> Complex Variables  1 semester > > Number Theory  1 semester
This is cute, but not fundamental.
Some very basic mathematics has been left out. Logic and set theory are very basic. Also measure theory, including probability theory; please do this as abstract, not confusing the issue by using unnecessary properties of the real line.
My main objection is that the program as you have outlined it spends too much time on tooearly use of computational procedures, and too little genralization. Doing special cases first does not help in understanding, and can lead to confusion later when the more general situation, often easily accessible at the time the special cases are introduced, comes up. See my comments above about general topology.
 Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN479071399 Phone: (317)4946054 hru...@stat.purdue.edu (Internet, bitnet) {purdue,puree}!snap.stat!hrubin(UUCP)
/// [End quote from Google groups archive]
 Best regards, Jean Mackerot



