Split Windows on Textmate

Here is a slightly frivolous answer, but only slightly. Suppose we were to divide mathematicians into two classes: those who could in principle happily pursue their research without knowing what cohomology is, and those for whom that would be completely unthinkable. (Of course, there is a spectrum in between, but let's not worry about that.) Now people in the latter class can be found in many many areas, from topology and geometry to algebra and number theory. There is a certain sense in which mathematicians of that kind have something very important in common, despite their differences, and form a mainstream from which discrete mathematics is mostly excluded.

However, it is also true that nowadays discrete mathematics is much more accepted by members of that mainstream as being an important subject. One sign of that is that there are papers appearing in Annals that almost certainly would not have been accepted twenty-five years ago. Another is that top universities tend to want at least some discrete mathematicians in a way that they used not to. I don't know whether we have got to the point where one could speak of an alternative mathematical mainstream, but I think that discrete mathematics is a well-established area that is broadly respected by more traditional mainstream mathematicians. (Perhaps another sign of that is that Lovasz is president of the IMU.)

There are two different questions here, one objective and one subjective. I will try to give my view, for what it's worth. Bear with me.

First, you are asking what is the publication history of discrete mathematics? (Even if I suspect you know this much better than I do). Well, originally there was no such thing as DM. If I understand the history correctly, classical papers like this one by Hassler Whitney (on coefficients of chromatic polynomials) were viewed as contributions to "mainstream mathematics". What happened is that starting maybe late 60s there was a rapid growth in the number of papers in mathematics in general, with an even greater growth in discrete mathematics. While the overall growth is relatively easy to explain as a consequence of expansion of graduate programs, the latter is more complicated. Some would argue that CS and other applications spurned the growth, while others would argue that this area was neglected for generations and had many easy pickings, inherent in the nature of the field. Yet others would argue that the growth is a consequence of pioneer works by the "founding fathers", such as Paul Erdős, Don Knuth, G.-C. Rota, M.-P. Schützenberger, and W.T. Tutte, which transformed the field. Whatever the reason, the "mainstream mathematics" felt a bit under siege by numerous new papers, and quickly closed ranks. The result was a dozen new leading journals covering various subfields of combinatorics, graph theory, etc., and few dozen minor ones. Compare this with the number of journals dedicated solely to algebraic geometry to see the difference. Thus, psychologically, it is very easy to explain why journals like Inventiones even now have relatively few DM papers - if the DM papers move in, the "mainstream papers" often have nowhere else to go. Personally, I think this is all for the best, and totally fair.

Now, your second question is whether DM is a "mainstream mathematics", or what is it? This is much more difficult to answer since just about everyone has their own take. E.g. miwalin suggests above that number theory is a part of DM, a once prevalent view, but which is probably contrary to the modern consensus in the field. Still, with the growth of "arithmetic combinatorics", part of number theory is definitely a part of DM. While most people would posit that DM is "combinatorics, graph theory + CS and other applications", what exactly are these is more difficult to decide. The split of Journal of Combinatorial Theory into Series A and B happened over this kind of disagreement between Rota and Tutte (still legendary). I suggest combinatorics wikipedia page for a first approximation of the modern consensus, but when it comes to more concrete questions this becomes a contentious issue sometimes of "practical importance". As an editor of Discrete Mathematics, I am routinely forced to decide whether submissions are in scope or not. For example if someone submits a generalization of R-R identities - is that a DM or not? (if you think it is, are you sure you can say what exactly is "discrete" about them?) Or, e.g. is Cauchy theorem a part of DM, or metric geometry, or both? (or neither?) How about "IP = PSPACE" theorem? Is that DM, or logic, or perhaps lies completely outside of mathematics? Anyway, my (obvious) point is that there is no real boundary between the fields. There is a large spectrum of papers in DM which fall somewhere in between "mainstream mathematics" and applications. And that's another reason to have separate "specialized" journals to accommodate these papers, rather than encroach onto journals pre-existing these new subfields. Your department's "encouragement" to use only the "mainstream mathematical journals" for promotion purposes is narrow minded and very unfortunate.

There's a curious sense in which almost no one really feels comfortably mainstream, regardless of how they stand with respect to the cohomological divide, or even of their community status. The Grothendieck phenomenon is rather an obvious example, but there are many others. If we venture outside of mathematics proper, Noam Chomsky, often referred to as the most cited intellectual alive, frequently speaks of himself as an outsider. (Specifically in relation to his linguistics, not his politics.)

Of course, it’s tempting to speculate about the honesty of such self-perception, but I tend to think of it as largely reflective of the human condition. It may also be that this kind of view goes well with a sort of rebellious energy conducive to creative intensity. For people who like literature, the sensibility is wonderfully captured in the novella `Tonio Kroeger’ by Thomas Mann. The irony is that almost anyone who reads the story is able to relate to the loner, as is also the case with the typical rebel in simpler dramas.

Why go far? Here we have Tim Gowers, an enormously respected mathematician by any standard, apparently presenting himself as a spokesman for the tributaries. In his case, I take it as the prototypical gentlemanly self-effacement one finds often in Britain.

At the very least, the whole picture is complicated.

The point is it’s probably not worth spending too much energy on this question. Administrative constraints, classifications, and selections are a real enough part of life within which we have to find some equilibrium, but serious mathematics has too much unity to be divided by the watery metaphor.

David Corfield once (good-humouredly) misquoted me with regard to the perceived distinction:

'Which do you like better, the theorem on primes in arithmetic progressions or the one on arithmetic progressions in primes?’

The original context of that dichotomy, however, was a far-fetched suggestion that there should be a common framework for the two theorems.

Added: The more I think of it, the more it seems that the original thrust of cohomology was very combinatorial, as might be seen in old textbooks like Seifert and Threlfall. The way I teach it to undergraduates is along the lines of:

space $X$ --> triangulation $T$ --> Euler characteristic $\chi_T(X)=V_T+F_T-E_T$ --> $T$-independence of $\chi(X)$ --> dependence of $V_T$ etc. on $T$ --> 'refined incarnation' of $V_T, E_T, F_T$ as $h_0$, $h_1$, and $h_2$, which are independent of $T$--> refined $h_i$ as $H_i$.

The emphasis throughout is on capturing the combinatorial essence of the space.