Topology: The slum of Combinatorics OR ``Don't show off too much, your specialty will soon be Trivialized''
By: Doron ZeilbergerFor many years, and even today, topologists considered themselves the cream of mathematics. They monopolized editorial boards of `prestigious'(i.e. snooty, e.g. Acta, Annals, Invent.) journals and would-be-prestigious journals (which are even more pathetic, e.g. Duke J. and Israel J.), as well as the Fields Medal committee. Most people have heard Whitehead's epithet that `Graph Theory is the slum of topology', and many have agreed.
I am pleased to be alive to witness the beginning of the trivialization of Topology. First came the Jones polynomials, that historically arose in another `fancy field': C* algebras, but this turned out to be a red herring, and thanks to the work of Lou Kauffman and others, they could have, and should have, arisen directly in elementary graph theory, being a very special case of the Tutte (chromatic) polynomials.
Now, even more dramatically, we have the Seiberg-Witten invariants. It was Nati Seiberg who made the initial breakthrough, in the context of `exactly-solvable-models' in quantum field theory. Ed Witten, who speaks fluently both the language of topology and that of physics, realized the revolutionary significance.
Myself, I don't know either languages. Nevertheless, I am almost sure that, just like the Jones polynomials, the Seiberg-Witten invariant would be reducible to combinatorics and/or high-school algebra. Similar developments will occur all over mathematics, and will make many heavy tomes written by arrogant `high-brow' mathematicians, all obsolete, and unnecessary.
In this process, computer algebra, that great implementer of high-school algebra, will play a central role.
The king (abstract math) is dead. Long live the King (Concrete Mathematics).
Added Feb. 4, 2002: read interesting feedback by Lucas Wiman
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