The Registar,
The Institute of Combinatorics and its Applications.

Dear Ralph,

The news item about the formation of CADCOM to promote 
combinatorial mathematics, which appeared in Volume 10 
of the Bulletin of the ICA, prompted me to put my thoughts 
on paper. The suggestions expressed in that item are in 
line with the ICA's objectives, which I support. Nevertheless,
I would put the emphasis a bit differently. My thesis is 
that combinatorics has a central role in mathematics which 
is beyond question; it is not combinatorics we should be 
promoting, but mathematics (and the tools which come 
naturally into our hands for this job will be combinatorial 
ones).

Let me offer some evidence for the first claim. The most 
exciting development in mathematics in the late 80s grew 
from the work of Vaughan Jones, for which he received a 
Fields Medal in 1990. His research on traces of Von Neumann 
algebras came together with representations of the Artin
braid group to yield a new invariant of knots, with 
ramifications in mathematical physics and elsewhere. Few 
areas of mathematics were unaffected. (See the citation by 
Joan Birman in the ICM proceedings [1] for a map of this 
territory.) Combinatorics was involved in this activity. 
There was the work of Francois Jaeger [3], who derived a 
spin model, and hence an evaluation of the Kauffman polynomial, 
from the strongly regular graph associated with the Higman-Sims 
simple group; and that of Dominic Welsh and his collaborators 
(described in his book [4]) on the computational complexity of 
the new knot invariants. Two things are apparent. First, this
work was done by combinatorial mathematicians (rather than by 
topologists or others "slumming"), and tied in naturally with 
the body of their work. Second, the background discrete 
mathematics already existed - in one case, the algebraic theory 
of association schemes, developed by Bose, Delsarte, and others, 
and the Higman-Sims graph itself, constructed from Witt's 3-design 
on 22 points; in the other, the complexity theory of counting,
especially #P-completeness, and the Tutte polynomial of a matroid.

Another example concerns the ubiquity of the Coxeter-Dynkin 
diagrams A_n, D_n, E_6, E_7, E_8. Arnol'd (see [2]) proposed 
finding an explanation of their ubiquity as a modern equivalent 
of a Hilbert problem, to guide the development of mathematics. 
He noted their occurrence in areas such as Lie algebras (the 
simple Lie algebras over C), Euclidean geometry (root systems), 
group theory (Coxeter groups), representation theory (algebras of
finite representation type), and singularity theory (singularities 
with definite intersection form), as well as their connection 
with the regular polyhedra. To this list could be added 
mathematical physics (instantons) and combinatorics (graphs 
with least eigenvalue -2). Indeed, graph theory provides the 
most striking specification of the diagrams: they are just the
connected graphs with all eigenvalues less than 2.

Many more homely illustrations could be offered. In the last 
few years, two of my "applied" colleagues (one working on 
non-linear dynamics, the other on Lie superalgebras) 
independently came up with the Catalan numbers in their research.

Nor is this just a modern phenomenon. Euler is our hero, 
of course, but combinatorics has been an integral part of 
the work of many others (Pascal, Leibniz, Cayley, Mobius, ...).

My point is that there is no natural boundary to "combinatorics" 
or "discrete mathematics"; its roots and influences spread 
throughout mathematics (and this is a claim with solid 
foundations). On the other hand, I think that mathematics 
as a whole has a boundary, albeit a semi-permeable membrane: 
it imports problems and exports techniques, but not the reverse 
to any great extent. I also believe that boundaries tend to
isolate a group, nation, or discipline. We are better served 
by inner strength and confidence than by fanaticism. We should 
always be ready to get up and speak as combinatorialists (though 
I would rather say as combinatorial mathematicians, with the 
emphasis on the second word), but we should speak for all of 
mathematics, our natural community.

In the present climate, with its stress on short-to-medium 
term projects, mathematics as a whole is especially vulnerable, 
since its resources so often find their application long 
after they are developed and in unexpected ways. (Einstein's 
use of Riemannian geometry is a classic example.) Discrete
mathematics may be one of the least threatened areas, because 
of its close ties with computer science. More reason, then, 
for us to speak up for mathematics!

Yours sincerely,
Peter J. Cameron

		References

1. J. S. Birman, The work of Vaughan F. R. Jones, pp. 9-18 in
Proceedings of the International Congress of Mathematicians, 
Kyoto 1990, Math. Soc. Japan/Springer-Verlag, Tokyo 1991.

2. F. E. Browder, Problems of present day mathematics, pp. 35-79 in
Mathematical Developments arising from Hilbert Problems, 
Proc. Symp. Pure Math. 28, Amer. Math. Soc., Providence, 1976.

3. F. Jaeger, Strongly regular graphs and spin models for 
the Kauffman polynomial, Geom. Ded. 44 (1992), 23-52.

4. D. J. A. Welsh, Complexity: Knots, Colourings and Counting,
London Math. Soc. Lecture Notes 186, Cambridge Univ. Press, 
Cambridge, 1993.