RE: A well kept secret?

["Larry Harper" <harper@math.ucr.edu>]

Dear All,



As one of MacLane's working mathematicians who follows the catlist, I would
like to add some thoughts about perceptions of category theory (CT). I
earned a Bachelor's in Physics at Berkeley in 1960 and went to grad school
in Mathematics at the University of Oregon the following fall. In Frank
Anderson's graduate algebra course I was first exposed to CT and hated it.
My background and ability in algebra were marginal anyway and to have my
first definition of tensor product be in terms of commuting diagrams was
disastrous. Fortunately, I got a summer job at the Jet Propulsion Lab and
one of my coworkers, Gus Solomon, gave me the classical constructive
definition of tensor products. Sammy Eilenberg came by Eugene and gave a
lecture on CT which did nothing to change my opinion of it. When I heard of
Serge Langs's characterization of CT as "abstract nonsense" it reinforced
what I already thought (See however,

                           http://en.wikipedia.org/wiki/Abstract_nonsense

which does not mention Serge Lang in the body of the article).



My fascination with, and love of, CT was ignited in 1966 when I was a
postdoc with Gian-Carlo Rota at the Rockefeller University. Ron Graham and I
were collaborating on a conjecture of Rota; that the lattice of partitions
of an n-set has the same property that Erwin Sperner had demonstrated for
the lattice of subsets of an n-set (the largest antichain is the largest
rank). We had some partial results on Rota's conjecture and in the course of
writing them up I realized that they implicitly involved a notion of
morphism for the Ford-Fulkerson maxflow problem. I thought this was a
promising insight and incorporated it with the other material. I gave the
finished paper to Ron for approval but when it came back to me it had been
rewritten and all mention of flowmorphisms eliminated. I took this as a
challenge to show that flowmorphisms could lead to further insight into
Sperner problems. The result, which I called The Product Theorem, was
natural conditions under which the product of Sperner posets must also be
Sperner. The key was to realize that the concept of normalized flow
introduced in Graham-Harper (which is stronger than the Sperner property) is
equivalent to a flowmorphism from the given poset to a chain (total order).
If two posets, P,Q, have normalized flows then product, being a bifunctor,
will induce a flowmorphism from their product to the product of their
chains. All I had to do then was to find natural conditions under which the
product of (weighted) chains has a normalized flow. The Product Theorem
generalized known theorems of  Sperner, deBruijn et al & Erdos. and has
since been applied to prove at least 3 new conjectures.



Having such a success with flowmorphisms motivated me to dig more deeply.  I
showed that flowmorphisms have pushouts and was asking about pullbacks
(though I did not use those terms because I did not know them) when (about
1971) my office mate at JPL, Dennis Johnson, introduced me to Saunders
MacLane's classic, Categories for the Working Mathematician. This was, of
course, a revelation and changed my mathematical universe.



I joined the faculty of the University of California at Riverside in the
fall of 1970. In alternate years I taught a 2-quarter graduate course on
combinatorics. Over the next 36 years it evolved into two independent
courses having a common thesis. One was on maximum flows in networks and
Sperner problems, the other on minimum paths in networks and combinatorial
isoperimetric problems. I believe it is no accident that maximum flow and
minimum path (aka dynamic programming) problems are central to algorithmic
analysis and that they both have nice notions of morphism. The common thesis
of the two courses is that morphisms can be effective in solving hard
problems. In 2004 the notes for one course were published under the title
Global Methods for Combinatorial Isoperimetric Problems. If I live long
enough its companion volume on Sperner problems will appear. It will show
how several steps in the eventual resolution of the Rota Conjecture were
illuminated by  flowmorphisms.



It has been a personal goal, since the early 1970s, to demonstrate the
existence and usefulness of morphisms for combinatorial problems. This often
comes down to questions of

            1) How to use symmetry to systematically simplify the problem?

            2) How to pass to a continuous limit?

I like to call this endeavor the relativity theory of combinatorics. Albert
Einstein asked "What are the symmetries of the universe and what do they
tell us about it?" To show the depth and subtlety of such questions,
consider that two of the leading mathematicians of his age, Henri Poincare
and Hendrick Lorentz, studied Lorentz transformations five years before
Einstein. However they both missed the epoch-making relation E = mc^2 that
is easily deduced from Lorentz transformations. In studying a problem
through its morphisms we need all the help we can get. CT is invaluable as
the road map to morphism country!



Regards,



Larry Harper

Professor Emeritus of Mathematics

University of California, Riverside



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