From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5385 Path: news.gmane.org!not-for-mail From: "Larry Harper" Newsgroups: gmane.science.mathematics.categories Subject: RE: A well kept secret? Date: Sat, 19 Dec 2009 17:00:41 -0800 Message-ID: Reply-To: "Larry Harper" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1261320761 26158 80.91.229.12 (20 Dec 2009 14:52:41 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 20 Dec 2009 14:52:41 +0000 (UTC) To: Original-X-From: categories@mta.ca Sun Dec 20 15:52:34 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1NMN95-0002mj-RD for gsmc-categories@m.gmane.org; Sun, 20 Dec 2009 15:52:32 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NMMZ0-0001FF-Bt for categories-list@mta.ca; Sun, 20 Dec 2009 10:15:14 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5385 Archived-At: 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]