From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4922 Path: news.gmane.org!not-for-mail From: "Ronnie Brown" Newsgroups: gmane.science.mathematics.categories Subject: Re: Famous unsolved problems in ordinary category theory Date: Wed, 3 Jun 2009 21:30:17 +0100 Message-ID: Reply-To: "Ronnie Brown" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain;format=flowed;charset="iso-8859-1";reply-type=original Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1244162782 6797 80.91.229.12 (5 Jun 2009 00:46:22 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 5 Jun 2009 00:46:22 +0000 (UTC) To: "Hasse Riemann" , Original-X-From: categories@mta.ca Fri Jun 05 02:46:19 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 1MCNZa-0006yL-Qd for gsmc-categories@m.gmane.org; Fri, 05 Jun 2009 02:46:18 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MCMzA-0006S7-Tt for categories-list@mta.ca; Thu, 04 Jun 2009 21:08:40 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4922 Archived-At: In reply to Hasse Riemann's question (see below): I remember being asked this kind of question at a Topology conference in Baku in 1987. It is worth discussing the background to this, as someone who has never gone for a `famous problem', but found myself trying to develop some mathematics to express some basic intuitions. Saul Ulam remarked to me in 1964 at my first international conference (Syracuse, Sicily) that a young person may feel the most ambitious thing to do is to tackle a famous problem; but this may distract that person from developing the mathematics most appropriate to them. It was interesting that this remark came from someone as good as Ulam! G.-C. Rota writes in `Indiscrete thoughts' (1997): What can you prove with exterior algebra that you cannot prove without it?" Whenever you hear this question raised about some new piece of mathematics, be assured that you are likely to be in the presence of something important. In my time, I have heard it repeated for random variables, Laurent Schwartz' theory of distributions, ideles and Grothendieck's schemes, to mention only a few. A proper retort might be: "You are right. There is nothing in yesterday's mathematics that could not also be proved without it. Exterior algebra is not meant to prove old facts, it is meant to disclose a new world. Disclosing new worlds is as worthwhile a mathematical enterprise as proving old conjectures. " It is like the old military question: do you make a frontal attack; or find a way of rendering the obstacle obsolete? I was early seduced (see my first two papers) by the idea of looking for questions satisfying 3 criteria: 1) no-one had previously asked it; 2) the question was technically easy to answer; 3) the answer was important. Usually it has been 2) which failed! Of course you do not find such questions where everyone is looking! It could be interesting to investigate how such questions arise, perhaps by pushing a point of view as far as it will go, or seeing a new analogy. "If at first, the idea is not absurd, then there is no hope for it." Albert Einstein It could be interesting to investigate historically: if (let us suppose) category theory has advanced without a fund of famous open problems, how then has it advanced? One aim of mathematics is understanding, making difficult things easy, seeing why something is true. Thus improved exposition is an important part of the progress of mathematics (even if this is ignored by Research Assessment Exercises). R. Bott said to me (1958) that Grothendieck was prepared to work very hard to make something tautological. By contrast, a famous algebraic topologist replied to a question of mine about his graduate text by asking: `Is the function not continuous?' He never gave me a proof! And I never found it! (Actually the function was not well defined, but that I could fix!) Grothendieck wrote to me in 1982: `The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps ...'. See also http://www.bangor.ac.uk/~mas010/Grothendieck-speculation.html The point I am trying to make is that the question on `open problems' raises issues on the nature of, on professionalism in, and so on the methodology of, mathematics. It is a good question to start with. Hope that helps. Ronnie Brown ----- Original Message ----- From: "Hasse Riemann" To: "Category mailing list" Sent: Tuesday, June 02, 2009 5:31 PM Subject: categories: Famous unsolved problems in ordinary category theory Hello categorists I don't know what to make of the silence to my question. This is the easiest question i have. I can't believe it is so difficult. It is not like i am asking you to solve the problems. There must be some important open problems in ordinary category theory. There are plenty of them in the theory of algebras and in representation theory, so there should be more of them in category theory. Especially if you broaden the boundaries a bit of what ordinary category theory is. Take for instance: model categories, categorical logic, categorical quantization, topos theory-locales-sheaves. But i had originally pure category theory in mind. Best regards Rafael Borowiecki [For admin and other information see: http://www.mta.ca/~cat-dist/ ] -------------------------------------------------------------------------------- No virus found in this incoming message. Checked by AVG - www.avg.com Version: 8.5.339 / Virus Database: 270.12.51/2151 - Release Date: 06/02/09 17:53:00 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]