Subject: Re: 2 on topology/geometry Date: Fri, 10 Aug 2001 09:48:21 -0400 (EDT) From: Jack Morava > > Subject: Re: NewYorker comment > Date: Wed, 8 Aug 2001 10:19:22 -0400 (EDT) > From: Tom Goodwillie > > Topology involves comparatively extreme distortion of objects. > Maybe some people (like Brayton) take this to mean that our subject > matter is soft, while others take it to mean either that we are > boisterous/hardheaded or that the rules we live by are > indulgent/libertarian. > > Tom Goodwillie > This reminds me somehow of Cliff Taubes' observation that his kids make their beds, but they make them up in the topological, rather than the smooth, sense... ______________________________________ Subject: topology in context Date: Fri, 10 Aug 2001 10:26:09 -0400 (EDT) From: hrm@math.mit.edu something for the group.... Jie and Tom have reminded me of a discussion I had with Jack Morava a little while ago about an ecological definition of topology: that is, a description of the subject in terms of the niche it fills in the ecology of mathematics. Our conclusion (or at least where we stopped) was that it is a test-bed, a place where a certain (but fairly wide) range of ideas can be tried out in their most unstructured form. It serves as a nursery. Once ideas have reached a certain degree of maturity, other fields adopt them (often without recognizing the extensive development they had already undergone, like stars forming behind a veil of dust). This has been the role of topology since the beginning: "point-set topology" seeks explicitly to study the concept of continuity in its most general and unstructured form. A little later a variety of primitive geometric forms - manifolds, simplicial complexes - emerged from the confusion of other disciplines. Homological algebra is everyone's favorite example of a general paradigm which saw first light in topology. I would argue that the concept of homotopy theory is now shedding its birth cowl and being observed and used in other richer mathematical disciplines. This aspect of topology accounts for why we seem always to be rediscovering just what it is we are dealing with. The evolution of our concept of a spectrum is a good case in point. The experimental nature of our subject also helps to explain why we don't have a unitary founding context or document, akin to SGA. It also explains something about our inclusive and (in my experience) supportive community. Haynes Miller