most recent 30 from http://mathoverflow.net 2013-06-19T00:54:59Z http://mathoverflow.net/feeds/question/59827 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59827/current-validity-for-erlangen-in-some-or-other-form Charles Matthews 2011-03-28T12:46:48Z 2011-03-28T12:46:48Z

<p>I’m referring, naturally, to Felix Klein and his Erlanger Programm (http://en.wikipedia.org/wiki/Erlangen_program). Colloquially put, “groups rule geometry OK”. To be more sophisticated about it, geometric languages are related by a type of Galois connection to the subgroup structure of a “master” group, which for Klein’s purposes could normally be thought of as the complex projective linear group. Euclidean geometry, for example, has a smaller group but a richer geometric language; complex projective geometry has fewer theorems but they are more powerful, for example in the way that parallel lines need not be exceptions, and conics intersect also in a way that can be described much more simply.</p> <p>Klein’s work, once a revelation, became later a Procrustean bed: a not uncommon fate for big ideas in mathematics. I was thinking about this just now from the aspect of pedagogy in geometry: Klein was interested in this aspect, I was brought up on “transformation geometry” myself, and read Coxeter’s “Introduction to Geometry” at about 17 which is a serious implementation. I don’t regret the transformation geometry. But I would feel that a doctrinaire view that “Klein was right” would be rather jarring.</p> <p>Which of the following ring true, then?</p> <p>(a) Lifting up from the homogeneous space to the group is usually progress in geometry.</p> <p>(b) Different ways to represent a homogeneous space as coset space represent opportunities.</p> <p>(c) There is a “theory of the classical groups” and it still informs our view of geometry.</p> <p>(d) Despite Euclid, the axiomatic method is really no more significant now in geometry than in any other subfield of mathematics.</p> <p>(e) G-structures on manifolds are a more inclusive way to relate geometry to Lie groups.</p> <p>(f) In fact structures on manifolds in general, and the pseudogroup concept, are a better explanation of geometry.</p> <p>(g) The privilege attached to the projective group is now misleading, since the general linear group is the correct starting point. The privilege attached to invariance/invariants is also a partial view.</p> <p>(h) Category theory is always going to unify more than group theory does, and we are still working through the consequences.</p> <p>(i) In particular topology has taught us about much more than the rubber sheet, and algebraic geometry that we didn’t know what a point was.</p> <p>More attitudes could be added here. But is the following the correct one: the criterion for a successful programmatic approach in mathematics is more than just being a philosophical “moment”? And, in that light, Klein actually did fall short?</p>