# What, precisely, does Klein's Erlangen Program state?

People write that the Erlangen Program is a "program" (like the "Langlands Program"), i.e. a series of related conjectures, which in this case were all solved. There are various intuitive accounts, describing that this program is about relating algebra and geometry, about relating transformation groups of spaces (Lie groups) and different geometries, invariants, etc.

What I haven't been able to find is a precise statement in modern language (e.g. not that of his original paper) of what Klein's conjectures were. What precisely were his conjectures, or equivalently, what results constitute their resolution? Or was there never truly a precise statement?

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I kind of tried this with the question mathoverflow.net/questions/59827/… . The observation underlying is surely that there is a "Galois connection" between the sets of theorems in geometries, and the (sub)groups determining the geometry. Mathematics of the 19th century was discursive, not axiomatic, so you don't always get the precision. – Charles Matthews Jan 15 '13 at 19:06
It's not entirely clear what you're asking. There's an English translation of his program at arXiv:0807.3161. – Chris Godsil Jan 15 '13 at 19:06
What, precisely, does the Langlands Program over number fields state? :) – user30379 Jan 15 '13 at 19:36
[What?! A user with nickname Berlusconi on a research level math site! It must surely be ironic... (Apologies if that happens to be the user's real name) ] – Qfwfq Jan 15 '13 at 22:53
I think this is tangential to the original question. – David Corwin Jan 16 '13 at 2:12

For the historical part of the question, what about reading the (professional) historians of mathematics, for instance, as a point of departure : Jeremy Gray, Felix Klein's Erlangen programme, in Landmark Writings in Western Mathematics, ed. I. Grattan-Guinness, Elsevier, p. 544-552, 2005, which explains the circumstances of the paper and its main contents, and gives a bibliography.
Indeed, the programme was not a series of conjectures ! It states a view of geometry in which a geometry was associated to a group of transformations (not uniquely, of course), and (a part often forgotten) this also should provide explicit invariants. It was important because 1) geometry was still currently understood as "geometry" and not "geometries" at the time, 2) it allows to classify them and show the analogies/identity between different geometries (some quite bizarre to-day). As said by others above, from our point of view, a lot was not included, of course, for instance, Riemannian geometry was explicitely out of it (it was of the achievements of the Elie Cartan generation, and specially Elie Cartan himself, to integrate Riemann to this picture). The influence of the programme has been also studied extensively. Best, C. Goldstein

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Note: I am not sure if you have seen this paper, but I post it here as an answer and I quote a couple of paragraphs, they are not my words! There is also this book.

The first section of Klein's Erlangen Program (E.P.) announced its main theme as follows (E.P., 67): "Given a manifold and a transformation group acting on it, to investigate those properties of figures [Gebilde] on that manifold which are invariant under [all] transformations of that group." In today's language, Klein proposed studying the concept of a homogeneous manifold: a structure $[M,G]$ consisting of a manifold $M$ and a group $G$ acting transitively on $M$. This contrasts sharply with Riemann's concept of a structure $[M;d]$ consisting of a manifold on which a metric $d(p,q)$ is defined by a local distance differential $ds^2 = \sum g_{ij}dx_idx_j$.

Two paragraphs later, Klein restated his proposal in a single terse sentence: "Given a manifold, and a transformation group acting on it, to study its invariants.'' Thus Klein was also proposing to apply to geometry the concept of an 'invariant' that Clebsch, Jordan, and their predecessors had previously applied to algebra, and there only to the full linear group.

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This answers seems to capture the rough understanding I had of the programme. I recall hearing an even more simplistic formulation to the effect that 'a geometry is determined by its symmetries.' If some idea of this sort really was articulated in the Erlangen programme, then it seems to have rested on a very restricted (one might even say old-fashioned) notion of geometry. From the Riemannian viewpoint, most geometries have no symmetries at all. That is, from a modern perspective, such a statement is clearly wrong. – Minhyong Kim Jan 16 '13 at 22:21
I think one can place Riemannian geometry into something like a Kleinian perspective if one replaces "M" by the pair "(M,g)" (or maybe a triple consisting of M, g, and a coordinate atlas), and lets G be the group $Diff(M)$ of diffeomorphisms of M (which acts on the pair (M,g) by pushforward). The point is then to study those features of a Riemannian manifold that are invariant with respect to change of coordinates. This is to the original Kleinian philosophy as the general principle of relativity is to the special principle of relativity. – Terry Tao Jan 28 '13 at 17:18
... the point being that modern geometry still relies very much on symmetries; the difference being that the symmetries we use tend to map one geometric space to a slightly different geometric space (e.g. a diffeomorphism $\phi: M \to M$ maps $(M,g)$ to $(M,\phi_* g)$) rather than necessarily mapping a geometric space to itself. That is, we use a symmetry groupoid (or category) rather than a symmetry group. – Terry Tao Jan 28 '13 at 17:22
... indeed, one could make the argument that category theory is the heir to the Erlangen program. – Terry Tao Jan 28 '13 at 17:24
@Terry Tao: I don't know if Klein would have agreed, but I'm sympathetic to the view that category theory inherits the Erlangen programme. If one allows the categorical generalization of symmetry, a version of the 'recover space from symmetries' view that is almost tautologically true is the Yoneda Lemma. In a different direction, there are a number of interesting theorems in algebraic and arithmetic geometry that allow the reconstruction of a space from an associated category. – Minhyong Kim Jan 29 '13 at 0:05

(Rewritten in response to David Corfield's comment below.)

A somewhat more modern take on the Erlanger Programm was given in Tarski's 1966 talk What Are Logical Notions? (published 1986), as described in the Wikipedia article on Tarski, which proposes a distinction between what is logical and what is non-logical. The idea that as one loosens the theory (say from Euclidean geometry to affine geometry to topology to...), the relevant automorphism group becomes larger and larger, so that maximal automorphism groups (symmetric groups) correspond to theories of maximal looseness, where one is left with purely logical notions.

However, it should be said that Tarski's idea was clearly anticipated by F.I. Mautner, writing in 1946; see here. For some commentary on this, see this post by David Corfield at the $n$-Category Café.

As shameless self-promotion, I'll mention that James Dolan and I dabbled a little in this as well; some results were described at the $n$-Category Café, here and here. There we give describe a Galois correspondence between subgroups of symmetric groups and complete theories, in categorical terms.

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Todd, as I discussed at the nCafe - golem.ph.utexas.edu/category/2012/01/… - the idea you attribute to Tarski of distinguishing the logical from the non-logical is there in Mautner. He was extending Weyl's Kleinian treatment of algebraic groups to symmetric groups. By the way, any chance that you and James might pursue that very interesting work? – David Corfield Jan 16 '13 at 10:09
You are right, David; thank you. I'll think about how to rewrite this answer to reflect that. I do plan on renewing discussions with James soon, and will bring this up. – Todd Trimble Jan 16 '13 at 12:41
Thanks for your kind words, Berlusconi. Yes, there's lots of food for thought at the Caf&eacute;. – Todd Trimble Jan 16 '13 at 19:30

A small complement:

It is good to have a look at what the contemporaries thought about Klein's program and how it articulated with the big deal of the time: classical invariant theory. In particular Franz Meyer's review mentioned in my answer to MO96140 is a useful reference in this regard (Klein's program is mentioned on pages 19, 42 and 45).

At the opposite extreme of the time axis, one can note that Klein's program continues to be a source of inspiration for mathematicians, see e.g. this article posted today on arXiv by Freed and Hopkins.

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