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March 13, 2009

Geometric Help Wanted

Posted by David Corfield

I’m giving a talk to some philosophers and would like to get across in a non-technical way the idea that geometry has gone, and continues to go, through profound changes. I wanted to touch on the Erlangen Program, and perhaps illustrate its idea with the following example:

Elementary plane geometry and the projective investigation of a quadric surface with reference to one of its points are one and the same.

Klein’s explanation is as follows:

If a quadric surface be brought into correspondence with a plane by stereographic projection, the surface will have one fundamental point, - the centre of projection. In the plane there are two, - the projections of the generators passing through the centre of projection. It then follows directly: the linear transformations of the plane which leave the two fundamental points unaltered are converted by the representation (Abbildung) into linear transformations of the quadric itself, but only into those which leave the centre of projection unaltered. By linear transformations of the surface into itself are here meant the changes undergone by the surface when linear space-transformations are performed which transform the surface into itself. According to this, the projective investigation of a plane with reference to two of its points is identical with the projective investigation of a quadric surface with reference to one of its points. Now, if imaginary elements are also taken into account, the former is nothing else but the investigation of the plane from the point of view of elementary geometry. For the principal group of plane transformations comprises precisely those linear transformations which leave two points (the circular points at infinity) unchanged.

David Rowe explains this as follows:

Consider the stereographic projection f:S 2()f: S \to \mathbb{P}^2 (\mathbb{C}) from a fixed point PP on a 2nd-degree surface SS in 3()\mathbb{P}^3 (\mathbb{C}). This mapping is one-to-one except for two points pp and qq in the range which are the image of two generators of SS that pass through PP. Now the group of linear transformations of 2()\mathbb{P}^2 (\mathbb{C}) that leave pp and qq fixed, when pulled back by f 1f^{-1}, yields the group of linear transformations of SS that fix PP, i.e., the restrictions to SS of those linear transformations of 3()\mathbb{P}^3 (\mathbb{C}) that leave both SS and PP invariant. Furthermore, since any two points of 2()\mathbb{P}^2 (\mathbb{C}) are projectively equivalent, one can take pp and qq to be the circular points at infinity. But, by Cayley’s principle, the linear transformations of 2()\mathbb{P}^2 (\mathbb{C}) that leave these points fixed are precisely the transformations of Euclidean plane geometry. (The Early Geometric Works of Sophus Lie and Felix Klein, 261)

He goes on to complain that the “vast majority of writers” derive their knowledge of the Program from secondary sources, which

…trivialize its content, in particular by restricting their discussion to examples in which familiar transformation groups act on real spaces. This interpretation necessarily overlooks all the deeper geometrical results–which are the only ones Klein bothered to present–since these all require that the base space be a complex manifold. (p. 264)

So what’s happening here? There’s an isomorphism between the group of Euclidean transformations and the subgroup of PGL(3,)PGL(3, \mathbb{C}) which fixes the quadric and a point on it? Any more illumination appreciated.

Posted at March 13, 2009 9:39 AM UTC

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Re: Geometric Help Wanted

The automorphism group of the quadric is SO(4) having Dynkin diagram D_2 which are just 2 points, that is the same as A_1xA_1. Geometrically this says that the quadric is isomorphic to P^1 x P^1 (the two families of lines on the quadric). If you fix a point on the quadric you also fix the two lines on it through that point, so the complement is A^1 xA^1 = A^2 (here the A^k is k-dml affine space) the affine plane.

Posted by: lievenlb on March 13, 2009 7:45 PM | Permalink | Reply to this

Re: Geometric Help Wanted

David asks:

So what’s happening here? There’s an isomorphism between the group of Euclidean transformations and the subgroup of PGL(3,)PGL(3,\mathbb{C}) which fixes the quadric and a point on it?

I’m wondering if you’ve ever thought about some similar but easier puzzles. If you haven’t, it might be good to start with this:

PGL(3,)PGL(3,\mathbb{R}) acts on P 2\mathbb{R}\mathrm{P}^2. Why is the subgroup that fixes a point of P 2\mathbb{R}\mathrm{P}^2 isomorphic to the group of affine transformations of 2\mathbb{R}^2?

If that’s too easy, please forgive me. From this you can boost your way up to:

PGL(3,)PGL(3,\mathbb{C}) acts on P 2\mathbb{C}\mathrm{P}^2. Why is the subgroup that fixes a point of P 2\mathbb{C}\mathrm{P}^2 isomorphic to the group of affine transformations of 2\mathbb{C}^2?

These examples ‘trivialize the content’ of the Erlangen program, but they’re necessary warmups before the fancier one you’re asking about.

Posted by: John Baez on March 13, 2009 10:08 PM | Permalink | Reply to this

Re: Geometric Help Wanted

I remember discussing your first example in an early Klein 2-geometry post. Just look at matrices of the form

(A 11 A 12 v 1 A 21 A 22 v 2 0 0 1). \begin{pmatrix} A_{11} & A_{12} & v_1 \\ A_{21} & A_{22} & v_2 \\ 0 & 0 & 1 \end{pmatrix}.

Posted by: David Corfield on March 15, 2009 9:47 PM | Permalink | Reply to this

Re: Geometric Help Wanted

Right. So, geometry wizards are entitled to mutter this magic spell: The projective plane minus a point at infinity is the affine plane — and projective transformations that preserve this point are precisely the affine transformations.

(Your point at infinity is the one with homogeneous coordinates x 3=0x_3 = 0, which is why most of the third row of your matrix must vanish — all but the entry that’s also in the third column.)

This sort of matrix business works in higher dimensions, too, which is why we’re entitled to mutter this: Projective nn-space minus a hyperplane at infinity is affine nn-space — and projective transformations that preserve this point are precisely the affine transformations.

But that’s a digression. The main point is that we understand how affine transformations of the affine plane form a subgroup of PGL(3)PGL(3).

Now, I don’t understand what Klein wrote, but as you suggest, David Rowe seems to be identifying Euclidean transformations of the Euclidean plane as a still smaller subgroup of PGL(3)PGL(3). First of all, we’ve seen that’s a sensible project. And second of all, he writes:

Now the group of linear transformations of 2(\mathbb{P}^2(\mathbb{C} that leave pp and qq fixed, when pulled back by f 1f^{-1}, yields the group of linear transformations of SS that fix PP, i.e., the restrictions to SS of those linear transformations of 3()\mathbb{P}^3(\mathbb{C}) that leave both SS and PP invariant. Furthermore, since any two points of 2()\mathbb{P}^2(\mathbb{C}) are projectively equivalent, one can take pp and qq to be the circular points at infinity. But, by Cayley’s principle, the linear transformations of 2()\mathbb{P}^2(\mathbb{C}) that leave these points fixed are precisely the transformations of Euclidean plane geometry.

This is not what I’d call prose of stellar lucidity, perhaps because my education is deficient when it comes to certain sorts of buzzwords.

For example, I’m not 100% sure what a ‘linear transformation of 2()\mathbb{P}^2(\mathbb{C})’ is — though I can guess it’s an element of PGL(3,)PGL(3,\mathbb{C}), since this group acts on 2()\mathbb{P}^2(\mathbb{C}) and preserves lines. I’d probably call elements of this group ‘projective linear transformations’.

(Oddly, the Wikipedia article on projective transformations seems to have no links from or to the page on the projective linear group — they seem to come from different centuries. I’m going to put in links, just so the 19th-century geometers can start talking to the 20th-century ones, and vice versa. If it’s true, someone should tell Wikipedia that ‘projective transformations’ are precisely elements of the ‘projective linear group’.)

Worse, I have no idea what a ‘circular point at infinity’ is — it sounds like an outpost of hell where people spend eternity after committing the sin of petitio principii.

So, all I can feebly do is imagine taking your description of the group of affine transformations of the plane as a subgroup of PGL(3)PGL(3), and restricting it still further to obtain Euclidean transformations! But here I will gloss over the distinction between PGL(3,)PGL(3,\mathbb{C}) and PGL(3,)PGL(3,\mathbb{R}), which seems to go against David Rowe’s desire.

Euclidean transformations of the plane look like this:

(A 11 A 12 v 1 A 21 A 22 v 2 0 0 1) \begin{pmatrix} A_{11} & A_{12} & v_1 \\ A_{21} & A_{22} & v_2 \\ 0 & 0 & 1 \end{pmatrix}

where now we demand that

(A 11 A 12 A 21 A 22) \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}

is a rotation/reflection. Or… conjugate to one. Matthew Emerton’s comment seems to suggest that we use diagonal matrices. These matrices, at least, are conjugate to rotations:

(e iθ 0 0 e iθ) \begin{pmatrix} e^{i \theta} & 0 \\ 0 & e^{-i \theta} \end{pmatrix}

Perhaps my feeble flounderings here will help you understand what he wrote.

Posted by: John Baez on March 16, 2009 12:17 AM | Permalink | Reply to this

Re: Geometric Help Wanted

John wrote:

Right. So, geometry wizards are entitled to mutter this magic spell: The projective plane minus a point at infinity is the affine plane — and projective transformations that preserve this point are precisely the affine transformations.

(Your point at infinity is the one with homogeneous coordinates x 3=0, which is why most of the third row of your matrix must vanish — all but the entry that’s also in the third column.)

This sort of matrix business works in higher dimensions, too, which is why we’re entitled to mutter this: Projective n-space minus a hyperplane at infinity is affine n-space — and projective transformations that preserve this point are precisely the affine transformations.

At first the first paragraph looks like it has a typo, i.e., as if you meant to say “projective line at infinity”, not “projective point”. That would better fit with the third paragraph. But actually, the statement is not incorrect, in the sense that the group of projective transformations which preserve a given point (at “infinity” or anywhere else) is isomorphic to the group of affine isomorphisms (automorphisms of the affine plane). In other words, maybe you meant exactly what you said in the first paragraph!

In a sense, the reason for this is trivial: by projective duality of the projective plane, there is a duality which interchanges points and lines. So that the group of projective plane automorphisms which preserve a given point would be isomorphic to the group of automorphisms preserving a given line. (And the group of projective transformations preserving a given line, say “the line at infinity”, is the group of affine automorphisms by restriction to the set-theoretic complement of the line.)

This duality can be made explicit via linear duality w.r.t. a chosen non-degenerate bilinear form. That is, points in the projective plane PP are identified with 1-dimensional subspaces LL of a 3-dimensional space VV, and lines in PP with 2-dimensional subspaces of VV. Given a chosen bilinear form BB on VV, say the usual “dot product” corresponding to the quadratic form x 2+y 2+z 2x^2 + y^2 + z^2 defined in terms of some coordinate system, we can form the assignment LL L \mapsto L^{\perp}, where as usual

L ={vV: wLB(v,w)=0}L^{\perp} = \{v \in V: \forall_{w \in L} B(v, w) = 0\}

This assignment takes lines in VV to planes in VV and vice-versa; or points in the projective plane PP to lines, and vice-versa. The duality also interchanges joins and meets of subspaces; or passing to the projective plane, joins of points and meets of lines there.

In case the ground field is algebraically closed, this duality can be described beautifully in terms of “duality with respect to a chosen conic”. So, let’s work over \mathbb{C}. The locus or zero-set of the quadratic form attached to BB defines a smooth conic CC in the projective plane ℂℙ 2\mathbb{CP}^2. (For example, for the homogeneous quadratic form x 2+y 2+z 2x^2 + y^2 + z^2, the zero set would be the planar locus of

x 2+y 2+1=0;x^2 + y^2 + 1 = 0;

here we have substituted affine coordinates x/zxx/z \to x, y/zyy/z \to y for projective ones x,y,zx, y, z.) Since the quadratic form is, up to an inessential scalar factor, retrievable from CC, it should be possible to describe the projective duality above, ϕ:LL \phi: L \mapsto L^{\perp}, purely in terms of geometry of the conic CC.

The geometry is beautiful; here is how it works. Given a point pp in the projective plane PP, let l 1l_1 and l 2l_2 be the two lines incident to pp and tangent to CC, say tangent at points q 1q_1 and q 2q_2 on CC. (This prescription is well-defined over an algebraically closed field.) Then ϕ(p)\phi(p) is the unique line through q 1,q 2q_1, q_2. (NB: If pp is already on CC, then ϕ(p)\phi(p) is the line through pp tangent to CC.) Similarly, given a line ll in PP, define ϕ(l)\phi(l) to be the point where m 1m_1 and m 2m_2 meet, where m 1m_1 and m 2m_2 are the tangent lines to the conic CC at the points where ll meets CC. Then ϕ\phi is clearly an involution, and corresponds to the linear duality defined on the 3-dimensional space from which the projective plane descends.

I first learned this from a nice book by C. Herbert Clemens, A Scrapbook of Complex Curve Theory.

Posted by: Todd Trimble on March 16, 2009 5:44 PM | Permalink | Reply to this

Re: Geometric Help Wanted

It may be implicitly understood that everyone involved in this discussion understands the connection between the quadric and the plane, but just in case I will spell it out:

If we take the projective plane and blow it up at two points, then the proper transform of the line that joins those two points is a P^1 with self-intersection -1, which by Castelnuovo’s criterion can then be blown down. If one thinks about what surface one arrives at in this way, it is a product of two P^1’s, that is, a quadric (as was remarked above).

The two points we blew up are the two “fundamental points” in the plane. The point in the quadric obtained in the application of Catelnuovo’s criterion is the “fundamental point” in the quadric.

The subgroup of PGL(3,C) that fixes the two fundamental points in the plane is thus isomorphic to the subgroup of the group of automorphisms of the quadric that fixes the fundamental point in the quadric. (Such an automorphism will
automatically fix the two P^1’s passing through the point, because they are determined canonically as the lines in each of the two rulings that pass through the point.)

The subgroup of PGL(3,C) in question can be determined as follows: let’s think of the projective plan as being obtained from the affine plane by adding a projective line at infinity. Take the two points to be points at infinity that lie “at the end” of the x -axis and y-axis respectively. Then we have to compute all projective transformation that preserve this pair of points. They will also preserve the line joining these two points, i.e. the line at infinity, and thus will be affine transformations of the affine plane.

Thus we have to work out which affine transformations preserve the direction of each of the x-axis and y-axis, or else switch the two. Well, any translation does. Thus it remains to work out which affine transformations that preserve the origin also preserve the x and y axes, i.e. which elements of GL_2(C) preserve the two coordinate axes. Such an element just has to be a diagonal matrix, or the product of such a matrix with the matrix (0 1, 1 0) — which switches the two axes.

So the subgroup in question is generated by diagonal matrices in GL_2(C) and the matrix w = (0 1, 1 0), together with all translations. It is a semi-direct product of C* x C* x W with C x C. (Here W denotes the group of order two generated by w.)

To see the connection with Euclidean geometry, one has to replace C by R, but in the right way. I guess that one should look at elements of PGL(3,R) acting on the projective plane over R that preserve not two real points at infinity (which would give the obvious analogue of the above group, namely R* x R* semidirect product with R x R), but rather a pair of conjugate complex points at infinity. This should then give R* x S^1 x W semidirect product R x R, will be the group of rigid affine transformations of the plane, together with dilations. (The R* gives dilations, the S^1 gives rotations, and w is a reflection.) (Note also that both R* and S^1 are real forms of the complex group C*.)

I think that if one takes these two imaginary points and blows them up and then blows down the line joining them, and looks at what real quadric you end up with, it is a sphere. And of course removing one point from a sphere does give a plane (by stereographic projection).

So it seems that if we look at the subgroup of PGL(3,R) that preserve a sphere and one point on it, this should give the usual Euclidean group acting on the plane, when we stereographically project the sphere from the given point onto the plane. Is something that one can just verify directly?

Posted by: Matthew Emerton on March 14, 2009 7:04 AM | Permalink | Reply to this

for philiosphers?

How can this discussion be ‘popularized’ enough so a philosopher can get something out of it?

Posted by: jim stasheff on March 14, 2009 2:12 PM | Permalink | Reply to this

Re: for philosphers?

Jim wrote:

How can this discussion be ‘popularized’ enough so a philosopher can get something out of it?

Or even a mathematician!

I suspect there’s something fairly simple going on here. But so far everyone who knows what’s going on is using terminology that I don’t understand, like “fundamental point”, “circular point at infinity”, or “Castelnuovo’s criterion”.

Posted by: John Baez on March 16, 2009 1:02 AM | Permalink | Reply to this

Re: for philosphers?

Some elaboration:

Let’s begin by considering the usual stereographic projection from the sphere, i.e. the set of solutions in R^3 to the equation

x^2 + y^2 + z^2 = 1 ,

onto the plane, R^2.

More precisely, we project from the north pole of the sphere — the point (0,0,1) — to the tangent plane to the south pole of the sphere — the plane z = -1.

This map is well-defined everywhere on the sphere except at the north pole, and gives a bijection between the sphere minus the north pole, and the plane z = -1.

Now lets complexify everything. So we consider the complex solutions to

x^2 + y^2 + z^2 = 1

and project from the north pole — by which I still mean (0,0,1) — to the plane z = -1 (which is now a copy of C^2).

This map is of course still not well-defined at the north pole, but there are additional points at which it is not well-defined.

Namely, consider the plane z = 1, the tangent plane to the north pole. It is parallel to the plane z = -1, and so if we project from the north pole along any line lying in the plane z = 1, we will never intersect the plane z = -1.

This wasn’t worth mentioning in the real picture, because the tangent plane to a point on the sphere meets the sphere just at the point itself. But over C, the intersection of z = - 1 with the sphere is not just the north pole, but the union oftwo lines, l_1 and l_2, which intersect at the north pole. (This is an instance of intersections which are “missing” in the real picture, but which become apparent in the complex picture.)

So stereographic projection is in fact not defined along l_1 or l_2.

Now let us stay in the complex picture, but projectivize as well (so we add all appropriate points at infinity to all the geometric objects in play).

We now see that the planes z = 1 and z = -1 share a common line at infinity (since they are parallel), and the lines l_1 and l_2 meet this line at infinity in points P_1 and P_2 respectively. Thus we are saved: any point on l_i (other than the north pole itself) now has a well-defined stereographic projection: it projects to the point P_i, a point at infinity on the plane z = -1.

Thus in the projective picture, stereographic projection is again well-defined on the complement of the north pole, as a map to the (projective) plane z = -1. But note: it is no longer one-to-one: all the points on l_i map to the single point P_i.

Nor is it surjective: any line joining the north pole and the line at infinity on z = -1 (which is also the line at infnity on z = 1) lies in the plane z = 1. Since l_1 and l_2 are precisely the intersection of this plane with the sphere, we see that no other points on the sphere, other than those lying on the l_i, map to the line at infinity. Thus the image of stereographic projection contains no points on the line at infinity in z = -1 other than P_1 and P_2.

Now let’s consider the inverse to stereographic projection. This is not well-defined at the points P_1 and P_2 (since their preimages under projection are the entire lines l_1 and l_2). It is well-defined everywhere else, including at all the other points at infinity: it maps all the points on the line at infinity, other than P_1 and P_2, to the north pole.

Maps between surfaces like stereographic projection and its inverse, which are almost, but not quite, bijections, are called birational maps. The points at which such a map are not defined are called “fundamental points”. Thus stereographic projection has one fundamental point — the north pole. Its inverse has two fundamental points — P_1 and P_2.

I hope that this helps explain what Klein is saying at the beginning.

Note also that the fundamental points of one map occur exactly when a whole line is mapped to a point by the inverse map. (Thus: P_1 and P_2 are projected onto by l_1 and l_2, while under the inverse to stereographic projection, the whole line at infinity is taken to the north pole.) This is a general phenomenon in the birational geometry of surfaces, and Castelnuovo’s criterion tells you when a certain line can be “blown down” (as one says) to a point, so that that point becomes the fundamental point of the inverse to the blowing-down map (this inverse is called blowing-up, naturally enough). So the first part of my earlier post was just describing the geometry of the stereographic projection in the more abstract terms of blowing-up and down. One can just ignore it, and replace it by the above concrete description of the situation.

As for circular points at infinity: these are the points at infinity in the projective plane lying on the circle:

x^2 + y^2 = 1 .

Note that they are not real (the real solutions are the usual circle and are compact, so there are no real points at infinity). But there are complex points at infinity, two of them in fact. (If you like, Bezout’s theorem shows that they must exist: the degree two circle must meet the degree one line at infinity in two points. But of course, one can compute them explicitly. In homogeneous coordinates [x:y:z], they are [1:i:0] and [1:-i:0]. Note that they are complex conjugate, as must be, since they are complex solutions to a real equation.)

As I explained in my earlier post, the subgroup of PGL(3,R) fixing this pair of complex conjugate points at infinity is precisely the group of Euclidean transformations of the plane (rotations, reflections, translations, and scalings).

Posted by: Matthew Emerton on March 16, 2009 6:27 AM | Permalink | Reply to this

Re: for philosphers?

P.S. When Klein speaks of the centre of the projection, he means the north pole: i.e. the point that is being stereographically projected from. When he speaks of the generators passing through this point, he means what I called the lines l_1 and l_2 (and hence their projections are the points that I called P_1 and P_2).

(The reason for the terminology generators is that if we think of the quadric as being a one-sheeted hyperboloid rather than a sphere — which we may, since all quadrics are the same over the complex numbers — then it is a “ruled surface”, i.e. it can be “generated” by moving a line through space in a certain way. This is familiar to most of us from having seen “string art” constructions of hyperboloids. In fact, a hyperboloid of one sheet is ruled in two different ways, and so at every point there are two lines passing through the point and lying on the hyperboloid — one for each of the two rulings. For the north pole, these are what I called l_1 and l_2. One way to explain these two rulings is to note that a quadric is isomorphic as a surface to the product of two projective lines.)

Posted by: Matthew Emerton on March 16, 2009 6:36 AM | Permalink | Reply to this

Re: for philosphers?

Correction of a typo: in the paragraph defining l_1 and l_2, they should have been defined as the intersection (computed over the complex numbers) of the sphere with the plane z = 1 (not z = - 1, as I accidentally typed).

Posted by: Matthew Emerton on March 17, 2009 3:21 AM | Permalink | Reply to this

Re: for philosphers?

Thanks for your elaboration, Matthew! This looks like something a philosopher or even a mathematician can understand! I’ll ponder it when I get a chance…

Posted by: John Baez on March 16, 2009 7:31 PM | Permalink | Reply to this

Re: for philiosphers?

I would like to make another remark, because I just read over the original post again, and realized that David Rowe’s analysis could possibly create some confusion.

What we writes regarding the stereographic projection is completely correct. The points that he calls p and q are what I called P_1 and P_2, and the point P is what I called the north pole. My above elaboration pretty much just repeats what Rowe writes in a more long-winded way.

But his statement that “the linear transformations of P^2(C) that leave [the two circular points at infinity] fixed is precisely the transformations of the Euclidean plane geometry” is misleading.

I would naturally interpret this statement to be referring to “the subgroup of PGL(3,C) which leave p and q fixed”. But this group is not the group of motions of the Euclidean plane, but rather, is the *complexification* of that group. It is the subgroup of PGL(3,R) that leaves the points p and q fixed that gives the actual Euclidean group of transformations.

(As I wrote in my original post, the latter group is

S^1 x R* x W x R^2

(where some of the products are direct and some are semi-direct), while its complexification is the group

C* x C* x W x C^2 .

In some sense this latter group is less rich than the Euclidean group, because over the complex numbers, the circle S^1, which is the set of real solutions to

x^2 + y^2 = 1 ,

and the group of non-zero real numbers R*, which is the set of real solutions to

xy = 1 ,

become the same group.)

So I would take issue with Rowe’s assertion that the deeper geometrical interpretations of Klein’s program require the base space to be a complex manifold.

In fact, if we want to interpret Klein’s statement in a way which literally produces the group of Euclidean transformations in an unexpected context, we should say the following:

The subgroup of PGL(3,R) which leaves the sphere minus the north pole invariant is identified with the group of Euclidean transformations, when we identify the sphere minus the north pole with the plane via stereographic projection.

And that is a statement entirely about real manifolds!

It is not that Klein achieved great depth by working over complex numbers. Rather, in thinking about geometric concepts from the point of view of algebraic geometry (or in Klein’s time, what was called projective geometry), it is natural to introduce and make reference to the complex solutions of the equations under consideration, even if ones goal is to ultimately state a result over the real numbers.

I think this is quite a profound principle, which might be of interest to philosophers. It comes up constantly in number theory: to solve (or to get any understanding of) a Diophantine equation, it is normally crucial to understand very well the geometry of the complex solutions first. Perhaps the most impressive example of this principle at work is Mordell’s conjecture (Faltings’ theorem):

If f(x,y) is a polynomial in two variables over Q, and the complex Riemann surface cut out by f(x,y) = 0 has genus > 1, then f(x,y) = 0 has only finitely many pairs (x,y) of rational solutions.

In summary, I think that the actual deep geometrical phenomenon occuring in Klein’s example, and others, is the way that algebraic geometry as a geometric tool let’s you shift back and forth between different ground fields, so that phenomenon that are observed directly when working over one field (say C) can be brought to bear to influence what happens (or, if you want a different way of phrasing it, to make arguments about what happens) over another field (say R, or even Q), even though the original phenomena are no longer directly observable over that field.

Posted by: Matthew Emerton on March 17, 2009 3:53 AM | Permalink | Reply to this

Re: for philiosphers?

This is very helpful, thanks!

I see there’s an Oberwolfach Seminar later this year on ‘The Erlangen Program, Myths and Realities: Geometry and Group Theory, 1870-1920’, which has Rowe as one of the organisers.

Another organiser is Igor Dolgachev, who has lots of notes at his site, including 395 pages on classical algebraic geometry.

What do you think of the following passage from Rowe:

The foregoing analysis of the early geometric works of Klein and Lie is far from exhaustive. Nevertheless, it should suffice to make clear that during this relatively brief period they developed a wealth of interesting ideas, techniques, and results that are all but forgotten today. (p. 264)

I can’t give you the details of Rowe’s account, but we seem to have one of those charges that a chunk of earlier ways of thinking has been lost to us. Of course, we shouldn’t expect all mathematicians to understand this stuff, but does your contemporary algebraic geometer have any great trouble understanding Klein, Cayley, Plücker, etc. once they cotton on to the terminology? Is there anything going on back then for which we don’t have good ‘cognitive control’?

Posted by: David Corfield on March 17, 2009 12:45 PM | Permalink | Reply to this

Re: for philiosphers?

Dear David,

It could well be that there are certain specific points in Klein and Lie that are not well known today. I don’t know a lot about what is in Lie’s own writings (as opposed to later developments of Lie theory), but from Hans Wussing’s book “The genesis of the Abstract Group Concept” I got the impression that Lie did think more generally about continuous groups (including infinite dimensional groups) than one might guess from the later descriptions of his work. (On the other hand, there is lots of contemporary work on this topic, and I would be surprised if whatever ideas Lie had aren’t incorporated into it in some way.)

But regarding nineteenth century work on algebraic geometry, I don’t think there is very much lost knowledge, at least in the algebraic geometry community as a whole. For example, in one of the later volumes of Serre’s collected works, there is a letter of Serre (I forget to whom) in which he gives a very nice account in contemporary language of some of Klein’s work on from his book on the icosahedron.

In the 70’s (I think) there is a famous paper of Griffith’s (I think it appeared in Inventiones) building on ideas of Abel.
Lefschetz and Hodge also built on the ideas of Abel and later ideas of Picard, as did the Italian school, and Zariski and Weil very consciously built in turn on this work, casting it into more algebriac language. Another text that could be interesting is “The Eightfold way”, a collection of papers specificaly about Klein’s work. I should also mention that Klein and Fricke’s work on modular functions and modular forms is completely incorporated into the contemporary theory of these topics.


Sorry for the long summary, but my own feeling is that algebraic geometry really does have a continuity both in its subject matter, and its internal understanding of its subject matter, and I wanted to marshall a bit of evidence to support this viewpoint. Igor Dolgachev himself has a wealth of knowledge on classical geometry, and I would be interested to hear his views on the subject.

There are specific aspects of 19th century geometry which are considered somewhat mysterious today. For example, I think that some of the intersection theory computations made by Zeuthen and Schubert remain mysterious (in that people don’t know really how they arrived at their answers). On the other hand, some of their computations are known to have been incorrect. (They probably relied a lot on rather formal techinques, such as applications of Bezout’s theorem — or more general counting theorems along this line — that ignored the possibility of certain degenerate configurations that could disturb the accuracy of the count. This is often safe, but occasionally not, as we now know today. So while we can be impressed by what they did, and perhaps puzzled at which precise method they used, it’s not my impression that they had great chunks of knowledge and understanding that are lost now.)

Overall, my sense is that algebraic geometers are aware of the richness of the past of their subject (not all individually of course, but collectively, as a group of researchers), and have made continual efforts over the decades, as their subject developed, to go back to past literature and comb it for ideas that have been temporarily forgotten or misunderstood, if only for the hope of finding a technique that will help them solve open problems of current interest.

So, while I don’t want to speak for any particular geometer’s view of their subjecct, thinking about the field as a whole, and taking into account my impressions of the knowledge and viewpoints of the various leaders of the field over the decades, I have to disagree with Rowe’s statement rather strongly.

What may be true is that there are certain points of view or intuitions that some 19th century geometers had that are now lost. After all, some of them were very great mathematicians, and every great mathematician brings certain individual ideas to their work which are not always well-communicated in their writing or passed on to their students, and which can then be lost once they pass from the scene. I don’t want to argue that it is not worthwhile to reexamine their work and see what we have missed. I think their is always value in going over the work of earlier great mathematicians, even if the general understanding is that it has all been incorporated into later developments. But this is quite different to saying that the 19th century work on geometry is “all but forgotten today”.

Posted by: Matthew Emerton on March 17, 2009 1:40 PM | Permalink | Reply to this

Re: for philiosphers?

Here is another comment:

What is happening in Klein’s example that makes it interesting? What new insights or techniques are being exhibited?

Well, here is one way to abstract his situation. We have a map f: X —> Y (stereographic projection from the quadric to the plane). It is not quite an isomorphism, but it is very close, so for the sake of abstraction, let me pretend that it is. Then what Klein observes is that the map f induces an isomorphism between Aut(X) and Aut(Y). This seems to be what he himself wants to emphasize. The geometric details of the particular situation were, I would guess, well-known to his contemporaries.

I don’t know the history well enough to know if this is the first occurence of this idea, but it seems quite possible that it might be. And this is a very important and powerful idea, which underlies many, many important ideas in contemporary mathematics. (One of them, that should be close to the hearts of the readers of this blog, is the idea that it is okay to only have objects defined up to canonical isomorphisms, because the canonical isomorphisms will preserve all the structure of interest.) But it is not one that is lost. On the contrary, it’s hard to imagine an idea that is closer to our contemporary way of thinking about mathematical structures.

Posted by: Matthew Emerton on March 17, 2009 1:55 PM | Permalink | Reply to this

forgotten today?

Who was Maurer? and when? reference?
Wiki provides only Maurer Cartan and attributes it to Cartan 1904

Posted by: jim stasheff on March 17, 2009 2:53 PM | Permalink | Reply to this

Re: forgotten today?

Dear Jim,

I just looked at Maurer-Cartan on wikipedia, which had a link to a very short entry on Ludwig Maurer (1859-1927). It refers to a work of Borel from 2001 on the history of Lie groups for details of his mathematical contributions. So (although I don’t know the details myself) this doesn’t seem like someone who has been forgotten.

I will look at Borel’s book if I get a chance.

Posted by: Matthew Emerton on March 17, 2009 3:24 PM | Permalink | Reply to this

Re: forgotten today?

Thanks
Don’t know how I missed Ludwig
but with that I should be able to find the entry. I don’t have access to Borel’s book
so will appreciate it when you have time - no rush.

Posted by: jim stasheff on March 18, 2009 2:28 PM | Permalink | Reply to this

Re: forgotten today?

There is anohter Ludwig Maurer, a musician
so if you google, type
Ludwig Maurer math
and you will have access to Borel’s paper directly

Posted by: jim stasheff on March 18, 2009 2:36 PM | Permalink | Reply to this

Re: forgotten today?

The key work of L. Maurer was cited in the beautiful historical note (appendix) in N. Bourbaki’s (Dieudonne I guess in that case) “Lie groups and Lie algebras”, vol. 1. It is

L. Maurer, Ueber allgemeinere Invarianten-Systeme, Sitzungsber. Muench. (= Muench. Ber.) 18 (1888) 103-150.

A long abstract of Maurer’s paper can be found online at

http://jfm.sub.uni-goettingen.de

Posted by: Zoran Skoda on March 19, 2009 5:56 PM | Permalink | Reply to this

Ludwig Maurer

From outside the blog, i also learned of
Search Results

1.
Essays in the History of Lie Groups and Algebraic Groups
by Armand Borel,

who reviews Maurer’s work and in English

(ich spreche zehr schlect Deutsch)

Posted by: jim stasheff on March 19, 2009 9:56 PM | Permalink | Reply to this

Re: Geometric Help Wanted

I agree very much with the perception of continuity in the
development of Algebraic Geometry. I just wanted to add the remark
that topos theory is a further continuation:

Matthew wrote:

> In summary, I think that the actual deep geometrical phenomenon
> occuring in Klein’s example, and others, is the way that
> algebraic geometry as a geometric tool let’s you shift back and
> forth between different ground fields, so that phenomenon that
> are observed directly when working over one field (say C) can
> be brought to bear to influence what happens (or, if you want a
> different way of phrasing it, to make arguments about what
> happens) over another field (say R, or even Q), even though the
> original phenomena are no longer directly observable over that
> field.

This dynamical viewpoint is precisely what Grothendieck captures
when he describes an algebraic variety in terms of its functor of
points (a functor from commutative rings to sets): a variety has
different ‘stages of definition’, or ‘generalised elements’,
depending on which field or ring it is over. Topos theory is a
direct descendant of this fundamental idea. (Of course
Grothendieck and Lawvere were very conscious about these
classical roots.)

(By the way, both the quadric surface and the projective plane,
as well as the respective blowups in question, are actually
functors pulled back from the category of commutative monoids —
i.e. they are varieties over F_1. The isomorphism can be
observed already on this level and boils down to combinatorics.
I mention F_1 just because of the functor-of-points viewpoint
(see Toen-Vaquié theory of F_1) — in fact those surfaces are
just toric surfaces and the argument is classical: the projective
plane is the toric variety given by the complete fan in R^2 with
three rays: (1,0), (0,1) and (-1,-1). The three cones in between
correspond to the three T-equivariant points, or the three points
defined over F_1. Blowing up two of these points amounts to
inserting two new rays: (-1,0) and (0,-1). (Note that each
blowup increases the number of points by 1: this is because the
projective line over F_1 has two points.) On the other hand, the
smooth quadric surface is, as mentioned, P^1 x P^1. This is the
toric variety given by the fan in R^2 comprising the four
coordinate rays. Blowing up one point is to insert a new ray,
which we can take to be (-1,-1). This is the same toric surface
as the one we got from P^2.


> We have a map f: X —> Y (stereographic projection from the
> quadric to the plane). It is not quite an isomorphism, but it is
> very close, so for the sake of abstraction, let me pretend that
> it is.

Perhaps it is worth mentioning that notions of sameness weaker
than identity (and even weaker than isomorphism) also seem to
originate in algebraic geometry. Certainly group theory
originates in geometry, but also set theory got crucial input
from geometry: according to Lawvere, it was from Steiner that
Cantor learned about the notion of isomorphism crucial for his
foundation of set theory. (Steiner knew that a cuspidal plane
cubic is not isomorphic to the projective line although there is
a bijection between their geometric points: there is no
non-constant algebraic map from the cuspidal cubic to the line.)

(Steiner, one of the most famous geometers of his time,
unfortunately is also famous for his erroneous use of Bézout’s
theorem to count conics tangent to five given conics. (Since the
conics tangent to one conic form a degree-6 hypersurface in the
P^5 of all conics, Steiner concluded via Bézout, that there are
6^5 conics tangent to the five given ones. The error is that the
intersection of those hypersurfaces is never transversal: it will
always contain the surface of double lines, since a double line
has non-simple intersection with any conic. Chasles corrected
the error by identifying the excess: in modern language this
amounts to blowing up the bad intersection so that it becomes
transversal.)

> There are specific aspects of 19th century geometry which are
> considered somewhat mysterious today. For example, I think that
> some of the intersection theory computations made by Zeuthen and
> Schubert remain mysterious (in that people don’t know really how
> they arrived at their answers). On the other hand, some of their
> computations are known to have been incorrect.

Since the advent of modern intersection theory in the mid
eighties, I think Schubert’s work is mostly understood: most of
it can be interpreted as computations in cohomology rings of
Grassmannians, or as excess intersection computations in
parameter spaces of curves. Zeuthen’s work is much more
mysterious. It does not help that his main treatise is written
in Danish. Luckily he is generous with tables of numbers, and
most of these numbers were verified in the late nineties using
Gromov-Witten theory (notably by Ravi Vakil). I was not aware of
any error in Zeuthen’s book. I know that a few series of numbers
have not yet been verified (e.g. characteristic numbers of
rational quartics with triple point or tacnode), but since these
numbers serve as auxiliary results for the main numbers (smooth
quartics) it would seem unlikely that they could contain errors.


Not sure if any of this is of any help.

Closing parentheses.))

Cheers,
Joachim.

Posted by: Joachim Kock on March 19, 2009 10:53 AM | Permalink | Reply to this

Re: Geometric Help Wanted

Dear Joachim,

Thanks for your comment, and in particular for clarifying my comments on intersection theory. It was indeed Steiner’s famously incorrect calculation that I was (mis)remembering, and I can well believe that there are no errors in Schubert of Zeuthen.

Best wishes,

Matthew

Posted by: Matthew Emerton on March 19, 2009 1:50 PM | Permalink | Reply to this

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