Line geometry to relativity theory

0:00 And so that's where I'll start here. The famous collaboration, very brief as it was, and they were both quite young of course, and took place in Berlin. And here are a few dates about Sophos Lee, who was a few years older than Klein. He came to Berlin just by chance that Klein was there as a post-doctoral student. We got to know one another and we didn't go off right away and noticed that they were the only ones interested in the sort of geometry they were doing, so they then went to Paris together and we talked about that briefly, and you'll see after 1872, the year that Klein wrote the Hellen program, Lee returns to Norway, Professor Schuster is very isolated, his work is in Belmau and so forth. And then Klein, who in 1886 goes to Göttingen, he arranges for Sofus Lee to be his successor in Leipzig. Sofus Lee, in Leipzig, and that's in fact connected with the French mathematics. Lee, for an instructor, is a late bloomer. He took quite a while before he became a creative mathematician. He had some interest in mathematics, or I think he heard some interest in mathematics, in Norway. It's interesting that we actually heard one of the very first courses in Galois theory that was offered by Silo, but some of this media is a very geometric picture, algebraic ideas never played a very good part in it, we didn't appreciate much of what was going on in Galois theory at that very time, so the inspirations are highly geometrical and the figures I have here are to me the most important. In the French tradition of projective geometry, Monsignor, Charles, and then Plouca, the term.

2:30 So Plouca is in many ways considered the founder of line geometry, and this is a topic I'll get to a bit later on the significance of the word. And line geometry is a highly algebraic subject with Plouca, who is one of the founders of projective analytic geometry. There are different ways to think about it and do that, but one obvious thing to realize is that if you look at an algebraic equation in line coordinates, then this will cut out a three-parameter family of lines in space, and you can think about the degree of the line complex associated with the local cone, so the number of lines that pass through a point. Typically, what we'll look at later on is the simplest case, the first degree or linear line complex, and then all the lines that pass to a point lie in a plane. In the second degree case, you get an actual cone as in a different kind of section. So Klein absorbs this work of Kluge and Deitsch, as you see in 1868, and Klein is the one responsible for publishing work on line geometry in that year. But then he connects with Alfred Kletch, in the opinion, a terribly important mathematician, actually, in the 19th century, he's largely overlooked much of the time, a huge, huge influence on Klein, less so on Lee, but Kletch is someone who does invariant theory, and there's a stress that's important as part of Klein's background. So, in Berlin, where Klein and we meet, Kuma is holding a seminar which just happens to be on races, two-parameter counterpart to line geometry, intersecting line complexes, you know, races. So, in this seminar, the German was really up to presenting all the new ideas he had, and so basically his work was presented in the seminar by Klein, and that's part of it.

5:00 The standard story is that in Berlin they meet and they go to Paris, and this is a quotation from Klein's first volume of Klein's Collected Words, which I'll refer to in a second. There are several places in this talk because there he gathers up all of the work related to the Ballard program. And here he just describes in commentary about being in Paris until the outbreak of the time of the Russian War. And spending all his time together. And that Camille Folland's Traité had just appeared at this time. So he writes about it that it appeared to them like a book with a second seal on exactly the work that they were able to understand at the time. Typically, though one focuses on this convergence in group theory, so typically the Erlanger program is connected to group theory, and when you try to suggest it for Klein, that's less important, but this is a simple way to think about the inspiration coming from group theory, because for Chardin, who is someone who is in communication with Kletch, by the way, so Francis Chardin can come up with things that Kletch was working on as well. There's an important connection here. But I think Dabu is actually a more important figure, especially for me, if you can kind of get on with that, but Dabu is someone I... The standard story tells about how the two of them basically learn about group theory, divide up the subject, and go on and dominate this field, especially the application of groups to geometry. Erhard Schultz thinks another work of Scholdan is more important, and that's the study of the use of motions in space. Again, these questions of influence are always tricky, and Erhard can be useful. This is, I think, important to realize, though, that there are sources which show us, in fact, that what's going on is very, very complicated.

7:30 It's so surprising to have sources that see a different picture of what's going on in just a little bit of root theory. So the letters that Klein wrote to Lee to describe this rich mathematics to an American age, after Lee leaves, he eventually goes back to Norway, of course, and it becomes very difficult for Klein to follow his work, although he's very interested in it, and he comes back in the summer of 1882. Klein gets an appointment at the University of Erlangen, and he writes his Erlangenprogram in that context. I tried to find these letters for a long time and I knew that they existed because England refers to them in the seventh volume of these works. And I finally found out where they were, they were in Mainz. These things were in his possession because he married his granddaughter Lillian somehow. However, they would never show me these letters and so it took a long time before I ever saw them and they're purchased then by the Oslo University Library and will hopefully be published someday. But it's very difficult to follow what they're writing about. So the programmatic statement about the importance of groups and the variance, and that's where it rocks out in the very beginning. Before I show you a quote from the Erlanger Programme, I do want to call your attention to a very misleading thing about the title Programme. I mean, we tend to think that this is a research program or something like that. In fact, it was just simply a formal thing. We had to write a programmatic piece. Programme sind einschritten für die Führungspolitik.

10:00 There are several Erlanger programs. Every new professor who came to Erlanger had to write one. And so that's where the title comes from. It doesn't have to do with the research program. And, of course, the part that becomes famous is the part leading to geometry in groups. And here I have the quote in German, in case you're fond of the language. Groups in space that change the whole present space. And he begins talking about physical spaces and how you can look at various properties that are unchanged. Positional properties, lengths, and so forth, orientation of space, and he describes this as the help tool as far as he's concerned if you're looking at physical space, because if you see transformations that we want to look at, and going, taking that as background, he then says, well, we can peel that away, that we can get down to the mathematical core and just say any old manifold. And that's our space, and we're interested in a group that acts on it, and now we don't have any distinguished groups. In fact, in a metaphor, each group is equally justified. How does group mean, like, maximum group? No, it means the Euclidean group, so it can increase that. Okay, but there's just subgroups of this group, no? No, I mean, so when he starts out with something physical, and it's important because I want to emphasize what I have, but he's a mathematician too, so let's abstract from that and take something much more general, and we can talk about that too for the purposes of human algebra and geometry. So you've got different things going on. And this is the famous statement, the general task is to give a mathematical transformation. Then you are to determine those from unchanged. And then the final thing, you develop invariant theory proper to that even transformation.

12:30 These are global groups. So he has two basic models in mind for this more general thing. I mean, I've already talked about it. We can talk about metric geometries, but dimension n on whatever. And then, of course, projected geometry, although it was very new at the time, that he helped to push forward with the idea that the metric geometries can be divided, fastened onto an absolute figure. So this is an idea he got from Cayley, and of course Cayley was the first to project a geometry this way, and Klein was able to then widen that out and recover non-infinite geometries by looking at different types of conics that remain invariant. This is very, very famous stuff that Klein did right before he wrote the Atlanta program, and that obviously motivates it, so there's nothing new here that I'm saying, this is pretty standard. Catering, of course, is a huge expert on invariant theory, and I do want to stress the importance in Klein's thinking of ex-collection in invariant theory. There's a systematic way of talking about doing these things, but not like a program, but like a collection. Geometrically significant. He quotes Cayley somewhere or other in which Cayley said, if I take 15 lectures on all of mathematics and one of them on determinants, and that goes back to something that Kletch shows, which is that in projected geometry you can get all of the basic invariants and covariants from the determinants. And so this is... What is really a huge growing thing at the time, analytic projected geometry with invariance and covariance, was new is that Klein is going to at least hold out the possibility you could do this for other groups at that time. The projected group was the only one that was really looked at in this sense. So an example that Hesse worked out beforehand to give you an idea of how these things work in algebraic geometry. The equation which you get is the determinant of the second partials of homogenies. It can form that determinant. The second partials set equal to zero, you get a new curve.

15:00 And if you intersect that curve with the first curve, you get the inflection points. And then Tessa showed these in pretty configuration. This is very famous stuff. Now to the literature very quickly here. A lot has been done in certain directions on the work of Klein and Lee, and the article by Hans Freudenhau in the PSB is a good place to start with this thoughtful piece of Tom Hawkins, as I mentioned his several things on Lee's work, he and I published his early work of Lee and Klein in a Vassar conference, which in the meantime is a large-scale non-mathematical biography of Thomas Lee. With lots of sort of funny opinions, if anyone's interested in reading my review of it, but it's maybe relevant for us in the end, I would like to get to the theme of how he's in Leipzig and he has a bunch of French students and how kind of he's in between the Spanx rivalry between French and German mathematics and so in this book of course the French come out to be very good at that. I've also written a little bit about the These are all relationships that you cannot ignore in the course of talking about the Elagha program and how it's received and what happens in 1893, etc. So I'll give you this as well briefly. So Hawking is the one who's really done very profound work. Me here, under his work, but I do have a file and I want to say a word about it. In doing so, he writes about the Elagha program in various contexts. One of which I'll describe very briefly, but you can find more in the book. Here in his article on Mathematica, he pointed out for the first time that in the 1880s not so many people knew about the program, including von Jackett. But he was told about it when we came to Paris and we talked and he learned about it himself as we.

17:30 It's a full picture of how this theory heads off. And the key figures that I have here are those leading in the killing of Elie Patin and Hermann Leib. I obviously can't talk about this book, which is a huge achievement. I must say it's one of the major studies in the history of mathematics. But it's relevant for my talk here because it does bring in the way in which the Italians basically met direct ideas in the Erlacher program. Juan Guerrero didn't know it, but Gerardo Segre did, and he picks up on these ideas, projected geometry, essentially, in higher dimensions, with an invariant figure, and all the nifty things you can do with that. So Segre starts publishing about this, and his student Gino Pano does an Italian translation he had on a program published in 1890, and that's the beginning of a wider awareness of what happens. But in fact, in the early 90s, Gino Pfeiffer goes together and works closely with Klein, and you can find many of his traces of strong influence here. So, Klein is interested in reviving these ideas and does so to a certain degree. He has been doing other things. The Italian translation then sparks translations in French and English and so forth. Klein wanted to put out a German translation as well, but not just a German translation. He wanted to republish all of the early work that Ian redid with the long commentary. So, Fanon comes to Göttingen and one of the places you see a lot going on here is in the German encyclopedias because Fanon and Zeke write major articles in them. There is a volume on geometry, also in Enriquez, who is another Italian very influenced by these ideas. So you see traces of influence here, and these go all the way up into Catan's work, which is often how it is expanded version of Gino Pano's article. So Pano in 1907 writes this article to the German encyclopedia. About the role of group theory in geometry, and this is expanded by Elie Kaltan and in major companies, and you see some very direct lines of influence, so they take this way beyond what Felix Kahn is doing.

20:00 So here's my brief overview of Klein's own engagement with these ideas in speech, the one which we're working closely with Lee, leading up to the publication of the Erlanger Program in speech. He was heavily immersed in lines here, geometry. Then there's this period beginning in the 1890s, but this is also very much in a time period when Klein's interest in physics, so he's promoting rigid body mechanics and line geometry and that connection. This is the stuff that I think is being very unknown and so I'll come back to it. And then of course somewhat well better known I think is what he does in relativity theory, which is an example for him anyway. But he does things in general relativity, which are very, very interesting, playing with ideas all through here, which are very important for physics. And again, the encyclopedia is a place to look for a lot of this stuff. So in the mechanics volumes, several of these volumes are all lumped together in volume four, of course. So this is what's going on in writing about vector analysis. And then later on, when Wolfgang Pauling writes his Aristotle on relativity, Klein comes back in and gives him all kinds of tips about this, because the Elana program shows up there in a big way and gives many ideas about what's going on. So this is a place where we see a point based on what's going on. So here I just made four points now about the Elana program that I think are sort of well known. The first of these is I don't believe the Eleanor program had very much to do with these theories of continuous groups. I think that's a separate track, and I'll say a bit more about that. The invariant theory is what I was stressing out. The search for groups, in fact, what you see in time's work is you've got the groups already. It's invariant theory that's actually what you're going after.

22:30 So the example we talked about a moment ago, the Hult group, is actually hugely important. He wants to do physics with us. And so we, in the 1890s, when he was doing rigid body mechanics and so forth, he got a good example. And this is all tied then to a larger physical motivation, and this is, you can find it already back in 1870. So he has it all along in fact. And you can see some of this if you just look at how he packages the Erlanger program in his collected works. So he gets, Klein has the big advantage of most mathematicians die in a week's work, or no one cares anymore. In his case, he, even though the inflation was killing Germany, they still had enough money to do Felix Klein's collected works in a lavish schwingheim without a lot of money to do it. And he puts together a section of works of the Elana program, which contain two works that precede the Elana program, then in the 1890s and afterward, the middle period, he includes two works, one of which is just a report on his lectures on higher geometry, These are very important lectures which are later published. And then, something probably no one reads anymore, but it's actually a huge clue to what's going on, is 1902's review of survivor falls deemed as true. This is actually very important to understand why it's up to. And the final period, of course, is no surprise, the relativity stuff. It has four papers. I'm not going to talk about these in any detail for lack of time, but I will say just a few things about this last period because I don't want to neglect it altogether. So what I have to say is the following things. First of all, of course, in Göttingen there's a huge interest in Einstein's theory going back to Minkowski and Hilbert and electron theory. Then Klein really first gets involved around 1960.

25:00 In 1916, when he begins giving lectures on the mathematical foundations of relativity, these eventually get published in the second volume of his well-known lectures on 19th century mathematics. And that's after his death, of course, that Kuhlman publishes these. So these lectures that Klein is giving during wartime take place at a time when Emi Noether is in Göttingen, Klein and Gruber, and Einstein of course have been doing work, and Klein is putting in some very difficult stuff, but he has the help of various assistants, especially Emi Noether and Hermann Hermeyer. And so this is kind of part of a little industry of interest in Göttingen and the topic. Emi Noether was originally an assistant to Gruber, who was working on unified field theory. So she was an expert on differential invariance, and he had big ideas about how to get a rule function that would enable him to become additional equations for all of physics, so that was her original thing, but then energy conservation turns out to be a very puzzling issue in general relativity theory, and Hilbert, Einstein, and Bowden all published on this, and they come up with different versions of it. This is an ideal sort of thing for climate. He knows this. Let's figure out what is the mathematics and what is the physics and can we figure out where these, we get in great trouble with the assistants and we have to, of course, develop all the kind of mathematical identities that you can derive from a general, generally invariant, variational problem, etc. Very involved and the new thing is since you have general covariance going on here, you've got four free parameters and so you get certain things which obviously aren't physical in the form of it. And that had to be sorted out and it was very difficult. So there's lots of confusion and Klein does some major work to help unpack that. An Italian named Palatini is, I think, the first to really come up with a sound, generally covariate approach to variational methods,

27:30 so that you know that what you're getting is actually generally covariate. But without solving the previous problem, right? About the definition of the energy in generality. Yeah, that's a separate issue. That's for another team a bit later. I'm saying too many things at once. No, thanks. But one point that's interesting here is that in the back of Klein's mind is Mies' theory. He knows that Mies' theory is well accommodated to this, and no one apparently had ever used Mies' theory beforehand on a variational problem where you get the Lagrangian corrections out of formalism and so forth. But now you have to interpret, well, what are these identities between these different Lagrangian... So this is where he's coming from and as I said I don't want to go into any details here but it's extremely very much very much appreciated because she really shows how certain quantities come up in these variational problems and she was also able to show which of these things are physically meaningful and which not so that was a client place at her work a great deal but It's not nearly forgotten for several decades, actually. So that's a little bit about what happens with ideas related to the realm later. Let's now go back here quickly and try to trace a few threads and find physical information. So as I try to claim here, this goes up there. And you can find such threads over 50 years' time. There's really a lot here linking physics to geometry. There are special types of quadratic line complexes that you get by essentially just letting a line intersect the faces of a coordinate system. Let me show you here with a formula how easy this is.

30:00 We can set x, y, and z equal to zero and let those be the coordinate planes, and the fourth plane will be the plane of infinity, and then take any line that intersects that fixed cross-ratio, that if I apply these transformations, these three family of transformations, it's obvious that that coordinate tetrahedron state cross-ratio is an invariant, so there I get from this very simple procedure family of lines that satisfy this property that they need to face a fixed cross-ratio. There's, of course, a simple group action on the line here that we're, Lee does amazing things with this, as he goes, takes him very deeply into it and shows you that here in a second, but this is the stuff they worked on originally, and you should all present to the Paris Academy and so forth, and they tried doing, well, that's people complexes, but those are objects which are The self invariant under the subgroup of this one-parameter group of transformations that leads to a certain figure, a curve or whatever, invariant is called a W-curve. So within this context, you get these certain types of objects and Klein studied these in a plane and a bit in space where they gave up after a while because it got so complicated. The differential equation associated with these tetrahedral complexes, they are of this type. The beauty of this is you have such control over the individual curves and so forth, and you apply the so-called logarithmic transformation to the individual curves, and then you could just essentially transform them out. And so something which looks like this gets transformed away, but the spaceports just vanish and end up with the two partials you just saw.

32:30 The integration of this differential equation becomes a very simple matter because he has all this information about the group invariance at hand. And he takes that and he generalizes it right away. He starts looking at, well, okay, if I have a three-parameter group available, like the one we showed, if I only have two parameters, then I'm producing something like the second case, a one-parameter group. There's a lot of work to do, but there's an intimate interplay between group invariance and group theory, and all of these things are contact transformations, so this is what general theory develops out of all of this stuff, and geometers, of course, love things like contact transformations, so this is all very, very visual mathematics. And it leads him to his line to sphere transformation, which is surely the most, the nature thing that inspired both Lee and Klein at this time when they came to Paris. However, this work is not well appreciated in Germany, at first anyway, and Berlin is certainly not appreciated by people like Weierstrass and and so forth. So Klein and Lee are very marginal figures in Germany for quite some time, and the fact that Klein was able to bring Lee to Leipzig was something that caused a huge, huge controversy. He says on other things that Lee thought very, very geometrically, and unfortunately to try to promote his mathematics, he started dressing it up in Berlin-style analysis with power theories, representations, and things like this. And so the three-volume collective works, or three-volume study of transformation groups, written, of course, by Hubert Gaynor, based on Lee's ideas, is really rather foreign to the way Lee thought about them. In fact, Lee never ever retrieved his own work. If you read Scheffern's lectures, you'll see that the sphere is much more geometrical.

35:00 In view of time, I'm going to skip over to this part here pretty much, but I just want to say, you know, there is this influence on theory on mathematical catalogs. We addressed that in geometry that Klein does, and this is important in the context of Klein's work. Or we classify the three space and apply them to use that to connect with complex variables and so forth. And so he develops all this in his famous book published in 1884 about the Icosahedron equations. It's interesting that he does this work together with Paul Grévin, who was a tremendously algorithmic thinker. So there are absolutely different types of thinkers who somehow get to collaborate on this. And in the introduction to this work, he thanks Lee and Gordama, and he talks about how, and this is where one of the places this legend comes up about splitting up group theory, he talks about how back in their youth, we find, studying groups and all of this, and they're divided into continuous groups that divide the subject up, and he talks about... The importance of all of this. And Lee was very moved by this tribute that Klein writes in such correspondence and things such as. And he really ought to republish it. And now it will be better understood. But of course he doesn't republish it until the day that he wanted to do it. That of course is a massive thing and Lee by that time is no longer interested. He changes his mind in the 1890s. He's getting quite unhappy about this promotional. He's unhappy about a lot of things, in fact. He's going through a hard time. And it leads to this major break between the two of them. This is a factor in the Eland program and the break in the end of a partnership.

37:30 So that's the personal element. It has to be taken into account just to understand what's going on here at all. So what about line geometry and sphere geometry? Well, again, I'm going to be very brief about these. So, Jukka Alvorn is introduced as the founder of line geometry and linear line complexes. Out of the Erlanger program itself, you see the emphasis on the fact that you can use an arbitrary space element. You don't need points, you don't need lines in space. You don't need to use any dimension in space, it's irrelevant. Kuma surfaces are extremely important in this context, but not necessarily a good part of the integration. There are a number of things that are complicated, but they're sitting around here as the singularity surfaces or second degree complexes, that it means you have a point and the cone collapses, and you no longer have a quadratic cone, you just have a planar cone. So it's sort of a determinant that you look at all those points and it turns out... So there's a huge context for studying these objects that were new at the time within the context of line geometry. But Klein is very interested in the physical aspects of this. He took a lecture course, a six-hour lecture course in 1871, where he talks in detail about the Fresnel wave surface, which is a specific case. Comber surface and conical diffraction and all these nice properties that it has and so forth. So there's a physical context here throughout. And the line-to-sphere transformation, that's the really huge thing that we discovered, is there's a way of transferring structures from one setting to another, so the surfaces which you get as envelopes of lines and other surfaces which you get as envelopes of spheres can be connected and carried over things like the lines of curvature of one to the other. Principle tangent curve for the other, etc. So within this very critical and rich context, there's a carryover, so transference principles.

40:00 These are referred to in the Erlanger program. Tom Hawkins writes about them in a very kind of group-theoretic way, because they go all the way up to Catan, and actually there's a bunch of testes and protagons and conceits and so forth. The line is very group-theoretic, but for Klein, I would argue, it's all very geometric, and so here's my point. Klein says in the etymological run that geometry boils down in this situation to, well, if I have two spaces, R and R prime, I know the automorphism group of the one, say R, if I've got a bijection between the two, then I can simply carry over the map to the other, I've got a group action on the other, and those two things are the same. Well, then we're tempted to say, yeah, then you have the same geometry, and so you don't need to look any further, but that's not what it's about. But it's about this invocation to now look at what you can carry over from these two things, the lines, your geometry, and these types of things that are all very, very much colored by the idea that there are analogous theorems that you can get in a surprising way, and you said, it's an old idea to project the geometry dualities and just some formalism, and it's a nice way to get new content, new results, and new connections. That's, I think, what's going on. And so, if you have a complete knowledge of the invariant theory of a given group, then you can systematically go after the results in a different setting. That's the message that we have on the program, is that we need to be ready for motivation. So that's a major point I want to make here. Not just think about the groups, but think about the invariant theories. This is something that Tom Hawkins writes about, and it simply shows how he said he had wandered into something more modern, we would say. Through the work of Fanon and Catan, the issue does come up about, for example, if you change the space element, or whatever, that changes the dimension, maybe, and can you really talk in a meaningful, sharp way about the equivalent geometries? Fanon says yes, and Catan actually connects that with the structure theory of degrees, and so it becomes an important thing to do here, but it's certainly not something that's fine thought about. Tom Hawkins doesn't try to press, but he does either. They somehow find meaning in Klein's text that Klein certainly reads. So this is a new program, classifying groups and so forth, that I simply don't think is part of summarizing here what I say.

42:30 Now, something else here too, because this is really about illustrating other types of mathematics that show you how physical and geometric Klein does things. I mean, his work on Riemann surfaces, I'll just show you some pictures here. Physical mathematics. Instead of working on the complex plane, Klein wanted to know what does the complex variable theory look like on a surface, where you know the genus in advance, the genus is a hugely important thing in the context, and come on bio, we'll pick this up later and use this as a foundation. So you use a little Maxwellian plane to begin with, so it's kind of like functions and how they fit together, you put the two together, you get a complex. So here's a diagram where you're forming a three-month surface, a closed surface, let's say, of the conjugated harmonic functions on it, and the surface has genus 2, and then below it has genus 1, so something is happening where there's going to be singularities that will emerge in the process of the deformation of the specific lines. He's looking at all these defamation types of actions and the way they connect to the various models. And of course he writes about how you should think about models in the context of algebraic geometry. We want pretty models that are symmetrical, that impress itself literally down the line. So we start with a clutch surface. You want to study cubic surfaces which have, you can see it well or not, And so you put this in a very symmetric configuration, and this idea here was I'm going to deform these things and try to classify the entire continuum of cubic surfaces by way of deformation. And so he writes about here, starting with this case here, where you have the maximal number of singularities, or tonical points, situated at the vertices of a tetrahedron. Down below is a triangle of lines, so of the 27 lines, three of them stay put down below. The other 24 fall into the six edges, lines fall into the edges.