Not long ago David Morrison (UCSB) came to the mathematics department here at USC to give a colloquium.

This was a treat for me for many reasons. Here are three:

- It’s always good to see Dave. He’s one of the people I’ve known in the field was since my very first postdoc when I was learning to survive in the big bad world on my own after graduate school. I mostly could not understand a word he or anyone there else said in those days (IAS Princeton, right in the belly of the beast that was the Mirror Symmetry movement at the time) as I had no training in the areas that everyone seemed to be interested in, and so spent rather a lot of time hiding and trying to play catch-up, but it did not stop me from making some good and lasting friendships while I learned the local lingo. Dave’s one of the nicest guys you can meet, and as a bonus, is quite marvellous (and patient) at explaining both mathematics and physics to people in clear, well-motivated physical language (I don’t know many mathematicians who can do this, frankly).
- Mathematics department colloquia in many places can have quite a different character compared to physics ones, I’ve found. They are often a rather pleasantly simple and old-fashioned affair – in a good sense. This was one of those times. Dave simply picked up a piece of chalk, walked to the board, and spoke while he wrote equations for about 50 minutes. No fancy graphics, computer technology,or even hand-written slides getting in the way between the pure ideas and the audience. He also did that thing that I also like doing when I am giving a particularly successful lecture or seminar – talk essentially entirely without notes for the duration. (The trick: prepare your notes carefully beforehand to the extent that you don’t actually need them. It can make for a really good talk for everyone involved.)
- The content itself was a pleasure to attend. The main punchline (or one of them) was familiar to me, but it was rather good to hear the setup, and particularly good to hear it presented to a mathematical audience, and to learn of the particular examples he chose to illustrate the punchline.

He was talking about techniques for repairing singularities in geometries. There are purely mathematical reasons for studying this issue (which he discussed in part), and there are also physical ones. I’ll talk here about the latter. The physical reasons, which often motivate my own work in this subject, have to do with understanding the fate of certain key physical scenarios in nature, involving gravity. Einstein’s beautiful theory of gravity -General Relativity- runs into trouble in certain situations, generating singularities in certain solutions that require some other theory to take over and describe the physics. The two most common places where this happens are inside black holes, and also at the beginning of our entire universe. These singularities are believed to not represent the end or the beginning of physics, but of our current theories’ ability to describe the physics (two things that, puzzlingly, are often confused with each other). The geometry of spacetime itself breaks down in the GR description of Einstein, and so it is of great interest to understand how -even in principle- whatever takes over from that sick geometry arises and can be described mathematically. That’s the first step to being able to incorporate it into a working dynamical physical theory, and perhaps then ultimately understand the answer to questions such as “What happened at the beginning of time?” or “What is the fate of matter that falls into a black hole?” (In my opinion, what is likely to occur when we do understand how to describe the physics there is that we may not have even been asking the right questions – especially those cosmological ones about the beginning of time, the origins of the universe, and so forth.)

The general idea (and how it actually works in many examples) is illustrated in the little sketch I drew for you to the right (while sitting here in my Sunday hideout cafe). Think of the blob as representing some curved space, or possibly spacetime, perhaps our universe. Either as time goes by, or perhaps as one moves in some direction (or both), the geometry develops a singularity – the little pointy bit. In gravitational physics as we know now (Einstein’s General Relativity), the description ends and the equations cease to be any use in saying what happens next. What is possible is that the physics carries on happily, and there might be a description of it using new techniques – eventually one can even end up back in a situation where everything is nice and smooth. But to do that one needs a bigger framework that allows for a description of the entire process. The outcome,in physics for example, would be that while the geometrical description using Relativity works for some of this, and tells you that some physical quantities have just gone crazy, the larger framework tells you that if you keep track of the right physical variables, the physics is quite readily accessible all the way through the singular region.. in fact, from a physical perspective, the situation is therefore not singular at all.

Dave’s talk was about the modern history of understanding some of this from a mathematical perspective – following a lot of his own work on these issues predating strings in the context of complex manifolds (or better, algebraic varieties, as they’d say in the trade) through to his work in string theory. What they understood was that from a purely mathematical perspective, there are examples of just the sort of scenario that I described and illustrated above. Essentially, instead of describing one by one each of the geometries in the picture above, you end up describing a continuous family of geometries, where there’s a parameter, let’s call it [tex]a[/tex], that gives a geometry for every value it can take. As a not-too-inaccurate simplification of the story, the parameter [tex]a[/tex] ends up describing the size of some piece of the geometry, where [tex]a=0[/tex] is the case that is singular. So the family is then the geometries with a range of sizes for this piece, and when it goes to zero the pinch develops. Insight into how to describe what happens on the “other side” of the singularity was rather like realizing that [tex]a[/tex] could take negative values, and that there was a meaningful family of geometries for those values too, where there was a piece whose size was equal to [tex] – a\ .[/tex] That obscures a lot of richness, but gets some of the story right – the point being that how you naturally go through an apparently singular situation can be given a natural mathematical embedding into a larger problem where the singularity is put into a larger context and seen to be naturally part of a family of geometries.

There was a puzzle, however (and certain aspects of this were news to me, and very interesting historically and otherwise). The geometries that were given this nice resolution were as I said already, *complex* geometries, which is to say that their coordinates were naturally grouped into pairs to make complex numbers, and many essential properties of the geometry respect this complex structure in an essential way. The parameter [tex]a[/tex] that I described above for resolving the singularities was *real*. This meant that things were rather messy indeed, and in some sense, not very natural from a mathematical perspective. This was a puzzle.

Now at this point in the story, I could see where our speaker was going. String theory turns out to give you a way of complexifying the resolution parameter and making the entire resolution problem very natural (and doing a lot more besides). What I did not know however was the following. I simply had always assumed that mathematicians had thought of a number of ways of getting around the problem, and that string theory picked one of them, but I learned from Dave that this is simply not what happened. There *were* no solutions to the puzzle that were found, using mathematics alone. Eventually, string theory in the 80s began to make things clearer and supplied the natural resolution to the problem.

It was the beginning of what might be called (and in fact *is* called) Stringy Geometry. The point is that strings are not points, and specifically, their extended nature means that in addition to being able to see the usual geometrical properties of a space that the theory like General Relativity can see, the strings can see other, intrinsically stringy, data. There is a quantity in the theory that is called the Kalb-Ramond field (or just the “B-field”) that can be used to measure how much the string can winds on or wraps a piece of the geometry, in essence. The parameter [tex]a[/tex] that measures the size of a piece of the space that collapses when the geometry becomes singular, is essentially joined by another parameter, [tex]b[/tex], that sort of measures how much the strings have wound or smeared themselves on that piece of the space. The upshot is that [tex]a[/tex] and [tex]b[/tex] naturally combine themselves into a complex parameter that naturally describes the resolution process, solving the puzzle that the Mathematicians faced.

This remarkable story of string theory (a theory of physical, dynamical objects) playing a key role in the mathematics to the extent that it led (and still leads) the researchers to solve mathematical puzzles, develop new techniques, and give new insights into how some of this new mathematics may be relevant to the physics of our universe one day (we’re some long way off from answering the cosmological questions, and have made remarkable process in some of the physics questions about black holes, although with quite a lot left to do there too) turns out to be just the beginning. The above complexification of the resolution problem leads naturally to the remarkable phenomenon of Mirror Symmetry, for example. There was already a set of complex parameters known to geometers which control what’s called the “complex structure” of the geometry – sort of describing the shape of the geometry. The newly discovered string-endowed “size” parameters I described above supplied a new set of complex parameters a geometry could have. The discovery of Mirror Symmetry was the discovery that the geometries come in mirror pairs, and the physics/mathematics of the strings moving on one of these pairs could be described in terms of moving on the other one in the pair. The difference? The complex and size parameters, respectively, of one space play the role of the size and complex parameters, respectively of the other space in the mirror description. This also opens up the full facility of the powerful idea/phenomenon that we use in string theory a lot that’s called “duality”, where physical properties that are difficult to describe in one picture are more easily described in the dual picture – the dual pictures in Mirror Symmetry are the strings propagating on one or other of the Mirror pairs. It is that same duality technique that is teaching us to use ten dimensional string theories to say important things about strongly coupled gauge theories in four dimensions that may one day give powerful insights into experimental issues in nuclear physics, as I’ve described in an earlier post.

We are, as I said, a long way from definitively applying this to definite physics questions about the real world, and constraining how it works enough to predict consequences for experiments we might do. However, the progress on the issues has been remarkable and encouraging. We’ve learned some key things of use for addressing how we might go beyond Einstein – not just words and ideas, but concrete computational framework with several detailed and sharp examples. That’s one of the reasons string theory is so attractive to those who work on it, as compared to other approaches to these physics problems, at least so far. In a very real sense, the key point is that back then strings began to reveal to us that spacetime geometry is probably not fundamental, as it is in Einstein’s General Relativity. It has also told us quite concretely how this comes about, in this framework. Rather than geometry largely controlling the physics (through spacetime, and all that takes place in it) here’s something deeper going on, as is hinted at by the fact that the strings can choose to describe their physics in two completely different geometrical settings, connecting together geometries that – outside of a stringy context – simply have nothing to do with each other. *There’s something deeper than geometry at the core, which probably isn’t geometry at all.*

It is my own feeling that we’ve still only scratched the surface here, and that string theory (and the larger theory we know it to be part of called M-theory which includes all we’ve learned about the role of other extended objects – Branes – besides strings (I’ve spoken of them here before)) might lead us to a description of physics that is fully independent of the crutch of geometry. If we get there -and I hope we do- I think we’ll then have a better chance of addressing a lot of the physics issues concretely. Right now, we’ve been supplied with some pretty marvellous hints that are extremely tantalizing and encouraging.

Well, I’d better stop there, since my one remaining reader* -thanks for sticking with it- probably needs to get on with other things. No doubt I’ll come back to these issues again.

-cvj

*Estimated based on the post being longer than four paragraphs, and hardly anybody seems to have the patience to read anything of any length these days…

Excellent post! By the way I also really enjoyed David Morrison presentation style… so simple and beautiful.

Some years ago, Lasenby, Doran and Gull of the Cambridge geometry group, rewrote general relativity using Clifford algebra instead of tensors. The result was subtly different from GR. Instead of being defined on an arbitrary manifold that would allow wormholes etc., it built on a flat Minkowski spacetime.

Since it’s built from Clifford algebra, it is considerably easier to do stuff with elementary particles. I would think string theorists would be all over it, but it seems like the stuff got ignored.

Hestenes wrote about the methods extensively and his papers may be easier to understand, see his website.

Now the point of all this is that you have a choice. You can define gravity using tensors and symmetry principles. The result is what the mainstream lives with; articles on arXiv that read like bad science fiction. Or you can use the mathematics of particle theory to model general relativity. You will get exactly the same results so long as you stay outside of black holes, but no bad science fiction papers. AND NO SINGULARITIES IN SPACETIME, except the usual singularities at mass points that you already learned to deal with in flat space E&M. And you get to use mathematics that fits the elementary particles.

I mean really, the best people in geometric algebra worked this stuff out. They’ve been steadily grinding out peer reviewed papers using the techniques. And it’s why I wrote the simulation in Painleve coordinates, these are the coordinates that fit a black hole to a Minkowski background.

Gees. You didn’t get any responses. Maybe I *was* your one reader (not that I understood it all, but it was still fascinating). Tagging it now.

Hi Clifford,

thanks a lot for this post – that’s very interesting! About your nice series of illustrations: the similarity to a drop of water falling from a tap is probably not a coincidence? I’m not sure about this, but I think I remember that the mathematical description of the separation of drops also involves singularities developing in the standard mathematical formalism for the handling of the surface of the drop? I that true, and is there are connection to singularities in GR beyond vague analogy? Was this discussed in the talk?

Best regards, Stefan

Very nice, superbly informative, and just at the right level, at least for me, who know little Algebraic Geometry and less Physics.

We (Okay I) demand more posts like this ðŸ™‚

I am not familiar with complex manifolds and Mirror symmetry, but know about T-duality.

How can you conclude “Thereâ€™s something deeper than geometry at the core, which probably isnâ€™t geometry at all.”? It seems to me that at least T-duality is a consequence of using a 1-dimensional object(string) rather than a point to describe the geometry.

As a retired and thoroughly obsolete solid-state physicist I am just fascinated by this work. Solid-state physics is thoroughly quantum-mechanical, requiring one to abandon classical physics and “ordinary” (and comfortable) thought processes. It seems inevitable that fundamental physics has to get beyond the usual, comfortable concepts of space and time (including General Relativity) and get to something deeper.

It will surely be hard to visualize this deeper understanding but it’s not so easy to visualize quantum tunneling either. It does get more natural with experience in solving actual problems and therein lies the hope. You write very well and I look forward to your future posts.

Thanks

Hi Metal,

Well, that’s

myconclusion, based on a host of calculable phenomena. You get to pick: You’re free to conclude other things. Alternatives include saying that there are X numbers of geometries describing a given situation, and that they are dual to each other and so forth, and the one you use to describe the physics depends upon your point of view, the probe you’re using, and so forth. That’s possibly a fine way to proceed.I think it is simpler to declare that the focus on the geometry is the problem (especially when there are also intermediate situations where there’s no reliable geometry at all, but the string theory continues to make physical sense).

-cvj

Very nice post. I was wondering if you know of any good introductory papers to stringy geometry?

There are too many errors in your book “D-Branes” and it’s

hard to take it as a textbook; do you have a erratum list

to the book?

Thank you very much!

Elias:

I can’t think of anything off the top of my head. I seem to remember that Kentaro Hori wrote a book a few years back called Mirror Symmetry. I do not know what level he pitched it at.

d-brane:

So sorry. Yes, there are some typos, but people have found it very useful, nonetheless, I hear. I maintained an erratum page for a while (google for it) but it is a bit out of date since I incorporated some of it into a later printing.

Best,

-cvj

Stefan… actually it was a coincidence, and from what you point out, a nice one. I don’t know anything about the mathematics of droplets, but it seems likely that there’d be a singularity, indeed. No, Dave did not talk about this, as far as I could tell.

best,

-cvj

On speaking without notes:

The first and last professor of mine who did this regularly was David Benney, who taught multivariable calculus my freshman year. It was remarkable, perhaps more so in retrospect: just ideas flowing from one head to many others through slate and chalk intermediates!

More recently, I’ve heard Greg Chaitin give a clear and entertaining presentation without slides, notes

ora blackboard: he’s the one-man show of computational complexity.Thanks for keeping it interesting Clifford.

I am always trying to get the “visual models” of such proposals in terms of the B Field. Nigel Hitchin

Can you tell me, if the Dynkin diagrams and the points on a Sylvestor surface/ Cayley model have some value when looking at this subject?

Also, if it would be wrong to see “UV coordinates of a Gaussian arc” can be seen in this light as well?

Hah. I feel caught. I read the first few paragraphs, scanned to the end, and noticed the footnote having predicted my actions.

Chris,

I’m sure you’re in the good company of very many others!

-cvj

I did a quick Google Scholar search, and it looks like models of fluid pinching do exhibit singularities, at least in the case of vanishing viscosity (see,

e.g.,Jens Eggers in theSIAM Journal on Applied Mathematics). Pugh and Shelley have a model in which pinching singularities form in thin jets of fluid, driven by surface tension. Maybe something interesting is going on there.Blake,

Thanks.

Singularities in fluid dynamics and other areas of physics are certainly not uncommon. I think Stefan was asking whether there was a deeper connection.

Cheers,

-cvj

Your image looks like a depiction of some sort of time-reversal, perhaps?

Me old fogy physicist now likes to find out what happened in cosmology since I attended differential geometry lectures in the early sixties. Trying to find books available about this topic I realized that the “mathematical apparatus”, including now string theory, still seems to be guiding the research. But I feel a certain reluctance in assuming that nature keeps always following those mathematical models. That’s why I googled “avoid singularities in space time” and found your post. So, I guess what I am looking for are some speculative (necessarily) ideas that include loosening this reliance on mathematical formalism. “Inflation” seems to me just an example where a parameter of a mathematical model had to be sacrificed/ adjusted to make it fit observations. There of course are other, now sacrosanct, parameters which could be questioned. You see, I need help. Are there any crazy ideas out there, of some merit?

Idle questions beginning at a birthday party:

How do algebra and geometry “break apart?” For example a

high school geometry class and a high school algebra class

are followed by an analytical geometry class. There are other examples.

I noticed that Carl Brannen mentioned a Clifford Algebra.

Is there a Clifford Geometry?

I also noticed a Stringy Geometry.

Is there a Stringy Algebra?

Is something “which probably isnâ€™t geometry at all”

also something which probably isn’t algebra at all?

If algebra and geometry are “broken apart” isn’t there a

“cost” with respect to the physics?

Is “breaking apart” a “free lunch”?

Doesn’t some “real” experimental data have come in?

How do we slip into a new paradigm?

Suppose we change an axiom? That could change geometry, old hat.

When I think of changing axioms I think of something such

as changing five axioms to six axioms. I don’t think of a

“moving axiom”. For example, when I think of changing from the classical Hamiltonian to the Schroedinger equation I think of having h or not having h. I’ve never read of a variable h.

Where did this h come from? It came from new experiments, observations, and additional information.

Hooray for physics.

Some variables appeared to have made it across the jump from classical mechanics to quantum mechanics. Time made it across but it was never it’s old self again. Space also took a hard knock. For some of the potential functions it was a piece of cake. Coulomb’s Law didn’t need no stinking h.

Geometry changed so much that it could hardly be recognized.

Geometry was still there because the potential functions needed it, but the wave functions needed a new geometry a

lot more. Did the new quantum geometry via Hilbert space

appear to be something “which probably isnâ€™t geometry at all”?

So, how about them singularities? How about a “new h”?

Is there an experiment missing, observations missing, or

information missing? Won’t physics go beyond Beyond Einstein because we measure something new or because we measured something that wasn’t what we thought it was? Do we need some new experiments about singularities or do we need to know what we’re looking at, or both? Do we

expect the 21st century physics to produce as great a change

in physics as did the 20th century? Do we expect the new theories to be strange and difficult to understand?

Bill

give a clear explanation as to how gravitation comes about, with illustrattions

You might try reading any one of a huge number of clear expositions on the subject of Einstein’s General Relativity.

-cvj

I’m sorry for reviving this post, but I can’t help saying “This is Great”!!! In addition, it’s beautiful that this is kept in archives and available for a lot of people.

Though I don’t know if it is (still) useful, because of the question in comments concerning introductiory papers on stringy geometry, I want to mention the review by Greene, hep-th/9702155 . It surveys a lot of interesting and essential things and references to the original papers on stringy geometry may be found there.

Thanks!

-cvj