## This Week's Finds in Mathematical Physics (Week 101)

#### John Baez

Darwinian evolution through natural selection is an incredibly powerful way to explain the emergence of complex organized structures. However, it is not the only important process that naturally gives rise to complex structures. Maybe when we study biology we should also look for other ways that order can spontaneously arise.

After all, there are plenty of complex structures in the nonbiological world. When it snows, we see lots of beautiful snowflakes with similar but different hexagonal structures. Do we conclude that snowflakes evolved to be hexagonal through natural selection? No.

But wait! Maybe in some sense a hexagonal snowflake is "more fit" in certain weather conditions. Perhaps this shape is more efficient at getting water molecules to adhere to it than other shapes. We can think of different snowflakes as engaged in "competition" for water molecules, and the ones that grow fastest as the "winners". In fact, the exact shapes of snowflakes in a snowstorm depend crucially on the temperature, humidity and so on... so who the "winners" are depends on the environment, just as in Darwinian evolution!

A biologist will reply: fine, but this is still not "Darwinian evolution". For Darwinian evolution in the strict sense, we require that there be a "lineage". Darwinian evolution applies only to entities that reproduce and pass some of their traits down to descendants. The idea is that over the course of many generations, traits that aid reproduction will accumulate, while traits that hinder it will be weeded out. Snowflakes don't have kids. A one-shot competition for resources, followed by melting into oblivion the next day, is not what Darwinian evolution is about.

Okay, okay, so it's not Darwinian evolution. But it's still interesting. It's showing us that Darwinian evolution is just one of various ways that order can arise. So we shouldn't study Darwinian evolution in isolation. We should study all the ways that systems spontaneously generated complex patterns, and see how they relate. If we do that, perhaps we'll see a bunch of interesting relationships between physics and chemistry and biology. Also, maybe we'll get a better handle on how life arose in the first place... that curious transition from chemistry to biology.

If I wasn't so hooked on quantum gravity I would love to work on this stuff. It's obviously cool, and obviously a lot more practical than quantum gravity. The origin of complexity a very hot topic these days. But alas, I am just an old-fashioned guy in love with simplicity. Whenever I see a new journal come out with a title like "Complex Systems" or "Journal of Complexity" or "Santa Fe Institute Studies in the Science of Complexity", I heave a wistful sigh and dream of starting a journal entitled "Simplicity".

Actually, the fun lies in the interplay between complexity and simplicity: how complex phenomena can arise from simple laws, and sometimes obey new simple laws of their own. I like to hang out on the simple end of things, but that doesn't stop me from enjoying the new work on complexity. At one point I got a big kick out of Manfred Eigen's work on "hypercycles" --- systems of chemicals that catalyze each others formation. (You may remember Eigen as the discoverer of the "Eigenvalue"... in which case I pity you.) Presumably life started as some sort of hypercycle, so the mathematical study of the competition between hypercycles may shed some light on why there is only one genetic code. There is a lot of nice math of this type in:

1) Manfred Eigen, The Hypercycle, a Principle of Natural Self-Organization, Springer-Verlag, Berlin, 1979.

Another name that comes up in this context is Ilya Prigogine, mainly for his work on non-equilibrium thermodynamics and the spontaneous formation of patterns in dissipative systems. The following are just a few of his many books:

2) G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems: from Dissipative Structures to Order Through Fluctuations, Wiley, New York, 1977.

Ilya Prigogine, From Being to Becoming: Time and Complexity in the Physical Sciences, W. H. Freeman, San Francisco, 1980.

Ilya Prigogine, Introduction to Thermodynamics of Irreversible Processes, 3d ed., Interscience Publishers, New York, 1967.

A bit more recently, the work of Stuart Kauffman has dominated the subject. It's really him who has pushed for the unified study of the whole gamut of methods of spontaneous generation of order, particularly in the context of biological systems. He's written two books. The latter, in particular, includes a lot of math problems just waiting to be tackled by good mathematicians and physicists.

3) Stuart A. Kauffman, At Home in the Universe: the Search for Laws of Self-Organization and Complexity, Oxford University Press, New York, 1995.

Stuart A. Kauffman, The Origins of Order: Self-Organization and Selection in Evolution, Oxford University Press, New York, 1993.

If non-Darwinian forms of spontaneous pattern-formation can be important in biology, can Darwinian evolution be important in non-biological contexts? Well, as I mentioned in "week31" and "week33", the physicist Lee Smolin has an interesting hypothesis about how the laws of nature may have evolved to their present point by natural selection. The idea is that black holes beget new "baby universes" with laws similar but not necessarily quite the same as their ancestors. Now this is extremely speculative, but it has the saving virtue of making a lot of testable predictions: it predicts that all the constants of nature are tuned so as to maximize black hole production. Smolin has just come out with a book on this, which also happens to be a good place to learn about his work on quantum gravity:

4) Lee Smolin, The Life of the Cosmos, Crown Press, 1997.

Interestingly, Stuart Kauffman and Lee Smolin have teamed up to write a paper on the problem of time in quantum gravity:

5) Stuart Kauffman and Lee Smolin, A possible solution to the problem of time in quantum cosmology, preprint available as gr-qc/9703026.

Right now you can also read this paper on John Brockman's website called "Edge". This website features all sorts of fun interviews and discussions. For example, if you look now you'll find an intelligent interview with my favorite living musician, Brian Eno. More to the point, a discussion of Kauffman and Smolin's paper is happening there now. As a long-time fan of USENET newsgroups and other electronic forms of chitchat, I'm really pleased to see how Brockman has set up a kind of modern-day version of the French salon.

6) Edge: http://www.edge.org

Okay. Now... what's even more fashionable, trendy, and close to the cutting edge than complexity theory? You guessed it: homotopy theory! Currently known only to hippest of the hip, this is bound to hit the bigtime as soon as they figure out how to make flashy color graphics illustrating the Adams spectral sequence.

Last week I went to the Workshop on Higher Category Theory and Physics at Northwestern University, and also, before that, part of a conference on homotopy theory they had there. Actually these two subjects are closely related: homotopy theory is a highly algebraic way of studying the topology of spaces of various dimensions, and lots of what we understand about "higher dimensional algebra" comes from homotopy theory. So it was a nice combination.

Lots of the homotopy theory was over my head, alas, but what I understood I enjoyed. It may seem sort of odd, but the main thing I got out of the homotopy theory conference was an explanation of why the number 24 is so important in string theory! In bosonic string theory spacetime needs to be 26-dimensional, but subtracting 2 dimensions for the surface of the string itself we get 24, and it turns out that it's really the special properties of the number 24 that make all the magic happen.

I began to delve into these mysteries in "week95". There, however, I was mainly reporting on very fancy stuff that I barely understand, stuff that seems like a pile of complicated coincidences. Now, I am glad to report, I am beginning to understand the real essence of this 24 business. It turns out that the significance of the number 24 is woven very deeply into the basic fabric of mathematics. To put it rather mysteriously, it turns out that every integer has some subtle "hidden symmetries". These symmetries have symmetries of their own, and in turn THESE symmetries have symmetries of THEIR own - of which there are exactly 24.

Hmm, mysterious. Let me put it another way. It probably won't be obvious why this is another way of saying the same thing, but it has the advantage of being more concrete. Suppose that the integer n is sufficiently large - 4 or more will do. Then there are 24 essentially different ways to wrap an (n+3)-dimensional sphere around an n-dimensional sphere. More precisely still, given two continuous functions from an (n+3)-sphere to an n-sphere, let's say that they lie in the same "homotopy class" if you can continuously deform one into another. Then when n is 4 or more, it turns out that there are exactly 24 such homotopy classes.

Now that I have all the ordinary mortals confused and all the homotopy theorists snickering at me for making such a big deal out of something everyone knows, I should probably go back and explain what the heck I'm getting at, and why it has to do with string theory. But I'm getting worn out, and your attention is probably flagging, so I'll do this next time. I'll say a bit about homotopy theory, stable homotopy theory, the sphere spectrum, and why Andre Joyal says we should call the sphere spectrum the "integers" (thus explaining my mysterious remark above).

Deep, deep infinity! Quietness. To dream away from the tensions of daily living; to sail over a calm sea at the prow of a ship, toward a horizon that always recedes; to stare at the passing waves and listen to their monotonous soft murmur; to dream away into unconsciousness.... - Maurits Escher