December 5, 1999

This Week's Finds in Mathematical Physics (Week 142)

John Baez

I was recently infected by a meme - a self-propagating pattern of human behavior. Now I want to pass it on to you! I like this particular meme because it's so simple. It's even simpler than the parasites described on my webpage:

1) John Baez, Subcellular life forms, http://math.ucr.edu/home/baez/subcellular.html

I wrote this webpage when I was trying to understand some of the simplest self-reproducing entities: viruses, viroids, virusoids, plasmids, prions, and various forms of junk DNA. Viroids are especially simple. Unlike a virus, a viroid doesn't even have a protein coat: it's just a naked RNA molecule! So instead of actively breaking into the host cell, it must passively wait to be absorbed. Then somehow it hijacks the machinery of the cell nucleus to reproduce itself. Theodore Diener discovered the first viroid in 1971: the potato spindle tuber viroid, which makes potatoes abnormally long and sometimes cracked. At first people doubted the possibility of a life form smaller than a virus. But by now the complete molecular structure of this viroid has been worked out. It consists of only 359 nucleotides - or in other words, about 12,000 atoms!

But since a meme relies on the complex apparatus of human culture to reproduce itself, it can get away with being even simpler than a viroid. It can even be the simplest sort of thing of all: an abstract mathematical structure defined by a short list of axioms!

A good example is the game of tic-tac-toe. It's not very interesting, but it's just interesting enough to keep propagating itself through human children, who are highly susceptible to the charm of simple games. Most children soon develop an immunity to tic-tac-toe, just like measles and mumps - but only after passing it on to some other child.

Unfortunately, the meme that infected me is a lot harder to shake, because it's a lot more interesting. I'm talking about the game of Go.

This game is played on a 19 x 19 square grid. Each player starts with a large supply of stones - black for the first player, white for the second. They take turns putting a stone on a grid point. A group of stones of one color "dies" and is removed from the board when it is surrounded by stones of the other color. More precisely, we say a stone is "dead" when none of its nearest neighbors of the same color have nearest neighbors of the same color which have nearest neighbors of the same color which... have nearest neighbors that are still vacant grid points.

There are also two subsidary rules, designed to keep silly things from happening.

First, you are not allowed to put a stone someplace where it will immediately die, unless doing so immediately kills one or more of the other player's stones - in which case their stones die, and yours lives.

Second, if putting down your stone kills a stone of the other player, but they could immediately put that stone back and kill yours, leading to an infinite loop, we say that "ko" has occurred. In this case, the other player is forbidden from putting their stone back right away.

How do you win? Simply put, the goal is to end up with as much "territory" as possible. Territory includes grid points occupied by stones of your color, and also vacant grid points that the other player could not occupy without their stones eventually dying. (In practice, Go players do not fight to the bitter end, so territory also includes stones of the other color that are "doomed to die".)

That's basically it!

The cool part is that starting from these simple rules, a whole world of strategy unfolds, full of specific tricks - but also quite general philosophical lessons about "power", "territory", and "threat". In a good game, both players start by efficiently marking out some territory, putting stones down in a widely separated way that looks random to the beginner, but in fact is delicately balanced between being too conservative and too ambitious. The midgame starts when both players start trying to surround each other and threaten to kill stones. But be careful: threatening to kill stones can be better than actually killing them, and the difference between "surrounding" and "being surrounded" is rather subtle! The endgame comes when territory is almost fully demarcated, with only a few squabbles around the edges. The endgame game proves unexpectedly difficult for beginners, since one can snatch defeat from the jaws of victory even at this stage.

A well-developed Go game is said to be like a work of art, with all opposing forces neatly balanced in a harmonious pattern. As a mathematical physicist, it reminds me of the Ising model at a phase transition, when there are as many black grid points as white ones, and arbitrarily large clusters of both colors. Perhaps there's even a real relation to the theory of "self-organized criticality", in which a system spontaneously works its way to the brink of a phase transition.

People say Go was developed in China between 4 and 6 thousand years ago. Its early history is obscure, but it is said to have started, not as a game, but as a tool for divination and the teaching of military strategy. I'm no expert, but to me Go seems like a nice illustration of yin-yang philosophy - the idea that the dynamic complexity of the universe arises from the dialectic interplay of binary opposites. For a good introduction to what I'm talking about, you can't beat the I Ching - the "Classic of Changes", a Chinese divination text compiled in the 9th century B.C., but containing material that probably dates back at least a few centuries earlier. This book describes the significance of 64 "hexagrams", which are patterns built from 6 bits of information, like this:

                        ______
                        __  __
                        ______
                        ______
                        __  __
                        __  __

In fact, I bet some you are smirking and wondering why I didn't learn Go much earlier! The reason is that I've always avoided playing games, except for the "great game" of mathematical physics. I only tried playing Go the weekend before last, while visiting my friend Bruce Smith up in San Rafael after giving a talk on quantum tetrahedra at Stanford. Bruce explained Go to me and showed me how it was philosophically interesting. But most importantly, he showed me a computer program that plays Go. Computers aren't great at Go, but they're good enough to beat an amateur like me, so they're good to learn from at first, and for some reason I prefer to play a computer than another person - perhaps because computers don't gloat.

The computer program I played against is called "GNU Go". You can download it free from the internet, thanks to the Free Software Foundation:

2) GNU Go, http://www.gnu.org/software/gnugo/devel.html

You can adjust the size of the board and also the handicap - the number of stones you get right away when you start. To use this program in a UNIX environment you need an interface program called "cgoban", which is also free:

3) CGoban, http://www.inetarena.com/~wms/comp/cgoban/

On Windows you can use an interface available from the GNU Go webpage.

For more information on Go start here:

4) American Go Association, http://www.usgo.org/resources/

You can find lots of go books listed at this website. Personally I found these books to be a nice introduction to the game, but they may be hard to find:

5) The Nihon Kiin, Go: The World's Most Fascinating Game, 2 volumes, Sokosha Printing Co., Tokyo, 1973.

When you get more advanced, there are a lot of books to read, with fun titles like "Get Strong at Invading", "Reducing Territorial Frameworks", and "Utilizing Outward Influence". It pays to study "joseki", or openings:

6) Ishida Yoshio, Dictionary of Basic Joseki, 3 volumes, Ishi Press International, San Jose, California, 1977.

It's also good to study "tsume-go", or "life and death problems", where you figure out which player can win in various configurations. A mathematician would call this the "local" analysis of Go:

7) Cho Chikun, All About Life and Death, 2 volumes, Ishi Press International, San Jose, California, 1993.

Ishi Press puts out a lot of other books on Go, but I haven't been able to get ahold of them yet. I'm sort of fascinated by one that talks about a difficult abstract concept called "thickness", since I suspect this is a global rather than local concept:

8) Ishidea Yoshio, All About Thickness: Understanding Moyo and Influence, Ishi Press International, San Jose, California.

If you want to get mathematical about Go endgames, try this:

9) Elwyn Berlekamp and David Wolfe, Mathematical Go: Chilling Gets the Last Point, A. K. Peters, Wellesley Massachusetts, 1994.

If you want to get computational, try this:

10) Markus Enzenberger, The integration of a priori knowledge into a Go playing neural network, http://www.cgl.ucsf.edu/go/Programs/neurogo-html/NeuroGo.html

If instead you prefer to curl up with a good novel based on a game of Go, try this:

11) Yasunari Kawabata, The Master of Go, trans. Edward G. Seidensticker, Knopf, New York, 1972.

On a different note, here are two good editions of the I Ching:

12) The I Ching or Book of Changes, trans. Richard Wilhelm and Cary F. Baynes, Princeton U. Press, Princeton, 1969.

The Classic of Changes: A New Translation of the I Ching as Interpreted by Wang Bi, trans. Richard John Lynn, Columbia U. Press, 1994.

Okay. Enough culture - time for some math!

I was invited to Stanford University by David Carlton, who works on modular forms, and I found out from him and his friends that the Shimura-Taniyama-Weil conjecture has been proved! This might have been a nice scoop for This Week's Finds, but by now it's appeared in the Notices of the AMS, so everyone knows about it:

13) Henri Darmon, A proof of the full Shimura-Taniyama-Weil conjecture is announced, Notices of the American Mathematical Society, 46 no. 11 (December 1999), 1397-1401.

Andrew Wiles proved part of this conjecture in order to prove Fermat's Last Theorem, but the conjecture is actually much more interesting than Fermat's Last Theorem, and a proof of the whole thing was announced this summer by Breuil, Conrad, Diamond and Taylor.

What does the conjecture say?

Well, first you have to know a bit about elliptic curves. An "elliptic curve" is the space of solutions of an equation like this:

y² = x³ + ax + b

They come up naturally in string theory, and I've talked about them already in "week13" and "week124" - "week127". If all the variables in sight are complex numbers, an elliptic curve looks like a torus, but number theorists like to consider the case where the coefficients a and b are rational. By a simple change of variables you can then get the coefficients to be integers. Then it makes sense to work modulo a prime number p: in other words, to think of all the variables as living in the field of integers mod p, better known as Z/p. If you're smart, you can tell if an elliptic curve mod p is "singular" or not: being nonsingular is like being a smooth manifold. People say an elliptic curve has "good reduction at p" if it's nonsingular mod p. For any given elliptic curve, this is true except for finitely many primes.

Any elliptic curve E has finitely many points mod p. Let's call the number of points N(E,p) and set

a(E,p) = p - N(E,p)

If this list of numbers satisfies a certain condition, which I'll describe in a minute, we say our elliptic curve is "modular". The Shimura-Taniyama-Weil conjecture states that all elliptic curves are modular.

Okay, so what does "modular" mean? Well, for this we need a little digression on modular forms. In "week125" I described the moduli space of elliptic curves, which is the space of all different shapes an elliptic curve can have. I showed that this space was H/SL(2,Z), where H is the upper half of the complex plane and SL(2,Z) is the group of 2x2 integer matrices with determinant 1. A modular form is basically just a holomorphic section of some line bundle over the moduli space of elliptic curves. But if this sounds too high-tech, don't be scared! We can also think of it as an analytic function on the upper half-plane that transforms in a nice way under the action of SL(2,Z). Remember, any matrix

                          a b
                          c d

τ |→ (a τ + b)/(c τ + d)

For an analytic function f: H → C to be a "modular form of weight k", it must transform as follows:

f((a τ + b)/(c τ + d)) = (c τ + d)^k f(τ)

for some integer k. We also require that f satisfy some growth conditions as τ → ∞, so we can expand it as a Taylor series

f(τ) = ∑ a_n qⁿ

where

q = exp(2 π i τ)

is a variable that equals 0 when τ = ∞. The nicest modular forms are the "cusp forms", which have a₀ = 0, and thus vanish at τ = ∞.

Next, we can straightforwardly generalize everything I just said if we replace SL(2,Z) by various subgroups thereof. (This amounts to studying holomorphic sections of line bundles over some moduli space of elliptic curves equipped with extra structure.) For example, we can use the subgroup Γ₀(N) consisting of those matrices in SL(2,Z) whose lower-left entries are divisible by N. If we use this group instead of SL(2,Z), we get what are called modular forms of "level N". We define "weight" of such a modular form just as before, and ditto for "cusp forms".

And now we can say what it means for an elliptic curve to be modular! We say an elliptic curve E is "modular" if for some N there's a weight 2 level N cusp form

f(τ) = ∑ a_n qⁿ

normalized so that a₁ = 1, with the property that

a_p = a(E,p)

for all primes p at which E has good reduction.

So now you know what the Shimura-Taniyama-Weil conjecture says: all elliptic curves are modular! It's not obvious that this implies Fermat's Last Theorem, but it does, thanks to a trick invented by Gerhard Frey.

There turn out to be fascinating but mysterious relationships between the Shimura-Taniyama-Weil conjecture, something called the Langlands program, and topological quantum field theory:

14) Mikhail Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, in Functional Analysis on the Eve of the 21st Century, Vol. 1, Birkhaueser, Boston, pp. 119-151.

For this reason - and others - it's not so surprising that David Carlton and some of his buddies are interested in n-categories. In fact, Carlton caught a small error in the definition of n-categories due to James Dolan and myself - it turns out that the number "1" should be the number "2" at one particular place in the definition! Anyone who can spot a problem like that is friend of mine.

Even better, Carlton is now interested in understanding the (n+1)-category of all n-categories, which is crucial for really doing anything with n-categories. Makkai has a new paper on this subject, and I realize now that I've never mentioned this paper on This Week's Finds, so let me conclude by quoting the abstract. It's pretty long and detailed, and probably only of interest to n-category addicts....

15) M. Makkai, The multitopic ω-category of all multitopic ω-categories, preprint available at ftp://ftp.math.mcgill.ca/pub/makkai

"The paper gives two definitions: that of "multitopic ω-category" and that of "the (large) multitopic set of all (small) multitopic ω-categories". It also announces the theorem that the latter is a multitopic ω-category. (The proof of the theorem will be contained in a sequel to this paper.)

The work has two direct sources. One is the paper [H/M/P] (for the references, see at the end of this abstract) in which, among others, the concept of "multitopic set" was introduced. The other is the present author's work on FOLDS, First Order Logic with Dependent Sorts. The latter was reported on in [M2]. A detailed account of the work on FOLDS is in [M3]. For the understanding of the present paper, what is contained in [M2] suffices. In fact, section 1 of the present paper gives the definitions of all that's needed in this paper; so, probably, there won't be even a need to consult [M2].

The concept of multitopic set, the main contribution of [H/M/P], was, in turn, inspired by the work of J. Baez and J. Dolan [B/D]. Multitopic sets are a variant of opetopic sets of loc. cit. The name "multitopic set" refers to multicategories, a concept originally due to J. Lambek [L], and given an only moderately generalized formulation in [H/M/P]. The earlier "opetopic set" of [B/D] is based on a concept of operad. I should say that the exact relationship of the two concepts ("multitopic set" and "opetopic set") is still not clarified. The main aspect in which the theory of multitopic sets is in a more advanced state than that of opetopic sets is that, in [H/M/P], there is an explicitly defined category Mlt of multitopes, with the property that the category of multitopic sets is equivalent to the category of Set-valued functors on Mlt, a result given a detailed proof in [H/M/P]. The corresponding statement on opetopic sets and opetopes is asserted in [B/D], but the category of opetopes is not described. In this paper, the category of multitopes plays a basic role.

Multitopic sets and multitopes are described in section 2 of this paper; for a complete treatment, the paper [H/M/P] should be consulted.

The indebtedness of the present work to the work of Baez and Dolan goes further than that of [H/M/P]. The second ingredient of the Baez/Dolan definition, after "opetopic set", is the concept of "universal cell". The Baez/Dolan definition of weak n-category achieves the remarkable feat of specifying the composition structure by universal properties taking place in an opetopic set. In particular, a (weak) opetopic (higher-dimensional) category is an opetopic set with additional properties ( but with no additional data), the main one of the additional properties being the existence of sufficiently many universal cells. This is closely analogous to the way concepts like "elementary topos" are specified by universal properties: in our situation, "multitopic set" plays the "role of the base" played by "category" in the definition of "elementary topos". In [H/M/P], no universal cells are defined, although it was mentioned that their definition could be supplied without much difficulty by imitating [B/D]. In this paper, the "universal (composition) structure" is supplied by using the concept of FOLDS-equivalence of [M2].

In [M2], the concepts of "FOLDS-signature" and "FOLDS-equivalence" are introduced. A (FOLDS-) signature is a category with certain special properties. For a signature L , an L-structure is a Set-valued functor on L. To each signature L, a particular relation between two variable L-structures, called L-equivalence, is defined. Two L-structures M, N, are L-equivalent iff there is a so-called L-equivalence span M<---P--->N between them; here, the arrows are ordinary natural transformations, required to satisfy a certain property called "fiberwise surjectivity".

The slogan of the work [M2], [M3] on FOLDS is that *all meaningful properties of L-structures are invariant under L-equivalence*. As with all slogans, it is both a normative statement ("you should not look at properties of L-structures that are not invariant under L-equivalence"), and a statement of fact, namely that the "interesting" properties of L-structures are in fact invariant under L-equivalence. (For some slogans, the "statement of fact" may be false.) The usual concepts of "equivalence" in category theory, including the higher dimensional ones such as "biequivalence", are special cases of L-equivalence, upon suitable, and natural, choices of the signature L; [M3] works out several examples of this. Thus, in these cases, the slogan above becomes a tenet widely held true by category theorists. I claim its validity in the generality stated above.

The main effort in [M3] goes into specifying a language, First Order Logic with Dependent Sorts, and showing that the first order properties invariant under L-equivalence are precisely the ones that can be defined in FOLDS. In this paper, the language of FOLDS plays no role. The concepts of "FOLDS-signature" and "FOLDS-equivalence" are fully described in section 1 of this paper.

The definition of multitopic ω-category goes, in outline, as follows. For an arbitrary multitope SIGMA of dimension >=2, for a multitopic set S, for a pasting diagram ALPHA in S of shape the domain of SIGMA and a cell a in S of the shape the codomain of SIGMA, such that a and ALPHA are parallel, we define what it means to say that a is a composite of ALPHA. First, we define an auxiliary FOLDS signature L<SIGMA> extending Mlt, the signature of multitopic sets. Next, we define structures S<a> and S<ALPHA>, both of the signature L<SIGMA>, the first constructed from the data S and a , the second from S and ALPHA, both structures extending S itself. We say that a is a composite of ALPHA if there is a FOLDS-equivalence-span E between S<a> and S<ALPHA> that restricts to the identity equivalence-span from S to S . Below, I'll refer to E as an equipment for a being a composite of ALPHA. A multitopic set is a mulitopic ω-category iff every pasting diagram ALPHA in it has at least one composite.

The analog of the universal arrows in the Baez/Dolan style definition is as follows. A universal arrow is defined to be an arrow of the form b:ALPHA-----> a where a is a composite of ALPHA via an equipment E that relates b with the identity arrow on a : in turn, the identity arrow on a is any composite of the empty pasting diagram of dimension dim(a)+1 based on a . Note that the main definition does not go through first defining "universal arrow".

A new feature in the present treatment is that it aims directly at weak *ω*-categories; the finite dimensional ones are obtained as truncated versions of the full concept. The treatment in [B/D] concerns finite dimensional weak categories. It is important to emphasize that a multitopic ω-category is still just a multitopic set with additional properties, but with no extra data.

The definition of "multitopic ω-category" is given is section 5; it uses sections 1, 2 and 4, but not section 3.

The second main thing done in this paper is the definition of MltOmegaCat. This is a particular large multitopic set. Its definition is completed only by the end of the paper. The 0-cells of MltOmegaCat are the samll multitopic ω-categories, defined in section 5. Its 1-cells, which we call 1-transfors (thereby borrowing, and altering the meaning of, a term used by Sjoerd Crans [Cr]) are what stand for "morphisms", or "functors", of multitopic ω-categories. For instance, in the 2-dimensional case, multitopic 2-categories correspond to ordinary bicategories by a certain process of "cleavage", and the 1-transfors correspond to homomorphisms of bicategories [Be]. There are n-dimensional transfors for each n in N . For each multitope (that is, "shape" of a higher dimensional cell) PI, we have the PI-transfors, the cells of shape PI in MltOmegaCat.

For each fixed multitope PI, a PI-transfor is a *PI-colored multitopic set* with additional properties. "PI-colored multitopic sets" are defined in section 3; when PI is the unique zero-dimensional multitope, PI-colored multitopic sets are the same as ordinary multitopic sets. Thus, the definition of a transfor of an arbitrary dimension and shape is a generalization of that of "multitopic ω-category"; the additional properties are also similar, they being defined by FOLDS-based universal properties. There is one new element though. For dim(PI)>=2 , the concept of PI-transfor involves a universal property which is an ω-dimensional, FOLDS-style generalization of the concept of right Kan-extension (right lifting in the terminology used by Ross Street). This is a "right-adjoint" type universal property, in contrast to the "left-adjoint" type involved in the concept of composite (which is a generalization of the usual tensor product in modules).

The main theorem, stated but not proved here, is that MltOmegaCat is a multitopic ω-category.

The material in this paper has been applied to give formulations of ω-dimensional versions of various concepts of homotopy theory; details will appear elesewhere.

References:

[B/D] J. C. Baez and J. Dolan, Higher-dimensional algebra III. n-categories and the algebra of opetopes. Advances in Mathematics 135 (1998), 145-206.

[Be] J. Benabou, Introduction to bicategories. In: Lecture Notes in Mathematics 47 (1967), 1-77 (Springer-Verlag).

[Cr] S. Crans, Localizations of transfors. Macquarie Mathematics Reports no. 98/237.

[H/M/P] C. Hermida, M. Makkai and J. Power, On weak higher dimensional categories I. Accepted by: Journal of Pure and Applied Algebra. Available electronically (when the machines work ...).

[L] J. Lambek, Deductive systems and categories II. Lecture Notes in Mathematics 86 (1969), 76-122 (Springer-Verlag).

[M2] M. Makkai, Towards a categorical foundation of mathematics. In: Logic Colloquium '95 (J. A. Makowski and E. V. Ravve, editors). Lecture Notes in Logic 11 (1998) (Springer-Verlag).

[M3] M. Makkai, First Order Logic with Dependent Sorts. Research monograph, accepted by Lecture Notes in Logic (Springer-Verlag). Under revision. Original form available electronically (when the machines work ...).