
Well, folks, this'll be the last "This Week's Finds" for a while, since I'm getting rather busy preparing for my conference on knots and quantum gravity, and I have a paper that seems to be taking forever to finish.
1) Elliptic Curves by Anthony W. Knapp, Mathematical Notes, Princeton University Press, 1992.
This is a shockingly userfriendly introduction to a subject that can all too easily seem intimidating. I'm certainly no expert but maybe just for that reason I should sketch a brief "introduction to the introduction" that may lure some of you into studying this beautiful subject.
What I will say will perhaps appeal to people who like complex analysis or mathematical physics, but Knapp concentrates on the aspects related to number theory. For other approaches one might try
2) Elliptic Functions by Serge Lang, SpringerVerlag, 2nd edition, 1987.
3) Elliptic Curves by Dale Husemoeller, SpringerVerlag, 1987.
Okay, where to start? Well, how about this: the sine function is an analytic function on the complex plane with the property that
sin(z + 2π) = sin z
It also satisfies a nice differential equation
(sin' z)^{2} = 1  (sin z)^{2}
and for this reason, we could, if we hadn't noticed the sine function otherwise, have run into it when we tried to integrate
(1  u^{2})^{1/2}
The differential equation above implies that the integral is nice to do by the substitution u = sin z, and we get the answer arcsin u. If the sine function  or more generally, trig functions  didn't exist yet, we would have invented them when we tried to do integrals involving square roots of quadratic polynomials.
Elliptic functions are a beautiful generalization of all of this stuff. Say we wanted, just for the heck of it, an analytic function that was periodic not just in one direction on the complex plane, like the sine function, but in two directions. For example, we might want some function P(z) with
P(z + 2π) = P(z)
and also
P(z + 2πi) = P(z)
This function would look just the same on each 2πby2π square:
x x x x x x x x x x x x x x xso if we wanted, we could think of it as being a function on the torus formed by taking one of these squares and identifying its top side with its bottom side, and its left side with its right side.
More generally  while we're fantasizing about this wonderful doublyperiodic function  we could ask for one that was periodic in any old two directions. That is, fixing two numbers ω_{1} and ω_{2} that aren't just realvalued multiples of each other, we could hope to find an analytic function on the complex plane with ω_{1} and ω_{2} as periods:
P(z + ω_{1}) = P(z)
P(z + ω_{2}) = P(z).
Then P(z) would be the same at all points on the "lattice" of points n ω_{1} + m ω_{2} which might look like the square above or might be like
x x x x x x x x x x x x x x x x x x x xor some such thing.
Let's think about this nice function P(z) we are fantasizing about. Alas, if it were analytic on the whole plane (no poles), it would be bounded on each little parallelogram, and since it's doubly periodic, it would be a bounded analytic function on the complex plane, hence CONSTANT by Liouville's theorem. Well, so a constant function has all the wonderful properties we want  but that's too boring!
So let's allow it to have poles! But let's keep it as nice as possible, so let's have the only poles occur at the lattice points
L = {n ω_{1} + m ω_{2}}
And let's make the poles as nice as possible. Can we have each pole be of order one? That is, can we make P(z) blow up like 1/(z  ω) at each lattice point ω in L? No, because if it did, the integral of P around a nicely chosen parallelogram around the pole would be zero, because the contributions from opposite sides of the parallelogram would cancel by symmetry. (A fun exercise.) But by the Cauchy residue formula this means that the residue of the pole vanishes, so it can't be of order one.Okay, try again. Let's try to make the pole at each lattice point be of order two. How can we cook up such a function? We might try something obvious: just sum up, for all ω in the lattice L, the functions
1/(z  ω)^{2}
We get something periodic with poles like 1/(z  ω)^{2} at each lattice point ω. But there's a big problem  the sum doesn't converge! (Another fun exercise.)
Oh well, try again. Let's act like physicists and RENORMALIZE the sum by subtracting off an infinite constant! Just subtract the sum over all ω in L of 1/ω^{2}. Well, all ω except zero, anyway. This turns out to work, but we really should be careful about the order of summation here: really, we should let P(z) be 1/z^{2} plus the sum for all nonzero ω in the lattice L of 1/(z  ω)^{2}  1/ω^{2}. This sum does converge and the limit is a function P(z) that's analytic except for poles of order two at the lattice points. This is none other than the Weierstrass elliptic function, usually written with a fancy Gothic P to intimidate people. Note that it really depends on the two periods ω_{1} and ω_{2}, not just z.
Now, it turns out that P(z) really is a cool generalization of the sine function. Namely, it satisfies a differential equation like the one the sine does, but fancier:
P'(z)^{2} = 4 P(z)^{3}  g_{2} P(z)  g_{3}
where g_{2} and g_{3} are some constants that depend on the periods ω_{1} and ω_{2}. Just as with the sine function we can use the inverse of Weierstrass P function to do some integrals, but this time we can do integrals involving square roots of cubic polynomials! If you look in big nasty books of special functions or tables of integrals, you will see that there's a big theory of this kind of thing that was developed in the 1800's  back when heavyduty calculus was hip.
There are, however, some other cool ways of thinking about what's going on here. First of all, remember that we can think of P(z) as a function on the torus. We can think of this torus as being "coordinatized"  I use the word loosely  by P(z) and its first derivative P'(z). I.e., if we know x = P(z) and y = P'(z) we can figure out where the point z is on the torus. But of course x and y can't be any old thing; the differential equation above says they have to satisfy
y^{2} = 4x^{3}  g_{2} x  g_{3}
Here x and y are complex numbers of course. But look what this means: it means that if we look at the pairs of complex numbers (x,y) satisfying the above cubic equation, we get something that looks just like a torus! This is called an elliptic curve, since for algebraic geometers a "curve" is the set of solutions (x,y) of some polynomial in two complex variables  not two real variables.
So  an "elliptic curve" is basically just the solutions of a cubic equation in two variables. Actually, we want to rule out curves that have singularities, that is, places where there's no unique tangent line to the curve, as in y^{2} = x^{3} or y^{2} = x^{2}(x+1)  draw these in the real plane and you'll see what I mean. Anyway, all elliptic curves can, by change of variables, be made to look like our favorite one,
y^{2} = 4x^{3}  g_{2} x  g_{3}
There are lots of more fancy ways of thinking about elliptic curves, and one is to think of the fact that they look like a torus as the key part. In a book on algebraic geometry you might see an elliptic curve as a curve with genus one (i.e., with one "handle," like a torus has). One nice thing about a torus is that is a group. That is, we know how to add complex numbers, and we can add modulo elements of the lattice L, so the torus becomes a group with addition mod L as the group operation. This is simple enough, but it means that when we look at the solutions of
y^{2} = 4x^{3}  g_{2} x  g_{3}
they must form a group somehow, and viewed this way it's not at all obvious! Nonetheless, there is a beautiful geometric description of the group operation in these terms  I'll leave this for Knapp to explain..
Let me wrap this up  the story goes on and on, but I'm getting tired  with a bit about what it has to do with number theory. It has a lot to do with Diophantine equations, where one wants integer, or rational solutions to a polynomial equation. Suppose that g_{2} and g_{3} are rational, and one has some solutions to the equation
y^{2} = 4x^{3}  g_{2} x  g_{3}
Then it turns out that one can use the group operation on the elliptic curve to get new solutions! Actually, it seems as if Diophantus knew this way back when in some special cases. For example, for the problem
y(6  y) = x^{3}  x
Diophantus could start with the trivial solution (x,y) = (1,0), do some mysterious stuff, and get the solution (17/9,26/27). Knapp explains how this works in the Overview section, but then more deeply later. Basically, it uses the fact that this curve is an elliptic curve, and uses the group structure.
In fact, one can solve mighty hardseeming Diophantine problems using these ideas. Knapp talks a bit about a problem Fermat gave to Mersenne in 1643  this increased my respect for Fermat a bit. He asked, find a Pythagorean triple (X,Y,Z), that is:
X^{2} + Y^{2} = Z^{2},
such that Z is a square number and X + Y is too! One can solve this using elliptic curves. I don't know if Mersenne got it  the answer is at the end of this post, but heavyduty number theorists out there might enjoy trying this one if they don't know it already.
Some more stuff:
4) Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli spaces, by Jim Stasheff, preprint available as hepth/9304061.
One conceptually pleasing approach to string theory is closed string field theory, where one takes as the basic object unparametrized maps from circle into a manifold M representing "space", i.e., elements of
Maps(S^{1},M)/Diff^{+}(S^{1}).
A state of closed string field theory would be roughly a function on the above set. Then one tries to define all sorts of operations on these states, in order to define write down ways the strings can interact. For example, there is a "convolution product" on these functions which almost defines a Lie algebra structure. However, the Jacobi identity only holds "up to homotopy," so we have an algebraic structure called a homotopy Lie algebra. Physicists would say that the Jacobi identity holds modulo a BRST exact term. This is just the beginning of quite a big bunch of mathematics being developed by Stasheff, Zwiebach, Getzler, Kapranov and many others. My main complaint with the physics is that all these structures seem to depend on choosing a Riemannian metric on M  a socalled "background metric." Since string theory is supposed to include a theory of quantum gravity it is annoying to have this Godgiven background metric stuck in at the very start. Perhaps I just don't understand this stuff. I am looking around for stuff on backgroundindependent closed string field theory, since I have lots of reason to believe that it's related to the loop representation of quantum gravity. Unfortunately, I scarcely know the subject  I had hoped Stasheff's work would help me, but it seems that this metric always enters.
5) A geometrical presentation of the surface mapping class group and surgery, by Sergey Matveev and Michael Polyak, preprint.
This paper shows how to express the mapping class group of a surface in terms of tangles. This gives a nice relationship between two approaches to 3d TQFTs (topological quantum field theories): the Heegard decomposition approach, and the surgery on links approach.
6) Invariants of 3manifolds and conformal field theories, by Micheal Polyak, preprint.
The main good thing about this paper in my opinion is that it simplifies the definition of a modular tensor category. Recall that Moore and Seiberg showed how any string theory (more precisely, any rational conformal field theory) gave rise to a modular tensor category, and then Crane showed that any modular tensor category gave rise to a 3d TQFT. Unfortunately a modular tensor category seems initially to be a rather baroque mathematical object. In this paper Polyak shows how to get lots of the structure of a modular tensor category from just the "fusion" and "braiding" operators, subject to some mild conditions. I have a conjecture that all nonnegative link invariants (in the sense of my paper on tangles and quantum gravity) give rise to modular tensor categories, and this simplifies things to the point where maybe I might eventually be able to prove it. There are lots of nice pictures here, too, by the way.
X = 1061652293520
Y = 4565486027761
Z = 4687298610289
© 1993 John Baez
baez@math.removethis.ucr.andthis.edu
