This Is the Blog That No One Reads...

Tuesday, January 03, 2012

The Aronhold Method, Polarization
2.1 Polarizations
Before proceeding, let us recall, in a language suitable for our purposes, the usual Taylor-Maclaurin expansion. Consider a function $ F(x)$ of a vector variable $ x \in V $. Under various types of assumptions we have a development for the function $ F(x+y) $ of two vector variables. For our purposes, we may restrict our considerations to polynomials and develop
$$F(x + y) := \sum^{i = 0}_\infty F_i(x,y), $$ where by definition $ F_i (x, y) $ is homogeneous of degree $ i $ in $ y $ (of course for polynomials the sum is really finite). herefore, for any value of a parameter $ \lambda $, we have $ F(x + \lambda y) := \sum^{i = 0}_\infty \lambda^i F_i(x,y) $. If F is also homogeneous of degree $ k $ we have
$$ \begin{align}
\sum^{i = 0}_\infty \lambda^k F_i(x,y) &= \lambda^k F(x+y) = F(\lambda (x + y)) = \\
& F(\lambda x + \lambda y) = \sum^\infty_{i=0} \lambda^i f_i (\lambda x, y)
\end{align} $$
and we deduce that $ F_i(x, y) $ is also homogeneous of degree $ k - i $ in $ x $.

Sunday, January 01, 2012

how to write

To this end, I have christened
all statements (theorems, examples, definitions, etc.) and basic equations with
a proper name (using capital letters as with ordinary proper names). Instead
of saying “by Lemma 21.2.1(1), which of course you will remember,” I say
“by Nuclear Slipping 21.2.1(1),” hoping to trigger long-repressed memories of
a formula for how nuclear elements of alternative algebras slip in and out of
associators.

cokernel infinite case!

from wikipedia CokernelMain article: CokernelA subtler invariant of a linear transformation is the cokernel, which is defined as
$\mathrm{coker}\,f := W/f(V) = W/\mathrm{im}(f).$
This is the dual notion to the kernel: just as the kernel is a subspace of the domain, the co-kernel is a quotient space of the target. Formally, one has the exact sequence $0 \to \ker f \to V \to W \to \mathrm{coker}\,f \to 0.$
These can be interpreted thus: given a linear equation $f (v) = w$ to solve,
the kernel is the space of solutions to the homogeneous equation $f (v) = 0,$ and its dimension is the number of degrees of freedom in a solution, if it exists;

the co-kernel is the space of constraints that must be satisfied if the equation is to have a solution, and its dimension is the number of constraints that must be satisfied for the equation to have a solution.

The dimension of the co-kernel and the dimension of the image (the rank) add up to the dimension of the target space. For finite dimensions, this means that the dimension of the quotient space $W / f (V)$ is the dimension of the target space minus the dimension of the image.
As a simple example, consider the map $f\colon \mathbf{R}^2 \to \mathbf{R}^2,$ given by $f (x, y) = (0, y).$ Then for an equation $f (x, y) = (a, b)$ to have a solution, we must have $a = 0$ (one constraint), and in that case the solution space is $(x, b),$ or equivalently stated, $(0, b) + (x,0),$ (one degree of freedom). The kernel may be expressed as the subspace $(x,0) < V :$ the value of x is the freedom in a solution – while the cokernel may be expressed via the map $W \to \mathbf{R}^1, (a,b) \mapsto (a):$ given a vector $(a, b),$ the value of a is the obstruction to there being a solution.
An example illustrating the infinite-dimensional case is afforded by the map $g\colon \mathbf{R}^\infty \to \mathbf{R}^\infty, \{a_n\} \mapsto \{b_n\}$ with $b 1 = 0$ and $b n + 1 = a n$ for $n > 0$ . Its image consists of all sequences with first element 0, and thus its cokernel consists of the classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its co-kernel has dimension 1. Since the domain and the target space are the same, the rank and the dimension of the kernel add up to the same sum as the rank and the dimension of the co-kernel ( $\aleph_0 + 0 = \aleph_0 + 1$ ), but in the infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an endomorphism have the same dimension ( $0 \neq 1$ ). The reverse situation obtains for the map $h\colon \mathbf{R}^\infty \to \mathbf{R}^\infty, \{a_n\} \mapsto \{c_n\}$ with $c n = a n + 1$ . Its image is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1.

the seekers

In a manner of speaking, the legacy of renunciation of philosophy and methodology led much of the orthodox economics profession to behave in ways rather similar to the Seekers from 2008 onwards. The parallels between the Seekers and the contemporary economics profession are, of course, not exact. The Seekers were disappointed when their world didn’t come to an end; economists were convinced their Great Moderation and neoliberal triumph would last forever, and were disappointed when it did appear to come to an end. The stipulated turning point never arrived for the Seekers, while the unsuspected turning point got the drop on the economists. The Seekers garnered no external support for their doctrines, indeed, quitting their jobs and contracts prior to their Fated Day; the economists, on the other hand, persist in being richly rewarded by many constituencies for remaining stalwart in their beliefs. The public press was never friendly towards the Seekers; it only turned on the economists with the financial collapse. (There are already signs it may be reverting to its older slavish adoration, however.) But nonetheless, the shape of the reactions to cognitive dissonance was amazingly similar. The crisis, which at first blush might seem to have refuted most everything that the economic orthodoxy believed in, was in the fullness of time more often than not trumpeted from both the Left and the Right as reinforcing their adherence to neoclassical economic theory. Thus was made manifest the ‘spontaneous methodology of the economics profession’.

Friday, December 30, 2011

baker

via baker we get the interesting fact that
the number of numbers less than n divisible by prime p is given by [n/p], where [] is the floor function.
since p is prime, only members of the progression {k.p} are divisible by p, and there are [n/p] of these, since p goes into n n/p times, and we can ignore the fractional part. for example, 3 goes into 10 ten thirds times, but this means that 3 numbers (3, 6, 9) less than ten are divisible by three.

this can be extended to [n/ p^2] which is the number divisible by p squared, and so on.

This tells us that the p-component of n factorial, let's call it l, is the sum over j of [n / p to the j] over all j.

this leads to a proof thatthe binomial coefficient is an integer....

spectrum

December 8, 2003
This Week's Finds in Mathematical Physics (Week 199)
John Baez
I've had a really busy quarter, teaching 3 courses that all require serious thought on my part, so it's been a long while since I've been able to write an issue of This Week's Finds. But, back in September I went to a conference on homotopy theory and its applications at the University of Western Ontario, run by Dan Christensen and Rick Jardine. There were some really cool talks at this conference - my favorite was one by Jack Morava about elliptic cohomology, and I'm really sorry I missed his lectures on Galois theory, since I've been studying that lately. But, instead of trying to describe the talks, I think it would be better if I said a bit about "spectra", which are an important tool in homotopy theory.
The word "spectrum" has a lot of different meanings in mathematics and physics. In experimental physics it refers to the frequencies of light, sound or any other sort of wave emitted by an object. For example, if you send the light emitted by hydrogen through a spectrometer, you'll see a bunch of sharp lines at specific frequencies - the "discrete" spectrum" - along with a diffuse glow at all frequencies - the "continuous spectrum". The German high school teacher Balmer noticed that the sharp lines correspond to light with frequencies proportional to
1/n² - 1/m²
where n,m = 1,2,3,...These days, in theoretical physics the "spectrum" of something is the set of frequencies at which it can vibrate - or in quantum theory, the set of energies it can have, since an energy is just a frequency times Planck's constant. For example, Bohr took Balmer's formula and realized that a hydrogen atom must have a discrete set of allowed energy levels
-1/n²
When the atom hops from one energy level to another, it emits or absorbs light with energy equal to the difference of two such numbers! This accounts for the discrete spectrum of light emitted by hydrogen. The atom can also have any positive energy, and this accounts for the continuous spectrum.In quantum mechanics, observables like energy are described as self-adjoint operators on a Hilbert space. The "spectrum" of an observable A is the set of values it's allowed to have, and mathematically this is the set of numbers x such that the operator A - x has no inverse. For example, if A is a "Hamiltonian", the operator that describes the energy of a quantum system, its spectrum is just the set of allowed energies! The simplest case is when x is an eigenvalue of A: the eigenvalues of an operator form its "discrete spectrum". But, there can also be numbers in the spectrum that aren't eigenvalues, and these form the "continuous spectrum".
In mathematical physics, people talk about the spectrum not just of one observable but of a whole bunch of commuting observables, since commuting observables can be measured simultaneously without the Heisenberg uncertainty principle kicking in to limit the precision. The nice way to think of the spectrum of a bunch of operators uses the concept of "C*-algebra". If we've got a bunch of bounded operators on a Hilbert space that's closed under addition, multiplication and scalar multiplication, closed under taking adjoints and also closed in the norm topology, it's called a "C*-algebra". The "spectrum" of a C*-algebra A is the set of all homomorphisms
x: A → C, 
where C is the complex numbers. Though it's not immediately obvious, this sort of spectrum reduces to the previous one when A is the C*-algebra of operators generated by a single self-adjoint operator. So, it's a nice way to define the spectrum of a whole bunch of observables. This generalization is not very useful when the C*-algebra is noncommutative, since then it may not have many homomorphisms to the complex numbers. But if it's commutative, we know everything about it once we know its spectrum!This amazing fact is called the Gelfand-Naimark theorem. Here's the idea. There's an easy way to make the spectrum of a commutative C*-algebra A into a topological space: we say x_i → x precisely when
x_i(a) → x(a)
for all elements a of A. With this topology any element a of A gives a continuous complex function on the spectrum, defined by this clever formula:a(x) = x(a).
The physicist Chris Isham says he couldn't sleep all night when he first saw this formula, it's so darn clever! And, it turns out that any continuous function on the spectrum comes from an element of A via this formula! So, if you hand me the spectrum Spec(A) of a commutative C*-algebra A, I can recover A (up to isomorphism) by forming the C*-algebra of all continuous functions on Spec(A).
As you can see, the concept of spectrum is getting more abstract - but it still has close ties to the original idea. What once was a bunch of lines on a spectrometer has now become a topological space associated to a commuting collection of observables. The idea is that each point in this space is a way of assigning values to all these observables... just like each line in the spectrometer represents a particular frequency of light!
But the abstraction process doesn't stop here. In algebraic geometry, people want to think of any commutative ring as consisting of functions on some sort of space. For example, the commutative ring of real polynomials in two variables mod the relation
x² + y² = 1
is just another way of thinking about polynomial functions on the circle. How do we get the circle back from this commutative ring? Simple: just form the space of all homomorphisms from it to the real numbers!
It would be nice to have a recipe to take any commutative ring A and extract a space from it: its "spectrum". As we've seen, one option is to take the spectrum to consist of all homomorphisms to the complex numbers:
x: A → C
Another would be to use the real numbers:
x: A → R.
Both the real and complex numbers are "fields": commutative rings where we can divide by anything nonzero. But there are a lot of other fields, like Z/p where p is any prime number. So, instead of picking one field, a more evenhanded approach is to use all possible fields, and say a homomorphism to any one of these should give a point of the spectrum.Actually, since there are zillions of fields out there, a more manageable option is to look not at the homomorphism itself but its kernel: the set of elements a in A with
x(a) = 0.
The kernel of a homomorphism from A to any other ring is an "ideal": a set closed under addition and also multiplication by all elements of A. Even better, the kernel of a homomorphism from A to a field is a "prime" ideal, meaning it's not not all of A, and whenever the product of two elements of A lies in the ideal, at least one of them must be in the ideal. Conversely, given a prime ideal in A, there's always a field k and a homomorphism
x: A → k 
whose kernel is that prime ideal. So, it's reasonable to define the spectrum of A, Spec(A) to be the set of all prime ideals in A.This turns out to exactly match the previous definition of spectrum when A is a C*-algebra. But why the word "prime"? Well, in the commutative ring of integers, Z, most prime ideals come from prime numbers. If we take all the multiples of any prime number, we get a prime ideal, which is the kernel of the obvious homomorphism
x: Z → Z/p
There's just one other prime ideal in Z, namely all the multiples of 0. In other words, the set consisting of just 0 alone! This is the kernel of the homomorphism from Z into the rationals. For some fascinating reason I'd rather not explain now, this prime ideal is often called "the prime at infinity". It's different from all the rest, but the wise know it's usually good to keep it in.So, the spectrum of the integers is just the set of ordinary primes together with the "prime at infinity":
Spec(Z) = {2, 3, 5, 7, 11, ... ∞}
We seem to have gotten pretty far from physics by now, but in fact many people believe that taking this spectrum seriously from a physical viewpoint will be crucial to proving the Riemann hypothesis - a famous open conjecture related to the distribution of prime numbers. I don't have time to do justice to this, but the basic idea goes as follows.Suppose we have a quantum system whose Hamiltonian has this spectrum:
{ln 2, ln 3, ln 5, ln 7, ln 11, ....}
We can think of these as energy states of some sort of particle: the "primon".Now let's second quantize this system. The idea of second quantization is that we form a new system consisting of an arbitrary finite collection of noninteracting indistinguishable copies of the original system. For example, if the original system was some sort of particle, a state of the new system would consist of an arbitrary number of particles of this sort, treated as identical bosons. If second quantize our "primon", we'll get a system with energy levels that are arbitrary sums of entries from the above list. If we write them in increasing order, they look like this:
{0, ln 2, ln 3, ln 2 + ln 2, ln 5, ln 2 + ln 3, ln 7, ln 2 + ln 2 + ln 2, ....}
or in other words, just
{ln 1, ln 2, ln 3, ln 4, ln 5, ln 6, ln 7, ln 8, ....}
since every whole number can be built from primons in a unique way! Bernard Julia calls this new system the "free Riemann gas", since it's made of noninteracting primons - and in a minute we'll see it's related to the Riemann hypothesis.To see this, let's do some statistical mechanics with the free Riemann gas! As usual, at any temperature T the probability that this system will be in a state of energy E is proportional to
exp(-βE)
where β = 1/kT and k is Boltzmann's constant. But to get these numbers to add up to one as probabilities should, we have to normalize them, dividing by their sum, which goes by the name of the "partition function". The partition function for the free Riemann gas is:
-β
 ∑ exp(-β ln n)   =   ∑  n
 n                    n
the so-called "Riemann zeta function". It's well-defined for β > 1 - that is, low temperatures - but it blows up when β = 1. This means that the free Riemann gas has a "Hagedorn temperature": a temperature that it can't go above, because doing so would take an infinite amount of energy.Nonetheless we can analytically continue the Riemann zeta function around β = 1, and the Riemann hypothesis says that it can only vanish if β is a negative even integer or a number with real part equal to 1/2. And, precisely because the free Riemann gas is made of primons, this hypothesis has a lot to do with prime numbers! For example, it's equivalent to the assertion that the number of primes less than x differs from
∞ 
Li(x) =  ∫  dt/ln t  
         2
by less than some constant times ln(x) √x.All this is lots of fun. I urge the physicist reader to compute the free energy and specific heat of the free Riemann gas, and also to investigate the system where we treat the primons as fermions. But, the big question is whether we can use physics-inspired reasoning to prove the Riemann hypothesis!
In 1995, a step in this direction was taken by Bost and Connes. I'm not ready to really explain it, so I'll just tantalize you by dangling their abstract in front of you:

In this paper, we construct a natural C*-dynamical system whose partition function is the Riemann zeta function. Our construction is general and associates to an inclusion of rings (under a suitable finiteness assumption) an inclusion of discrete groups (the associated ax + b groups) and the corresponding Hecke algebras of bi-invariant functions. The latter algebra is endowed with a canonical one-parameter group of automorphisms measuring the lack of normality of the subgroup. The inclusion of rings Z provides the desired C*-dynamical system, which admits the zeta function as partition function and the Galois group Gal(Q^cycl/ Q) of the cyclotomic extension Q^cycl of Q as symmetry group. Moreover, it exhibits a phase transition with spontaneous symmetry breaking at inverse temperature β = 1.
Here's the reference:1) J.-B. Bost and Alain Connes, "Hecke Algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory", Selecta Math. (New Series), 1 (1995) 411-457.
The idea of the free Riemann gas was introduced most clearly by Julia, though there were many precursors:
2) Bernard L. Julia, Statistical theory of numbers, in Number Theory and Physics, eds. J. M. Luck, P. Moussa, and M. Waldschmidt, Springer Proceedings in Physics, Vol. 47, Springer-Verlag, Berlin, 1990, pp. 276-293. Summarized by Matthew Watkins in http://www.maths.ex.ac.uk/~mwatkins/zeta/Julia.htm
Matthew Watkins has a lot of other fascinating material about prime numbers and physics on his website:
3) Matthew Watkins, http://www.maths.ex.ac.uk/~mwatkins/
so this is the best place to start if you're a beginner wanting to learn more about this stuff. There are also a bunch of new popular books on the Riemann hypothesis, so if you're looking for good Christmas gifts, you might try one of these:
4) Marcus du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins, 2003.
5) Karl Sabbagh, The Riemann Hypothesis: the Greatest Unsolved Problem in Mathematics, Farrar Strauss & Giroux, 2003.
6) John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem Mathematics, Joseph Henry Press, 2003.
I haven't read any of them, but from reviews it sounds like the third one focuses on Riemann while the first two talk more about modern developments.
If you want something quite a bit more substantial but still not requiring a PhD, try this:
7) Jeffrey Stopple, A Primer of Analytic Number Theory: from Pythagoras to Riemann, Cambridge U. Press, Cambridge, 2003.
This is the only introduction to analytic number theory that's so simple that I feel I have a good chance of reading it all the way through.
There's also a lot of interesting work relating the Riemann zeta function to quantum chaos. Alas, I don't know how this is related to the "free Riemann gas" idea! But here's a nice easy introduction:
8) Barry Cipra, A prime case of chaos, in What's Happening in the Mathematical Sciences, vol. 4, American Mathematical Society. Also available at http://www.maths.ex.ac.uk/~mwatkins/zeta/cipra.htm
Finally, if you get stuck on the fermionic version of the free Riemann gas, read Julia's paper or this one:
9) Donald Spector, Supersymmetry and the Moebius inversion function, Communications in Mathematical Physics 127 (1990) 239-252.
Anyway, all this post up to now has been just a big joke - although everything I said is true. The joke is that all this stuff about different meanings of "spectrum" has nothing to do with the sort of "spectra" they were talking about at that conference on homotopy theory! Topologists like to study a completely different sort of spectrum... so now let me talk about those.
In topology, a "spectrum" is defined to be a sequence of pointed topological spaces, each of which is homeomorphic to the space of all based loops in the next. So, each space in a spectrum is an "infinite loop space": a space of loops in a space of loops in a space of loops in....
In "week149" I described how this sort of spectrum gives a generalized cohomology theory, and I mentioned a bunch of examples. I gave some more examples in "week150" and "week197". But I never described the cool way to construct spectra that Graeme Segal came up with - so let me do that now.
There's a cute way to get a space from a category that goes like this. First create a simplicial set from your category, with one 0-simplex for each object:
.
                       x
one 1-simplex for each morphism:
f
                .------>------.
                x             y
one 2-simplex for each composable pair of morphisms:
y
                       .
                      / \
                     /   \
                    /     \
                  f/       \g
                  /         \
                 /           \
                /             \
               /      fg       \
              .------->---------.
              x                 z
and so on ad infinitum. This is called the "nerve" of the category. Then, think of this simplicial set as a topological space - i.e., take its "geometric realization". The result is called the "classifying space" of the category. By the way, I described this construction in a lot more detail in "week117". I also explained how you can get every space, up to homotopy equivalence, as the classifying space of some category! But what I didn't say is this:If you start with a monoidal category, the group completion of its classifying space will be a loop space.
You can get any loop space this way.
If you start with a braided monoidal category, the group completion of its classifying space will be a double loop space.
You can get any double loop space this way.
If you start with a symmetric monoidal category, the group completion of its classifying space will be an infinite loop space.
You can get any infinite loop space this way.
Huh? There are lots of terms here that I haven't defined yet....
For starters, a "loop space" is the space of based loops in some pointed topological space. A "double loop space" is the space of based loops in the space of based loops in some pointed topological space, and so on. Secondly, all the above statements are only true up to homotopy equivalence. Third, I'm talking about various sorts of category here. A monoidal category is roughly a category with a tensor product. This gives its classifying space a product, making it into a topological monoid; turning this into a group by throwing in inverses is called "group completion". A braided monoidal category is roughly a monoidal category with an isomorphism
B_x,y: x ⊗ y → y ⊗ x
for any pair of objects; we require this isomorphism satisfy some rules motivated by thinking it as a "braiding", like this:
x            y
               \          /
                \        /
                 \      /
                  \    /
                   \  /
                     /
                    /
                   /  \
                  /    \
                 /      \
                /        \
               /          \
              y            x
A symmetric monoidal category is a braided monoidal category for which B_x,y is the inverse of B_y,x. Some more details on these category-theoretic notions can be found in "week121".Symmetric monoidal categories abound in mathematics, so we can easily use them to get lots of nice infinite loop spaces - and thence spectra and generalized cohomology theories!
For example, if we take the category of finite sets, with disjoint union as the "tensor product", and the obvious braiding, the group completion of its classifying space will be
Ω^∞ S^∞  =  lim  Ω^k S^k
         k → ∞
the limit of taking the kth loop space of the k-sphere! The corresponding spectrum is called the "sphere spectrum" and the corresponding generalized cohomology theory is called "stable homotopy theory".If we take the category of finite-dimensional complex vector spaces, with direct sum as the "tensor product", and the obvious braiding, the group completion of its classifying space will be
BU(∞) =   lim        BU(k)
              k → ∞
where BU(k) is the classifying space of the group of k x k unitary matrices! The corresponding spectrum is called the "spectrum for connective complex K-theory" and the corresponding generalized cohomology theory is called "connective complex K-theory". (Here "connective" refers to the fact that unlike some other K-theory you may be familiar with, the cohomology groups Kⁱ with i negative have been set to zero.)More generally, we can take the category of finitely generated projective modules of a ring R, again with direct sum as the tensor product and the obvious braiding. This gives something called "algebraic K-theory". More precisely, the homotopy groups of the resulting infinite loop space are called the algebraic K-theory groups K_i(R).
Yet another example comes from taking the category of finite CW complexes, with disjoint union as the "tensor product" and the obvious braiding. This gives a generalized cohomology theory called "A-theory", due to Waldhausen.
I would like to say more about this stuff sometime. There's a lot more to say! For example, there are some cool relations between the algebraic K-theory groups of the integers, K_i(Z), and the Riemann zeta function at odd integers, ζ(2n+1). (Hmm, so maybe the different sort of spectra are related!) There's also a lot of nice stuff about how algebraic K-theory is related to topology. You can learn about that here:
10) Jonathan Rosenberg, K-theory and geometric topology, available at http://www.math.umd.edu/users/jmr/geomtop.pdf
But, I'll stop here for now. For more on how different sorts of category can be used to get ahold of n-fold loop spaces, see:
11) C. Balteanu, Z. Fiedorowicz, R. Schwaenzl, and R. Vogt, Iterated monoidal categories, available at math.AT/9808082.

Addenda: Here's my reply to some questions, and also some comments by my friend Squark about my use of the term "the prime at infinity".
Rene Meyer wrote:
John Baez wrote:

 > The "spectrum" of a C*-algebra A is the set of all homomorphisms
 > x: A → C,
 > where C is the complex numbers.  
 >
 > There's an easy way to make the spectrum of a commutative
 > C*-algebra A into a topological space: we say x_i → x precisely when
 > x_i(a) → x(a)
 > for all elements a of A.  With this topology any element a of A gives
 > a continuous complex function on the spectrum, defined by this clever
 > formula:

 I don't understand what you mean by
 
 x_i → x
 x_i(a) → x(a)
That's a way of saying that the sequence x_i converges to x, or the sequence x_i(a) converges to x(a).
What has the index i to do with this?
It's the index for some sequence of homomorphisms, x_i.
x_i and x are the above mentioned homomorphisms, right?
x is a homomorphism, x_i is a sequence of homomorphisms, and I'm telling you when the sequence x_i converges to x.
Could you explain in a little more detail?
I was describing how to make the spectrum of a C*-algebra into a topological space. One way to do this is to say when a sequence x_i of points in the spectrum converges to some point x. So, I took a sequence of homomorphisms
x_i: A → C
and told you when it converges to a homomorphism
x: A → C
And here's what I said: x_i converges to x precisely when the sequence of numbers x_i(a) converges to the number x(a) for all a in A.[Experts will know that now I'm lying slightly. In general, to specify the topology of a space, it's not really good enough to just say when sequences converge; you need to say when nets converge. A net is like a sequence, but the index i can range over an arbitrary "directed set". I don't feel like defining a directed set right now; one can find this in any good introduction to point-set topology. The point is that there are some spaces that are not "first countable", meaning that some points don't have a countable base of neighborhoods. A countable sequence just isn't long enough to converge to such a point, unless it equals that point for all sufficiently large i. So in general we need nets, though for metric spaces sequences are sufficient. Luckily, the notation and basic theorems concerning nets look almost like those for sequences! So, I was actually talking about nets in my post above - but I was hoping that people who only knew about sequences would think I was talking about sequences, in which case they'd be slightly wrong, but not too far off.]
Squark wrote:
John Baez wrote:

> ...in the commutative
> ring of integers, Z, most prime ideals come from prime numbers.  If we
> take all the multiples of any prime number, we get a prime ideal, which 
> is the kernel of the obvious homomorphism
>
> x: Z → Z/p
>
> There's just one other prime ideal in Z, namely all the multiples of
> 0.  In other words, the set consisting of just 0 alone!  This is the
> kernel of the homomorphism from Z into the rationals.  For some
> fascinating reason I'd rather not explain now, this prime ideal is
> often called "the prime at infinity".

I don't quite agree. The 0 ideal corresponds merely to the generic
point of Spec Z, a usual thing for schemes. The "prime at infinity",
as far as I understand, comes from viewing Spec Z as an "affine
line" over some mysterious impossible field and then completing
it into a "projective line".

In more detail, for any actual affine line Spec K[x] where x is a
field one can use each point x₀ in K to define a norm on K(x),
the field of rational functions over K. This is the
non-Archimedean norm ||f|| = q^{(deg_x₀ f)} where deg_x₀ f
is the degree of the pole f has at x₀ (or minus the degree of the
zero). I think it's possible to prove K(x) has exactly one norm
except this one: q^{deg f} where deg f is just the rational function
degree. This norm corresponds to the "point at infinity", adding it
gives us the projective line KP¹ (deg f is precisely the degree
of the pole at the point at infinity). Note that the product of all of
these norms is 1.

The rational functions over Spec Z is Q. Each prime gives us a
norm on Q which turns out to be the p-adic norm (modulo the
choice of q, which is a subtler issue, but also solvable, I think).
However Q has another norm on it: the usual, Archimedean
norm! Since it is Archimedean, it cannot come out of the q^deg
construction (in more fancy terms it doesn't correspond to any
local ring with Q its field of fractions). However, one can play the
"as if" game and imagine it does correspond to a point at infinity
lying in some weird completion of Spec Z. The generic point, on
the other hand, is present already for Spec K[x], and it is a
distinct point from the point at infinity for KP¹.

There are other interesting things related to this. In particular, the
Cauchy completion procedure is possible to formulate in purely
algebraic terms. For algebraic curves such as Spec K[x] it gives
a ring of formal series around the given point - a sort of
improvement of the usual local ring. This is something useful on its
own in algebraic geometry, for instance this "improved local ring"
(I don't remember the real name :-) ) is the same for the
self-intersection point of the "figure 8 curve" and the curve
consisting of two intersecting lines. The usual local ring
distinguishes between the two cases, so it's in some sense "not
local enough".

For Spec Z we get the p-adic numbers Q_p at the prime points
and we should get R at the point at infinity. This would be very
cool, since otherwise R seems to be an entirely analytic object,
impenetrable by algebra.

Best regards,
 Squark
By the way, the reason Squark pointed out that the product of all norms of an element of K(x) equals 1, is that the same is true for the product of all p-adic norms of a rational together with its usual norm. So, the analogy is good.But anyway, I guess I should have spoken of "the generic point" instead of "the prime at infinity" when talking about the prime ideal {0} in Z. The "prime at infinity" is a more mysterious thing. To learn more about it, read this book:
12) M. J. Shai Haran, The Mysteries of the Real Prime, Oxford U. Press, Oxford, 2001.
It touches upon lots of interesting relations between number theory and mathematical physics.

Riemann's insight followed his discovery of a mathematical looking-glass through which he could gaze at the primes. Alice's world was turned upside down when she stepped through her looking-glass. In contrast, in the strange mathematical world beyond Riemann's glass, the chaos of the primes seemed to be transformed into an ordered pattern as strong as any mathematician could hope for. He conjectured that this order would be maintained however far one stared into the never-ending world beyond the glass. His prediction of an inner harmony on the far side of the mirror would explain why outwardly the primes look so chaotic. The metamorphosis provided by Riemann's mirror, where chaos turns to order, is one which most mathematicians find almost miraculous. The challenge that Riemann left the mathematical world was to prove that the order he thought he could discern was really there. - Marcus du SautoyGod may not play dice with the universe, but something strange is going on with the primes. - Paul Erdös

© 2003 John Baez
baez@math.removethis.ucr.andthis.edu

the road to knots as primes

from baez

On to number theory....
There's a widespread impression that number theory is about numbers, but I'd like to correct this, or at least supplement it. A large part of number theory - and by the far the coolest part, in my opinion - is about a strange sort of geometry. I don't understand it very well, but that won't prevent me from taking a crack at trying to explain it....
The basic idea is to push the analogy between integers and polynomials as far as it will go. They're similar because you can add, subtract and multiply them, and these operations satisfy the usual rules we all learned in high school:
x + y = y + x (x + y) + z = x + (y + z) x + 0 = x x + (-x) = 0
xy = yx (xy)z = x(yz) x1 = x
x(y + z) = xy + xz
Anything satisfying these rules is called a "commutative ring". There are also a lot of deeper similarities between integers and polynomials, which I'll talk about later. But, there's a big difference! Polynomials are functions on the line, whereas the integers aren't functions on some space - at least, not in any instantly obvious way.The fact that polynomials are functions on a space is what lets us graph them. This lets us think about them using geometry - and also think about geometry using them. This was the idea behind Descartes' "analytic geometry", and it was immensely fruitful.
So, it would be cool if we could also think about the integers using geometry. And it turns out we can, but only if we stretch our concept of geometry far enough!
If we do this, we'll see some cool things. First of all, we'll see that algebra is just like geometry, only backwards.
What do I mean by this? Well, whenever you have a map T: X → Y going from the space X to the space Y, you can use it to take functions on Y and turn them into functions on X. Since this goes backwards, it's called "pulling back along T". Here's how it goes: if f is a function on Y, we get a function T*(f) on X given by:
T*(f)(x) = f(T(x))
Moreover, functions on a space form a commutative ring, since you can add and multiply them pointwise, and pulling back is a "homomorphism", meaning that it preserves all the structure of a commutative ring:
T*(f + g) = T*(f) + T*(g) T*(0) = 0
T*(fg) = T*(f) T*(g) T*(1) = 1
Conversely, any sufficiently nice homomorphism from functions on Y to functions on X will come from some map T: X → Y this way! Here I'm summarizing a whole bunch of different theorems, each of which goes along with its own precise definition of "space", "map", and "nice".Some of these theorems are technical, but the basic idea is simple: we can translate back and forth between the study of commutative rings (algebra) and the study of spaces (geometry) and by thinking of commutative rings as consisting of functions on spaces. We get a little dictionary for translating between geometry and algebra, like this:
GEOMETRY ALGEBRA
spaces commutative rings
maps homomorphisms
But be careful: this translation turns maps into homomorphisms going backwards: it's "contravariant". This is really important in two ways. First, suppose we have a point x in a space X. This gives a map
i: {x} → X
This, in turn, gives a homomorphism i* sending functions on X to functions on {x}. Functions on a one-point space are like numbers, so i* acts like "evaluation at x". Moreover, i* will tend to be onto: that's the backwards analogue of the fact that i is one-to-one!Second, suppose we have a map from a space E onto the space X:
p: E → X.
If you know some topology, think of E as a "covering space" of X. Then we get a homomorphism p* from functions on X to functions on E. Moreover p* will tend to be one-to-one: that's the backwards version of the fact that p was onto!We can use these examples to figure out the analogue of a "point" or a "covering space" in the world of commutative rings! And the resulting ideas turn out to be crucial to modern number theory.
In "week199" I explained the analogue of a "point" for commutative rings: it's a "prime ideal". So, now I want to explain the analogue of a "covering space". This will expand our dictionary so that it relates Galois groups to fundamental groups of topological spaces... and so on.
But, we won't get too far if we don't remember why a "prime ideal" is like a "point"! So, I guess I'd better review some of "week199" before charging ahead into the beautiful wilderness.
What's special about the ring of functions on a space consisting of just one point? Take real- or complex-valued functions, for example. How do these differ from the functions on a space with lots of points?
The answer is pretty simple: on a space with just one point, a function that vanishes anywhere vanishes everywhere! So, the only function that fails to have a multiplicative inverse is 0. For bigger spaces, this isn't true.
A commutative ring where only 0 fails to have a multiplicative inverse is called a "field". So, the algebraic analogue of a one-point space is a field.
This means that the algebraic analogue of a map from a one-point space into some other space:
i: {x} → X
should be a homomorphism from a commutative ring R to a field k:f: R → k
Our translation dictionary now looks like this:
GEOMETRY ALGEBRA
spaces commutative rings
maps homomorphisms
one-point spaces fields
maps from one-point spaces homomorphisms to fields
It's worth noting some subtleties here. In the geometry we learned in high school, once we see one point, we've seen 'em all: all one-point spaces are isomorphic. But not all fields are isomorphic! So, if we're trying to think of algebra as geometry, it's a funny sort of geometry where points come in different flavors!Moreover, there are homomorphisms between different fields. These act like "flavor changing" maps - maps from a point of one flavor to a point of some other flavor.
If we have a homomorphism f: R → k and a homomorphism from k to some other field k', we can compose them to get a homomorphism f ': R → k'. So, we're doing some funny sort of geometry where if we have a point mapped into our space, we can convert it into a point of some other flavor, using a "flavor changing" map.
Now let's take this strange sort of geometry really seriously, and figure out how to actually turn a commutative ring into a space! First I'll describe what people usually do. Eventually I'll describe what perhaps they really should do - but maybe you can guess before I even tell you.
People usually cook up a space called the "spectrum" of the commutative ring R, or Spec(R) for short. What are the points of Spec(R)? They're not just all possible homomorphisms from R to all possible fields. Instead, we count two such homomorphisms as the same point of Spec(R) if they're related by a "flavor changing process". In other words, f ': R → k' gives the same point as f: R → k if you can get f ' by composing f with a homomorphism from k to k'.
This is a bit ungainly, but luckily there's a quick and easy way to tell when f: R → k and f ': R → k' are related by such a flavor changing process, or a sequence of such processes. You just see if they have the same kernel! The "kernel" of f: R → k is the subset of R consisting of elements r with
f(r) = 0
The kernel of a homomorphism to a field is a "prime ideal", and two homomorphisms are related by a sequence of flavor changing processes iff they have the same kernel. Furthermore, every prime ideal is the kernel of a homomorphism to some field. So, we can save time by defining Spec(R) to be the set of prime ideals in R.For completeness I should remind you what a prime ideal is! An "ideal" in a ring R is a set closed under addition and closed under multiplication by anything in R. It's "prime" if it's not all of R, and whenever the product of two elements of R lies in the ideal, at least one of them lies in the ideal.
So, we have something like this:
GEOMETRY ALGEBRA
spaces commutative rings
maps homomorphisms
one-point spaces fields
maps from one-point spaces homomorphisms to fields
points of a space prime ideals of a commutative ring
Now let's use these ideas to study "branched covering spaces" and their analogues in algebra. This week I'll talk about two examples. The first is very geometrical, and it should be familiar to anyone who has studied a little complex analysis. The second is more algebraic, and it's important in number theory. But, the cool part is that they fit into the same formalism!If you don't know what a branched covering space is, don't worry: we'll start with the very simplest example. We'll look at this map from the complex plane to itself:
p: C → C
p(z) = z²Except for zero, every complex number has two square roots, so this map is two-to-one and onto away from the origin. In fact, away from the origin you can visualize this thing locally as two sheets of paper sitting above one. But these two sheets have a global complication: if you start on the top sheet and hike once around the origin, you wind up on the bottom sheet - and vice versa! In topology we call this sort of thing a "double cover". When we include the point z = 0 things get even more complicated, since the two sheets meet there. So we have something trickier: a "branched cover". In general, a branched cover is like a covering space except that the different "sheets" can merge together at certain points, called "branch points".Now let's think about this algebraically. To keep from getting confused, let's write
z² = w
so that p is a map from the "z plane" down to the "w plane", sending each point z to the point z² = w. The ring of polynomial functions on the z plane is called C[z]; the ring of polynomial functions on the w plane is called C[w]. We can pull functions from the w plane back up to the z plane:
p*: C[w] → C[z]
and p* works in the obvious way, taking any function f(w) to the function f(z²).
Just as p is onto, p* is one-to-one! So, we can think of C[w] as sitting inside C[z], consisting of those polynomials in z that only depend on z²: the even functions. We say C[w] is a "subring" of C[z], or equivalently, that C[z] is an "extension" of C[w].
In this example we can get the bigger ring from the smaller one by throwing in solutions of some polynomial equations, so we call it an "algebraic extension". We've already seen some algebraic extensions, namely algebraic number fields, where take the field of rational numbers and throw in some solutions of polynomial equations. Algebraic extensions can be complicated, but this one is really simple: we just start with C[w] and throw in the solution of one polynomial equation, namely
z² = w
It turns out that quite generally, algebraic extensions of commutative rings act a lot like branched covering spaces. I probably don't have the technical details perfectly straight, but let's add this to our translation dictionary, because it's an important idea:
GEOMETRY ALGEBRA
spaces commutative rings
maps homomorphisms
one-point spaces fields
maps from one-point spaces homomorphisms to fields
points of a space prime ideals of a commutative ring
branched covering spaces algebraic extensions of commutative rings
Now let's have some fun: let's see how our algebraic concept of "point", namely "prime ideal", interacts with our branched double cover of the complex plane. There's something straightforward going on, but also something more subtle and interesting.The straightforward thing is that any point up on the z plane maps to one down on the w plane. We don't need fancy algebra to see this! But, it's worth doing algebraically. According to the fancy algebraic definition, a "point" in the spectrum of the commutative ring C[z] is a prime ideal. But as you might hope, these are the same as good old-fashioned points in the complex plane!
It works like this: given any point x in C, we get a homomorphism from C[z] to C called "evaluation at x", which sends any polynomial f to the number f(x). The kernel of this is the prime ideal consisting of all polynomials that vanish at x. These are just the polynomials containing a factor of z - x, so we call this ideal
<z - x>
So, we get some prime ideals in C[z] from points of C this way. But in fact there's a theorem that every prime ideal in C[z] is of this form! So, we get a one-to-one correspondence
Spec(C[z]) = C
Similarly,
Spec(C[w]) = C
Now let's think about our branched cover
p: C → C
in different ways. It starts out life as a map from the z plane down to the w plane. We can use this to pull back functions on the w plane up to the z plane:p*: C[w] → C[z]
But then, by general abstract baloney, the inverse image under p* of any prime ideal in C[z] is a prime ideal back in C[w]. This gives a map from Spec(C[z]) to Spec(C[w]). But this is just a map from the z plane to the w plane! And it's the same map p we started with. If you don't see why, it's a good exercise to check this.
So: we translated from geometry to algebra and back to geometry, and we got right back where we started. Note that each time we translated, our description of the map p got turned around backwards.
But there's a subtler and more interesting thing we can do with our branched cover. We can take a point down on the w plane and look at the points up on the z plane that map down to it!
Usually there will be two, but for the origin there's just one. This much is clear from thinking geometrically. But if we think algebraically, we'll see something funny going on at the origin. We can already see it geometrically: the origin is where the two sheets of our branched cover meet, so we call it a "branch point". But the algebraic viewpoint sheds an interesting new light on this.
What we'll do is take a prime ideal in C[w] and push it forwards via
p*: C[w] → C[z]
The resulting subset won't be an ideal, but it will "generate" an ideal, meaning we can take the smallest ideal containing it. This ideal won't be prime, but we can "factor" it into prime ideals: there's a fairly obvious way to multiply ideals, and we happen to be working with rings where there's a unique way to factor any ideal into prime ideals.Let's try it. First pick a number x that's not zero. It gives a prime ideal in C[w], namely
<w - x>
Next push this ideal forwards via p* and let it generate an ideal in C[z], namely
<z² - x>
This is not prime, but we can factor it, which in this case simply amounts to factoring the polynomial that generates it:
<z² - x> = <(z - sqrt(x)) (z + sqrt(x))>
= <z - sqrt(x)> <z + sqrt(x)>
We get a product of two prime ideals, corresponding to two points in the z plane, namely +sqrt(x) and -sqrt(x). These are the two points that map down to x.In this sort of situation, we say the prime ideal <w - x> "splits" into the prime ideals <z - sqrt(x)> and <z + sqrt(x)> when we go from C[w] to the extension C[z]. This is just an overeducated way of saying the number x has two different square roots.
But suppose x = 0. This doesn't have two square roots! Everything works the same except we get
<z²> = <z> <z>
We say the prime ideal <w> "ramifies" when we go from C[w] to the extension C[z]. We still get a product of prime ideals; they just happen to be the same. This is a way of making sense of the funny notion that the number 0 has two square roots... which just happen to be the same! Lots of mathematicians and physicists talk about "repeated roots" when an equation has "two solutions that just happen to be equal". This is just a way of making that precise.But all this algebraic machinery must seem like overkill if this is the first time you've seen it. It pays off when we get to more algebraic examples. So, let me sketch the simplest one.
Let Z be the ring of integers, and let Z[i] be the ring of Gaussian integers, namely numbers of the form a+bi where a and b are integers. Z[i] is an algebraic extension of Z, since we can get it by throwing in a solution z of the polynomial equation
z² = -1
This equation is quadratic, just like it was in the example we just did! Now we're throwing in a square root of -1 instead of a square root of some function on the complex plane. But if we take the analogy between geometry and algebra seriously, this extension should still give some sort of "branched double cover"
p: Spec(Z[i]) → Spec(Z)
What's this like?It's actually really interesting, but I'll just sketch how it works.
The points of Spec(Z) are prime ideals in Z. In "week199" we saw that except for the prime ideal <0>, these are generated by prime numbers.
Similarly, except for <0>, the prime ideals in Z[i] are generated by "Gaussian primes": Gaussian numbers that have no factors except themselves and the "units" 1, -1, i and -i. (A "unit" in a ring is an element with a multiplicative inverse; we don't count units as primes.)
The map p sends each Gaussian prime to a prime, and it's fun to work out how this goes... but it's even more fun to work backwards! Let's take primes in the integers and see what happens when we let them generate ideals in the Gaussian integers! This is like taking points in the base space of a branched cover and seeing what's sitting up above them.
For example, the prime 5 "splits". It has two prime factors in the Gaussian integers:
5 = (2 + i)(2 - i)
so in Z[i] the ideal it generates is a product of two prime ideals:
<5> = <2 + i> <2 - i>
This means that two different points in Spec(Z[i]) map down to the point <5> in Spec(Z), namely <2 + i> and <2 - i>. So we indeed have something like a double cover!On the other hand, the prime 2 "ramifies". It has two prime factors in the Gaussian integers:
2 = (1 + i)(1 - i)
but these two Gaussian primes generate the same prime ideal:
<1 + i> = <1 - i>
since if we multiply 1+i by the unit -i we get 1-i. So, in the Gaussian integers we have
<2> = <1 + i> <1 + i>
A repeated factor! This is just what happened to the branch point in our previous example: it had "two points sitting over it, which happen to be the same".So far, everything seems to be working nicely. But, besides splitting and ramification, there's a third thing that happens here, which didn't happen in our example involving the complex plane. In fact, this third option never happens when we're doing algebraic geometry over the complex numbers!
Here's how it works. Consider the prime 3. This is still prime in the Gaussian integers! It doesn't split, and it doesn't ramify. If we factorize the ideal generated by 3 in Z[i] we just get
<3> = <3>
It doesn't do anything - it just sits there! So, we say this prime is "inert".This may seem boring, but it's actually mysterious - and downright MADDENING if we take the analogy between geometry and algebra seriously. It's weird enough to have a "branched" cover where sheets merge at certain points, but at least in that case we can see they've merged: a prime ideal in our subring generates an ideal in the extension that's not prime, but is a product of several prime factors, some of which happen to be the same. But when a prime ideal in our subring generates a prime ideal in the extension, it's as if our "cover" has just one sheet over this point in the base space! And if this happens for a quadratic extension - as it just did - something seems to have gone horribly wrong with the nice idea that "quadratic extensions are like branched double covers".
Luckily, this puzzle has a nice resolution. We shouldn't have decategorified! When we started discussing "points" for a commutative ring, we saw they form a category in a nice way: there are points of different "flavors", with "flavor-changing operations" going between them. Then we freaked out and turned this category into a set by decreeing that two point are the same whenever there's a morphism between them. If we hadn't done this, we'd have seen more clearly how "inert" primes fit into a nice pattern along with "split" and "ramified" ones.
I'll probably talk about this more sometime, and also look more carefully at what happens to all the different primes when we go to the Gaussian integers - to show you that we are, indeed, doing number theory!
But for now, I just want to make a few comments about this idea of points of different "flavors".
In fact Grothendieck proposed an even more general idea of this sort in his second approach to "schemes", which is simpler but much less widely discussed than his first approach. Basically, he said that given a commutative ring R, we should not only consider points that are homomorphisms from R to any field, but also to any commutative ring. For each commutative ring k we get a set consisting of all "k-points" of R, namely homomorphisms
f: R → k
And, for each homomorphism g: k → k' we get a "flavor changing operation" that sends k-points to k'-points. So, we get a functor from CommRing to Set! He called such a functor a "scheme". We can get schemes from commutative rings as just described - these are called "affine schemes" - but there are also others, for example those coming from projective varieties.Anyway, here are some places to read more about number theory... mostly with an emphasis on the geometric viewpoint and the issue of "splitting, ramification and inertia".
For a really quick and friendly no-nonsense introduction, try this:
2) Harold M. Stark, Galois Theory, Algebraic Number Theory, and Zeta Function, in From Number Theory to Physics, eds. M. Waldschmit et al, Springer, Berlin, 1992, pp. 313-393.
To dig a lot deeper, try this book by Neukirch:
3) Juergen Neukirch, Algebraic Number Theory, trans. Norbert Schappacher, Springer, Berlin, 1986.
I already mentioned it, but it's worth mentioning again, because it's pretty elementary, and very clear on the analogy between "function fields" (fields of functions on Riemann surfaces) and "number fields" (algebraic number fields).
This book by Borevich and Shafarevich doesn't make the analogy to geometry explicit:
4) Z. I. Borevich and I. R. Shafarevich, Number Theory, trans. Newcomb Greenleaf, Academic Press, New York, 1966.
However, it has a nice concept of a "theory of divisors" for a commutative ring - and if you know a bit about divisors from algebraic geometry, you'll see that this is secretly very geometrical! They show how to classify algebraic extensions of commutative rings using a theory of divisors, and show how to get a theory of divisors using "valuations". This manages to accomplish a lot of what other texts do using "adeles", without actually mentioning adeles. I find this instructive.
This book goes much further in the geometric direction, but still without introducing schemes:
5) Dino Lorenzini, An Invitation to Arithmetic Geometry, American Mathematical Society, Providence, Rhode Island, 1996.
It's really great - very pedagogical! It develops number fields and function fields in parallel. You'll need to be pretty comfy with commutative algebra to work all the way through it, though.
If you want to learn about schemes - not the kind I just talked about, just the usual sort, which still includes cool "spaces" like Spec(Z) - try these:
6) V. I. Danilov, V. V. Shokurov, and I. Shafarevich, Algebraic Curves, Algebraic Manifolds and Schemes, Springer, Berlin, 1998.
7) David Eisenbud and Joe Harris, The Geometry of Schemes, Springer, Berlin, 2000.
Schemes have a reputation for being scary, but both these books try hard to make them less so, including lots of actual pictures of things like Spec(Z[i]) sitting over Spec(Z).
To wrap things up, I just want to mention two papers on subjects I'm fond of....
In "week172" I discussed Tarski's "high school algebra problem". This asks whether every identity involving 1, +, x, and exponentials that holds in the positive natural numbers follows from the eleven we learned in high school:
x + y = y + x (x + y) + z = x + (y + z)
xy = yx (xy)z = x(yz)
1x = x
x¹ = x 1^x = 1
x(y + z) = xy + xz
x^{(y + z)} = x^y x^z (xy)^z = x^z y^z x^yz = (x^y)^z The rules of this game allow only purely equational reasoning - not stuff like mathematical induction. The reason is that this is secretly a problem about "universal algebra" or "algebraic theories", as explained in "week200".It turns out the answer is no! In fact there are infinitely many more independent identities! Here is the first one, due to Wilkies:
[(x + 1)^x + (x² + x + 1)^x]^y [(x³ + 1)^y + (x⁴ + x² + 1)^y]^x =
[(x + 1)^y + (x² + x + 1)^y]^x [(x³ + 1)^x + (x⁴ + x² + 1)^x]^y I just found a paper, apparently written after "week172", which gives a very detailed account of this problem:8) Stanley Burris and Karen Yeats, The saga of the high school identities, available at http://web.archive.org/web/20070212200835/http://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PREPRINTS/saga.psIt includes some new results, like the smallest known algebraic gadget satisfying all the high school identities but not Wilkies' identity - but also more interesting things that are a bit harder to describe.
Also, here's a cool paper relating some of Ramanujan's work to string theory:
9) Antun Milas, Ramanujan's "Lost Notebook" and the Virasoro Algebra, available as math.QA/0309201.
A lot of Ramanujan's weird identities turn out to be related to concepts from string theory, suggesting that he was born about a century too soon to be fully appreciated... but this paper tackles an identity of his that nobody had managed to explain using string theory before.

Addendum: Here's something a friend of mine wrote, and an expanded version of my reply.
By the way, I very much liked your explanation of points and prime
ideals. Up until now I haven't seen a satisfactory explanation of why
points correspond to prime rather than maximal ideals, and
although I haven't completely digested what you wrote, it looks
like it might do the job...
Both here and in my discussion of spectra in "week199", I've been avoiding saying the things people usually say. People usually note that a maximal ideal is the same as the kernel of a homomorphism ONTO a field, while a prime ideal is the same as the kernel of a homomorphism ONTO an integral domain. (Recall that an integral domain is a commutative ring where xy = 0 implies that x or y is zero.) If we define the "points" of a commutative ring R to be its maximal or prime ideals, we can therefore think of these as the kernels of homomorphisms from R onto fields or integral domains.
However, defining points in terms of homomorphisms ONTO a given sort of commutative ring is rather irksome, because it doesn't tell us how points transform under homomorphisms of commutative rings, nor how they transform under the "flavor-changing operations" I was describing. The problem is that the composite of a homomorphism with an onto homomorphism needn't be onto!
So, what really matters is that a prime ideal is the same as the kernel of a homomorphism TO a field. To see how this follows from the usual story, note that any integral domain is contained in a field called its "field of fractions" - just as Z is contained in Q. Any homomorphism ONTO the integral domain thus becomes a homomorphism TO this field, with the same kernel. Conversely, any homorphism TO a field becomes a homomorphism ONTO its image, with the same kernel - and this image is always an integral domain.
Best,
jb