This Week's Finds in Mathematical Physics (Week 216)

John Baez

May 23, 2005
This Week's Finds in Mathematical Physics - Week 216
John Baez

There are lots of different things called "zeta functions" in mathematics
and physics. The grand-daddy of them all is the Riemann zeta function:

zeta(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + ....

This is deeply related to prime numbers, thanks to Euler's product formula

zeta(s) = product 1/(1 - p^{-s})

where we take a product over all primes. This formula is fun to prove:
just use the geometric series to expand each factor, multiply them out
and see what happens!

Using this, Riemann and von Mangoldt derived an explicit formula
for how many primes are less than a given number as a sum over the
*zeros* of the Riemann zeta function. Instead of showing you
this formula, I'll just urge you to watch a *movie* of how it works:

1) Matthew Watkins, Animation: the prime counting function
pi(x), http://www.maths.ex.ac.uk/~mwatkins/zeta/pianim.htm

Thanks to this formula, information about the Riemann zeta function
is secretly information about the distribution of primes!

For example, the Riemann Hypothesis says that when we analytically
continue the zeta function to the complex plane, the only zeros occur
at negative even integers and numbers with real part equal to 1/2.
And, knowing this would be equivalent to knowing that the number of
primes less than x differs from

Li(x) = integral_2^x dt/ln t

by less than some constant times sqrt(x) ln(x). Everyone feels
sure these facts are true. But, despite over a century of hard work
and a million-dollar prize offered by the Clay Mathematics Institute,
nobody has come close to proving them!

It's known that apart from the negative even integers, the only
place the Riemann zeta function can vanish is in the strip where

0 < Re(s) < 1

But, nobody has been able to show that all the zeros in this "critical strip"
lie on the line

Re(s) = 1/2

Of course, this can be checked in special cases. The current record
may belong to Xavier Gourdon, who on October 12th of 2004 claimed to
have shown - with the help of a computer - that the first TEN TRILLION
zeros in the critical strip lie on the line Re(s) = 1/2.

2) Xavier Gourdon, Computation of zeros of the Riemann zeta function,
http://numbers.computation.free.fr/C...oscompute.html

Alas, such monster calculations don't seem helpful for proving the Riemann
hypothesis. They're more useful when it comes to formulating and testing
conjectures about the *statistical properties* of the zeros.

The most famous of these traces its way back to a teatime conversation
between Hugh Montgomery and Freeman Dyson... you can read the story here:

3) K. Sabbagh, Dr. Riemann's Zeros, Atlantic Books, 2002, pp. 134-136.
Story about Hugh Montgomery and Freeman Dyson also available at
http://www.maths.ex.ac.uk/~mwatkins/zeta/dyson.htm

The upshot is that the distribution of spacings between Riemann zeros
closely mimics the spacings between eigenvalues of a large randomly
chosen self-adjoint matrix.

This suggests fascinating relations between the Riemann zeta function
and quantum physics. In fact, one popular dream for proving the
Riemann zeta function is to find a chaotic classical system whose
quantized version has energy levels related to the Riemann zeta zeros!

I would like to understand this stuff, but it all seems a bit
indimidating - especially since the coolest aspects are the ones
*nobody* understands.

Luckily, the Riemann zeta function has spawned a lot of other functions
called zeta functions and L-functions, and many of these are *simpler*
than the original one - or at least raise fascinating questions that
are easier to solve. Many of these are listed here:

4) Matthew R. Watkins, A directory of all known zeta functions,
http://www.maths.ex.ac.uk/~mwatkins/...afunctions.htm

Matthew R. Watkins, A directory of all known L-functions,
http://www.maths.ex.ac.uk/~mwatkins/...-functions.htm

Lately I've been talking about zeta functions with James Dolan and also
Darin Brown, whose work I mentioned last week. I feel some things are
starting to make sense, so I'd like to explain them before it turns out
they don't.

I'll just list some zeta functions, so you can see what we're dealing with:

A) The zeta function of a number field. A "number field" is something you
get by taking the rational numbers and throwing in some algebraic numbers.
One can define "algebraic integers" for any number field, and they act a lot
like the ordinary integers. So, one can define a zeta function for any
number field.

Technically, we do this by summing over all nonzero ideals I in our
number field:

zeta(s) = sum_I |I|^{-s}

where |I| is a number called the "norm" of I. We also have an Euler product
formula:

zeta(s) = product_P 1/(1 - |P|^{-s})

where we take the product over prime ideals.

For example, if our number field is the rational numbers, its algebraic
integers are the ordinary integers. So, each ideal consists of multiples
of some number n = 1,2,3,..., and its norm is just n, so we get:

zeta(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + ....

A more fun example is the number field Q[i], where we take the
rational numbers and throw in a square root of -1. Here the
algebraic integers are the so-called "Gaussian integers" Z[i],
namely guys like

a + bi

where a and b are ordinary integers.

In this example it's easiest to work out the zeta function using
the Euler product formula. If you ask one of your number theory
pals about prime ideals in the Gaussian integers, they'll say:

"Well, the Gaussian integers are a principal ideal domain, so every
ideal is generated by a single element. So, we can actually talk
about *prime numbers* in the Gaussian integers. And there are 3 cases:

INERT: An ordinary prime number of the form 4n+3 is also prime
in the Gaussian integers: for example, 3.

SPLIT: An ordinary prime numbers of the form 4n+1 is the product
of two complex conjugate primes in the Gaussian integers: for
example, 5 = (2+i)(2-i).

RAMIFIED: The ordinary prime 2 equals (1+i)(1-i), but here the
two factors give the same prime ideal, since (1-i) = i (1+i),
and i is invertible in the Gaussian integers.

The three cases are called "inert", "split", and "ramified", and
we get all primes in the Gaussian integers this way."

So, the zeta function of the Gaussian integers goes like this:

zeta(s) =

1/(1 - 2^{-s}) 1/(1 - 3^{-2s}) 1/(1 - 5^{-s}) 1/(1 - 5^{-s}) ...

I went just far enough to show you what happens for each kind of
prime. As you might expect, we get two factors for each prime
that splits.

I should explain the other details, but number theory is best
absorbed in small doses, especially if you're a physicist. The
main lesson to take home is this:

A number field is like a funny sort of "branched covering space" of
the set of ordinary primes. Sitting over each ordinary prime
there are one or more prime ideals in our number field:


2+i 3+2i
--- 1+i --- 3 --- --- 7 --- 11 --- --- GAUSSIAN INTEGERS
2-i 3-2i


-----2------3------5------7-----11------13----- INTEGERS


And, the zeta function records the details of how this works!

For more on this covering space philosophy see "week205" and "week213".
This geometrical metaphor lies behind a lot of the really cool work
on number theory.

B) The zeta function of a discrete dynamical system. A "discrete
dynamical system" consists of a set X together with a one-to-one
and onto function

f: X -> X

Here X is the set of "states" of some physical system, and f describes
one step of time evolution. For each integer n we get a function

f^n: X -> X

Since the integers are called Z, mathematicians would call our
discrete dynamical system a "Z-set".

Whatever you call it, its zeta function is defined to be:

zeta(s) = product_P 1/(1 - |P|^{-s})

where P ranges over all periodic orbits and |P| is the
*exponential* of the size of this periodic orbit.

This is like an Euler product formula with the periodic orbits
being the "primes". Just as every natural number can be uniquely
factored into primes, every discrete dynamical system can be
uniquely written as a disjoint union of periodic orbits.
This explains the exponential in the definition of |P| above:
primes like to multiply, while sizes of orbits like to add.

One nice thing about this zeta function is that when we take
the disjoint union of two discrete dynamical systems, their zeta
functions multiply. Another nice thing is that the zeta function
of a dynamical system completely describes it up to isomorphism,
at least when the set X is finite. Decategorification at work!

We can also rewrite this zeta function as a sum:

zeta(s) = sum_I |I|^{-s}

where I ranges over all formal products of periodic orbits,
and we define |P_1 ... P_n| = |P_1| ... |P_n|.

Even better, examples A) and B) overlap. I'll explain how later, but
the key is to associate to any number field and any prime p a discrete
dynamical system

f: X -> X

called the "Frobenius automorphism". This gives a zeta function.
It works best if we take the exponential of the size of each
periodic orbit using the base p instead of base e. Then, if we
multiply all these zeta functions, one for each ordinary prime p,
we get the zeta function of our number field!

C) The zeta function of a continuous dynamical system.
Now suppose X is some topological space and we have a time
evolution map

f_t: X -> X

for each real number t. We can define a zeta function

zeta(s) = product_P 1/(1 - |P|^{-s})

where P ranges over all periodic orbits and |P| is the
exponential of the "period" of P - meaning the time it takes
for points on this orbit to loop around back to where they started.

A famous example is when we have a Riemannian manifold. A free particle
moving around on such a space will trace out geodesics, giving us a
dynamical system. The analogue of primes are now "prime geodesics":
periodic geodesics that loop around just once.

The "covering space" philosophy described in example A) can now be
taken literally! If the Riemannian manifold M' is a covering
space of M, any prime geodesic P in M defines a deck transformation
of M'. This transformation acts on the set X of prime geodesics
sitting over P, so we get a one-to-one and onto map

f: X -> X

This is exactly like the "Frobenius automorphism" in number theory!

All this is particularly interesting when our manifold is a quotient
of the upper halfplane by a discrete group - see "week215" for more
on this. The reason is that some of these quotients are related to
number theory. So, we get some direct interactions with example A).

D) The zeta function of a graph. We can take the idea of
"periodic geodesic on a Riemannian manifold" and vastly simplify
it by looking at closed loops in a graph with finitely many
edges and vertices. We get a zeta function

zeta(s) = product_P 1/(1 - |P|^{-s})

where P ranges over all "prime loops" in our graph: loops that don't
backtrack or loop around more than once. Now |P| is the exponential
of the length of the loop.

The "covering space" philosophy still applies, since we can define
what it means for a graph G' to be a covering space of a graph G.
Any prime loop P in G defines a deck transformation of G'. This
acts on the set X of prime loops sitting over P, so we get a
one-to-one and onto map

f: X -> X

which again deserves to be called the "Frobenius automorphism".

E) The zeta function of an affine scheme. Given a commutative ring,
we can think of it as the ring of functions on some space. The zeta
function of the commutative ring then counts the points of this space.

To make this precise, we cleverly invent a kind of space called an
"affine scheme", which is secretly JUST ANOTHER NAME FOR A COMMUTATIVE RING.
So, any commutative ring R gives an affine scheme called Spec(R),
purely by our fiendish definition.

If we take a function and evaluate it at a point, we should get a number.
This should give a homomorphism from functions to numbers. But in
algebraic geometry, "numbers" can be elements of any field k. So, let's
say the "k-points" of Spec(R) are the homomorphisms from R to k.

(This is a bit nontraditional, but I really need this here.
For a more traditional alternative, see "week205".)

In particular, for any prime p we can take k to be the algebraic
closure of the field with p elements. Let X be the set of k-points
of some affine scheme Spec(R). Then comes something wonderful:
if x is a k-point, so is x^p, since "raising to the pth power" is an
automorphism of k. So, we get a map

f: X -> X

sending x to x^p. This is called the "Frobenius automorphism"!

Since f is a discrete dynamical system, we can define its zeta function
as in example B):

zeta(s) = product_P 1/(1 - |P|^{-s})

where P ranges over all periodic orbits, and |P| is the exponential
of the size of the periodic orbit, defined using the base p.

So far, this the zeta function of our affine scheme "localized at p".
If we multiply a bunch of factors like this, one for each ordinary
prime p, we get the zeta function of our affine scheme.

If you know about schemes that aren't affine - like projective
varieties, such as elliptic curves and other curves - you'll see
this definition works for them too.

If you know someone else's definition of the zeta function of a scheme,
it may not look like what I wrote here! But don't panic. The reason is
that people like to express the zeta function of a discrete dynamical system
f: X -> X in terms of the number of fixed points of f^n. When f is the
Frobenius automorphism, these are usually called "points defined over the
field with p^n elements". So, you'll see lots of formulas for zeta functions
phrased in terms of these....

Okay. Enough examples.

There are a lot more, but I think these are the simplest. I hope you see
how all these examples are just different expressions of the same idea.
To go further, I would tell you how there are versions of the Riemann
Hypothesis in all these examples, and also things called "L-functions",
and lots of theorems and conjectures concerning them, too!

It's a wonderful example of the unity of mathematics. But, it's also
a wonderful example of how this unity is obscured by people who zoom
in on their own favorite special case and its particular technical
complexities while never discussing the big picture. You wouldn't
believe how hard it's been for me to figure out what I just told you!
It's like trying to learn English by reading the US legal code, or
learning basic chord progressions by listening to Schoenberg.

If you're just trying to get started, here's one of the more readable
introductions:

5) Anton Deitman, Panorama of zeta functions, available as
math.NT/0210060.

Audrey Terras has a lot of nice slide presentations about the zeta functions
and L-functions of graphs:

6) Audrey Terras, Artin L-functions of graph coverings,
available at http://math.ucsd.edu/~aterras/artin1.pdf

More on L-functions, available at
http://math.ucsd.edu/~aterras/artin2.pdf

Here's a paper written in broken English but making a very serious attempt
to explain things to the nonexpert:

7) David Zywina, The zeta function of a graph, available at
http://math.berkeley.edu/~zywina/zeta.pdf

He gives a characterization of graphs whose zeta functions satisfy an
analogue of the Riemann Hypothesis. Strangely, this analogue involves
*poles* of the zeta function in the critical strip

0 < Re(s) < 1

Is this a real difference or the result of some difference in conventions?

Finally, I should explain some more technical stuff about zeta functions
and fixed points, just so I don't forget it. Suppose we have a discrete
dynamical system

f: X -> X

and let

|fix(f^n)|

be the number of fixed points of the nth iterate of f.

We can define a weird function like this:

Z(u) = exp(sum_{n>0} |fix(f^n)| u^n / n)

You'll often see formulas like this running around, especially when f is
some sort of "Frobenius automorphism". Sometimes people even call these
guys zeta functions. But what in the world do they have to do with zeta
functions???

Let's see. Suppose first that X consists of a single cycle of length k.
Then f^n has k fixed points when n is a multiple of k, but none otherwise, so:

Z(u) = exp( ku^k/k + ku^{2k}/2k + ku^{3k}/3k + ... )

= exp( u^k + u^{2k}/2 + u^{3k}/3 + ... )

= exp( ln(1/(1 - u^k)) )

= 1/(1 - u^k)

This is starting to look more like the zeta functions we know and love.
It looks even better if we pick some prime number p and define

zeta(s) = Z(p^{-s})

Then we get

zeta(s) = 1/(1 - p^{-ks})

which is precisely what we'd get using the definition in example B).

Furthermore, for both that old definition and our new one, the zeta
function of a disjoint union of discrete dynamical systems is the
*product* of the zeta functions of the parts. Since every discrete
dynamical system is a disjoint union of cycles, we conclude that the
definitions *always* agree:

zeta(s) = Z(p^{-s})

with

Z(u) = exp(sum_{n>0} |fix(f^n)| u^n / n)

is always equal to the zeta function defined in example B).

So, don't let anyone fool you - there aren't lots of completely
different kinds zeta functions! There's just a few kinds, and
we could probably boil them down to just ONE kind with some work.

Quote of the week:

"Some decades ago I made - somewhat in jest - the suggestion that one should
get accepted a non-proliferation treaty of zeta functions. There was
becoming such an overwhelming variety of these objects." - Atle Selberg

-----------------------------------------------------------------------
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If you just want the latest issue, go to

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Per Vognsen

Zeta functions are wondrous beasts! Thanks for this awesome issue of
TWF.

John Baez wrote:
> May 23, 2005
> This Week's Finds in Mathematical Physics - Week 216
> John Baez
>
[...]
> E) The zeta function of an affine scheme. Given a commutative ring,
> we can think of it as the ring of functions on some space. The zeta
> function of the commutative ring then counts the points of this space.

I realize this wasn't your focus but I feel compelled to mention the
Weil conjectures in connection with this. In the related case of an
algebraic variety V over a finite field F_q (where q = p^n for some
prime p) we have the exponential generating function

Z(T) = exp(sum(n positive) #(points of V over F_{q^n}) T^n / n!)

which is the so-called local zeta function of the variety. Presumably
the word "local" is used because the F_{q^n} are in some sense
increasingly local versions of F_q--so the local zeta function counts
points of V over increasingly local fields.

Part of what Weil conjectured was that Z(T) should always be a rational
function. In fact he conjectured a pretty specific form. All this is
extremely unexpected! It turns out that it would follow from a
Lefschetz-style fixed point formula for a sufficiently rich cohomology
theory. But the problem at the time was no such cohomology theory was
known--the only natural topology on V is the Zariski topology, which is
very pathological. So all the usual ways of setting up cohomology
failed to give much useful information. Grothendieck eventually figured
out a way around these problems by inventing etale cohomology. Deligne
used this to eventually prove the Weil conjectures.

I've been trying to learn some of this stuff from J. Milne's lecture
notes on etale cohomology at

http://www.jmilne.org/math/CourseNotes/math732.html

which are excellent but very heavy going (unsurprisingly). The first
chapter or two are relatively chatty--they are definitely worth
checking out if you are interested in a vague impression of what some
of this stuff is about.

Per

Ken S. Tucker

Dr. Baez et al...
Where's the physics application in this post?
I read it twice and I'm uncertain how it is
applicable to physics, help a dense guy:-).
Ken

[Moderator's note: Entire bottom-quoted post deleted. Please quote as
much of the post as necessary, but as little as possible, and add your
reply after that. -P.H.]

John Baez

In article <1117079187.198693.76220@z14g2000cwz.googlegroups.com>,
Per Vognsen <per.vognsen@gmail.com> wrote:

>Zeta functions are wondrous beasts! Thanks for this awesome issue of
>TWF.

You're welcome! I plan to write more about this stuff...
though I know from experience that as soon as I *plan* to
write about something in This Week's Finds, I decide it's
a chore and switch to writing about something else!

>I realize this wasn't your focus but I feel compelled to mention the
>Weil conjectures in connection with this.

Excellent! These were indeed on my mind...

>In the related case of an
>algebraic variety V over a finite field F_q (where q = p^n for some
>prime p) we have the exponential generating function
>
>Z(T) = exp(sum(n positive) #(points of V over F_{q^n}) T^n / n!)
>
>which is the so-called local zeta function of the variety.

Right. Part of what used to bug me immensely is that I didn't
see how this formula was related to other things called zeta functions.
That's why I'm so happy to learn about a very simple idea - the zeta
function of a discrete dynamical system - and to realize that the
above zeta function comes from a dynamical system where the time
evolution map is called the "Frobenius".

I'm sure you understand this already, but just for everyone else:
this is why "week216" I gave a proof that for any discrete dynamical
system

f: X -> X

we can define a function generalizing the one you wrote down:

Z(T) = exp(sum_{n>0} |fix(f^n)| T^n / n)

and then

zeta(s) = Z(p^{-s})

equals what I'd consider the "usual" zeta function of a dynamical
system, namely

zeta(s) = product_P 1/(1 - |P|^{-s})

where P ranges over all periodic orbits, and |P| is the exponential
of the size of the periodic orbit, defined using the base p.

>Presumably the word "local" is used because the F_{q^n} are in some sense
>increasingly local versions of F_q--so the local zeta function counts
>points of V over increasingly local fields.

Right. The fields F_{q^n} are "local fields" in a precise technical
sense, while the rational numbers are a "global field" - but this precise
technical sense is based on a simple beautiful idea, namely that the rational
numbers are functions on a space whose points are just the prime numbers
(together with the "real prime").

So, keeping track of the power of a prime that shows up in a rational number
is a way of studying that number "locally". And, just as global problems in
geometry get easier if we first tackle them locally, the same is true of all
problems in number theory!

>Part of what Weil conjectured was that Z(T) should always be a rational
>function. In fact he conjectured a pretty specific form. All this is
>extremely unexpected!

It was less unexpected to him than to most of us, because he knew
that there was a very simple formula relating the cohomology of
complex Grassmannians to the number of points of these Grassmannians
over finite fields. I talked about this formula at enormous length
in "week187" and previous weeks. I knew it was related to Weil's
conjectures. But, only recently did I learn that Weil was motivated
by precisely this (and other facts) to make his conjectures!

Aaron Bergman

In article <d764b9$kke$1@glue.ucr.edu>, baez@galaxy.ucr.edu (John Baez)
wrote:

> In article <1117079187.198693.76220@z14g2000cwz.googlegroups.com>,
> Per Vognsen <per.vognsen@gmail.com> wrote:
>
> >In the related case of an
> >algebraic variety V over a finite field F_q (where q = p^n for some
> >prime p) we have the exponential generating function
> >
> >Z(T) = exp(sum(n positive) #(points of V over F_{q^n}) T^n / n!)
> >
> >which is the so-called local zeta function of the variety.
>
> Right. Part of what used to bug me immensely is that I didn't
> see how this formula was related to other things called zeta functions.

(I think there shouldn't be a factorial there)

It's sort of weird, I think, because for a lot of application it seems
like the L-function is much more analogous to the Riemann zeta function.

Of course, Hartshorne (Ex C.5.4) has the following (paraphrased)

Given a Scheme X of finite type over Spec Z, let

Zeta_X(s) = \prod (1 - N(x)^-s)^-1

where the product is taken over all closed points x and N(x) is the
number of elements of the residue field k(x)= O_x / m_x. Doing this for
X = Spec Z gives you the usual zeta function.

Then, if X is of finite type over F_q, we can do the same thing and get

Zeta_X(s) = Z(X,q^-s)

To define Z(X,t), let N_r be the number of rational points of X x_F_Q
\bar{F_q} (the algebraic closure of F_q) over F_q^r. Then stick these
into

Z(X,t) = Exp(\sum N_r t^r/r)

When I'm more awake I might actually try to understand this and relate
this definition to the one your state (N_r is also the number of fixed
points of the Frobenius automorphism).

Aaron

John Baez

In article <slrnd95p5l.8r3.robert@localhost.localdomain>,
Robert C. Helling <robert@helling-dell600.iuhb02.iu-bremen.de> wrote:

>will there be a part II of this post where you explain the relation to
>Gutzwiller's trace formula and zeta functions of operators used to
>regularize QFT's?

Umm, right now I'm much more fascinated by other things, like trying
to understand the network of ideas related to zeta functions and
L-functions in number theory - stuff like the Weil hypotheses, the
Taniyama-Shimura-Weil conjecture, and the Langlands program. After
about a year of battering my head against these, they seem like
they're starting to make sense... either that or I've knocked myself silly!

I want to write more about this stuff. So, I can't promise to get around
to the other aspects of zeta functions that you mention. But, they *are
related*, so I should keep them in mind.

John Baez

In article <abergman-14F5A4.02463127052005@localhost>,
Aaron Bergman <abergman@physics.utexas.edu> wrote:

>When I'm more awake I might actually try to understand this and relate
>this definition to the one you state (N_r is also the number of fixed
>points of the Frobenius automorphism).

I think I understand this stuff now; the key is that it has absolutely
nothing to do with algebraic geometry, schemes, Frobenius automorphisms
or all that jazz. It's just a fact about a 1-1 and onto map

f: X -> X

We can define

zeta(s) = product_P 1/(1 - p^{-|P|s})

where P ranges over all periodic orbits of f and
|P| is the size of the periodic orbit.

We can also define

Z(u) = exp(sum_{n>0} |fix(f^n)| u^n / n)

where |fix(f^n)| is the number of fixed points of f^n.
Then we have

zeta(s) = Z(p^{-s})

I gave the proof in "week216" (using slightly different notation).

We can then apply this to algebraic geometry by letting f be a
Frobenius automorphism, and see that two different-looking definitions
of the "zeta function" of a scheme agree. But, we can also apply it to
dynamical systems and all sorts of other stuff.

------------------------------------------------------------------------

Michael Berry on the dreamt-of "Riemann dynamical system". He is:

"... absolutely sure that ... someone will find a clever way to make it
in the lab. Then you'll get the Riemann zeros out just by observing its
spectrum."

"Finding this system could be the discovery of the century." Berry says.
It would become a model system for describing chaotic systems in the
same way that the simple harmonic oscillator is used as a model for all
kinds of complicated oscillators. It could play a fundamental role in
describing all kinds of chaos. The search for this model system could
be the holy grail of chaos. Until [it is found] we cannot be sure of
its properties, but Berry believes the system is likely to be rather
simple, and expects it to lead to totally new physics. It is a
tantalising thought."

From:

Martin Gutzwiller, Quantum Chaos, Scientific American, January 1992,
available at http://www.maths.ex.ac.uk/~mwatkins/...ntumchaos.html

Allan Adler

baez@math.removethis.ucr.andthis.edu (John Baez) writes:

> Umm, right now I'm much more fascinated by other things, like trying
> to understand the network of ideas related to zeta functions and
> L-functions in number theory - stuff like the Weil hypotheses, the
> Taniyama-Shimura-Weil conjecture, and the Langlands program. After
> about a year of battering my head against these, they seem like
> they're starting to make sense... either that or I've knocked myself silly!
> I want to write more about this stuff. So, I can't promise to get around
> to the other aspects of zeta functions that you mention. But, they *are
> related*, so I should keep them in mind.

When I was a graduate student and a while afterwards, I wanted to be
very sure of never doing war related research. After getting involved
with number theory and algebraic geometry and mathematical logic, I
thought I had found remote corners of mathematics without discernible
military applications. Then I found out that number theory is important
in code breaking and that algebraic geometry and number theory are of
increasing importance in physics. I still haven't made up my mind
whether the physics they are important in is really physics or is really
mathematics, but I'm a lot less comfortable in my useless niche than I
used to be. I certainly want to know the answers to basic questions,
both in mathematics and in physics, but I'm also concerned with the fact
that there is always the potential to use it to bring ever greater forms
of devastation into existence and I don't want to be a part of that. I'm
not really sure how to deal with this renewed concern or how to really
measure the relevance of any particular area of research. I'd be
interested in knowing how others deal with these issues.

I guess that what started me wondering about this again was when the
physicists at IHES wanted to have a copy of the preliminary manuscript
for my book, Moduli of Abelian Varieties, which I had been happy to
think had nothing to do with reality and was just endless fun with
purely mathematical objects.

--
Ignorantly, Allan Adler <ara@zurich.csail.mit.edu>
* Disclaimer: I am a guest and *not* a member of the MIT CSAIL. My
actions and * comments do not reflect in any way on MIT. Also, I am
nowhere near Boston.