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This Week's Finds in Mathematical Physics (Week 218)

  1. Oct 11, 2006 #1
    Also available as http://math.ucr.edu/home/baez/week218.html

    June 5, 2005
    This Week's Finds in Mathematical Physics - Week 218
    John Baez

    Classes are over! Summer is here! Now I can finally get some work done!
    I'll be travelling to Sydney, Canberra, Beijing, Chengdu and Calgary, but
    mainly I want to finish writing some papers.

    First, though, I need to recover from a hard quarter. I need to goof off a
    bit! I spent most of yesterday lying in bed reading. Now I want to talk
    some more about number theory.

    Let's see, where were we? I had just begun to introduce the theme of
    L-functions and their corresponding automorphic forms. My ultimate goal
    is to understand the Langlands Conjectures well enough to give a decent
    explanation of what they say. Instead of simply stating them, I'd like
    to really make them plausible, and this will take quite an elaborate warmup.
    So, this Week I want to talk about some background.

    Actually, this reminds me of sometime Feynman wrote: whenever he worked on a
    problem, he needed the feeling he had some "inside track" - some insight or
    trick up his sleeve that nobody else had. Most of us will never be as good
    as Feynman at choosing an "inside track". But I think we all need one to
    convert what would otherwise be a dutiful and doomed struggle to catch up
    with the experts into something more hopeful and exciting: a personal quest!

    For anyone with a physics background, a good "inside track" on almost any math
    problem is to convert it into some kind of crazy physics problem. It doesn't
    need to be realistic physics, just anything you can apply physics intuition
    to! This is part of why string theorists have been so successful in cracking
    math problems. It also underlies Alain Connes' attempt to prove the Riemann

    1) Alain Connes, Noncommutative Geometry, Trace Formulas, and the Zeros of
    the Riemann Zeta Function. Ohio State course notes and videos at

    Alain Connes, Trace Formula in Noncommutative Geometry and the Zeros of
    the Riemann Zeta function, available as math.NT/9811068.

    2) Mathilde Marcolli, Noncommutative Geometry and Number Theory,
    available at http://www.math.fsu.edu/~marcolli/ncgntE.pdf

    Of course, Connes also has another "inside track", namely his theory of
    noncommutative geometry.

    By the way: a number theorist I know says he thinks Connes has essentially
    proved the Riemann Hypothesis, in the same way that Riemann "essentially"
    proved the Prime Number Theorem. Namely, he has reduced it to some facts that
    seem obviously true! Of course, it took about 40 years, from 1859 to 1896,
    for Riemann's plan to be fulfilled by Hadamard and De La Vallee Poussin. So,
    even if Connes' insights are correct, it may be a while before the Riemann
    Hypothesis is actually proved.

    For anyone with a background in geometry, a good "inside track" on almost any
    math problem is to convert it into a geometry problem. In the case of number
    theory this trick is old news, but still very much worth knowing. It's based
    on an analogy which I began discussing in "week198".

    The analogy starts out like this:


    Integers, Z Polynomial functions on the complex plane, C[z]
    Rational numbers, Q Rational functions on the complex plane, C(z)
    Prime numbers, P Points in the complex plane, C

    Why is this analogy good? Well, for starters:

    Every rational number is a ratio of integers.

    Every rational function is a ratio of polynomials.

    Better yet:

    Every integer can be uniquely factored into primes
    (modulo invertible integers, namely +1 and -1).

    Every complex polynomial can be uniquely factored into linear polynomials
    (modulo invertible polynomials, namely nonzero constants).

    There's one linear polynomial z-a for each point a in the complex plane,
    so PRIMES are like POINTS in the complex plane.

    We can make this precise using the concept of "spectrum", which I defined in
    "week199". Ignoring a certain little sublety which is discussed there:

    The spectrum of Z is the set of prime numbers.

    The spectrum of C[z] is the complex plane.

    This way of thinking lets us treat the spectrum of any algebraic extension of
    the integers, like the Gaussian integers, as a "covering space" of the set of
    prime numbers. I've already drawn this picture:

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

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

    But, now I'm saying that the "line" down below really acts like the
    complex *plane*. Taking this strange idea seriously leads to all sorts
    of amazing insights.

    For example, if you poke a hole in this "plane" at some prime, there's
    something like a little *loop* that goes around this hole! In other words,
    there's a sense in which the spectrum of Z has a nontrivial "fundamental
    group", which contains an element for each prime. Technically this group
    is called the Galois group Gal(Qbar/Q), and we get an element in it for
    each prime, called the "Frobenius automorphism" for that prime.

    Another cool thing is that we can study integers "locally", one prime at a
    time, just like we study complex functions locally. We can analyze functions
    at a point using Taylor series and Laurent series. And, we can stretch our
    analogy to include these concepts:


    Integers, Z Polynomial functions on the complex plane, C[z]
    Rational numbers, Q Rational functions on the complex plane, C(z)
    Prime numbers, P Points a in the complex plane, C
    Integers mod p^n, Z/p^n (n-1)st-order Taylor series, C[z]/(z-a)^n
    p-adic integers, Z_p Taylor series, C[[z-a]]
    p-adic numbers, Q_p Laurent series, C((z-a))

    All the weird symbols are just the standard notations for these gadgets.
    The analogy goes as follows:

    To study a polynomial "at a point" a in the complex plane,
    we can look at its value modulo (z-a), or more generally mod (z-a)^n.

    To study an integer "at a prime" p,
    we can look at its value modulo p, or more generally mod p^n.

    This is nice because the value of a polynomial modulo (z-a)^n is just its
    Taylor series at the point a, where we keep terms up to order n-1.

    We can also also take the limit as n -> infinity. If we do this to the
    integers mod p^n we get a ring called the "p-adic integers". For example,
    a typical 3-adic integer, written in base 3, looks like this:


    They're just like natural numbers in base 3, except they go on forever to the
    left! We add and multiply them in the obvious way, for example:

    + ......10201101012201201122010012

    If we take the same sort of limit for Taylor series, we get Taylor series
    that go on forever - in other words, formal power series.

    We can also ratios of p-adic integers, which are called p-adic numbers,
    and ratios of Taylor series, which are called Laurent series. A typical
    3-adic number, written in base 3, looks like this:


    Laurent series can be used to describe functions that have a pole at some
    point, like rational functions. Similarly, p-adic numbers can be used
    to describe rational numbers. Using more math jargon:

    For any point a in C, there's a homomorphism
    from the field of rational functions
    to the field of Laurent series,
    which sends polynomials to Taylor series.

    For any prime p, there's a homomorphism
    from the field of rational numbers
    to the field of p-adic numbers,
    which sends integers to p-adic integers.

    This lets us study rational numbers "locally" at the prime p using p-adic
    numbers, just as we can study a rational function locally at a point using
    its Laurent series. This technique can be quite useful. For example, a
    polynomial equation can have rational solutions only if it has p-adic
    solution for all primes p.

    We might hope for the converse, but then we would be ignoring a funny extra
    "prime" besides the usual ones... something called the "real prime"!

    The point is, besides being able to embed the rational numbers in the p-adics
    for any prime p, we can also embed them in the real numbers! This embedding
    is a bit different than the rest: it's based on a weird thing called an
    "Archimedean valuation", while the usual primes correspond to non-Archimedean

    I'm sort of joking here, since if you're more used to real numbers than
    p-adics, you'll probably find Archimedean valuations to be *less* weird than
    non-Archimedean ones. The Archimedean valuation on the rational numbers is
    just the usual absolute value, while the non-Archimedean ones are other
    concepts of "absolute value", one for each prime p. If we take limits of
    rational numbers that converge using the usual distance function |x-y|, we
    get real numbers; if we take limits that converge using one of the
    non-Archimedean versions of this distance function, we get p-adic numbers.
    But from the viewpoint of number theory, it's the Archimedean valuation
    that's the odd man out! It indeed does act very weird and different than
    all the rest. That's why someone wrote this book:

    3) M. J. Shai Haran, The Mysteries of the Real Prime, Oxford
    U. Press, Oxford, 2001.

    ... which you will see is deeply connected to mathematical physics.

    If we take this weird "real prime" into account, things work better.
    We sometimes get results saying that some kind of polynomial equations
    have a rational solution if they have p-adic solutions for all primes p
    and also a real solution. For example, Hasse proved this was true for
    systems of quadratic equations in many variables.

    Results like this are called "local-to-global" results, since they're
    analogous to constructing a function from local information, like its
    Laurent series at all different points.

    In 1950, in his famous PhD thesis, John Tate came up with a clever way to
    formalize this "Laurent series at all different points" idea in the context
    of number theory. To do this, he formed a ring called the "adeles".

    Indeed, this is what my whole discussion so far has been leading up to!
    Adeles are a really nice formalism, and you pretty much need to understand
    them to follow what people are doing in work on the Langlands Conjectures,
    or even simpler things, like class field theory. But, adeles seem like
    an arbitrary construction until you see them as an inevitable outgrowth
    of our desire to study integers "locally" at all different primes, including
    the real prime.

    The definition is simple. An adele consists of a p-adic number for each
    prime p, together with a real number... but where all but finitely many
    of the p-adic numbers are p-adic integers!

    This is the number-theoretic analogue of a Laurent series for each point in
    the complex plane, including the point at infinity... but with poles at only
    finitely many points! We could call such a thing an "adele for the rational

    Any rational function gives such a thing, just as any rational number gives
    an adele. And, we don't lose any information this way:

    There's a one-to-one (but not onto) homomorphism
    from the rational functions to the adeles for the rational functions.

    There's a one-to-one (but not onto) homomorphism
    from the rational numbers to the adeles for the rational numbers.

    So, our table now looks like this. For good measure, I'll combine it with
    the related table in "week205":


    Integers Polynomial functions on the complex plane
    Rational numbers Rational functions on the complex plane
    Prime numbers Points in the complex plane
    Integers mod p^n (n-1)st-order Taylor series
    p-adic integers Taylor series
    p-adic numbers Laurent series
    Adeles for the rationals Adeles for the rational functions
    Fields One-point spaces
    Homomorphisms to fields Maps from one-point spaces
    Algebraic number fields Branched covering spaces of the complex plane

    There's a *lot* more to say about this analogy, but I think this is enough
    for now. Again, one of my secret goals was to start getting you comfy with
    adeles and the idea of studying number theory "locally".

    For more on the geometrical side of number theory, I again recommend these:

    4) Juergen Neukirch, Algebraic Number Theory, trans. Norbert Schappacher,
    Springer, Berlin, 1986.

    5) Dino Lorenzini, An Invitation to Arithmetic Geometry, American
    Mathematical Society, Providence, Rhode Island, 1996.

    But now, back to the subject of "inside tracks" - sneaky ways to get the
    beneficial feeling that you have secret insights into some problem.

    For anyone with a background in categories, a good "inside track" on
    almost any math problem is to categorify it: to see that people are using
    sets where they could, and therefore *should*, be using categories or

    I've already hinted that zeta functions are an example of
    "decategorification". Now I'd like to make this more precise.

    Let's think about the zeta function of a set X equipped with a one-to-one
    and onto function

    f: X -> X

    If you're a physicist, you might call this a "discrete dynamical system",
    with f describing one step of "time evolution". If you're a mathematician,
    you might call this a "Z-set". After all, for any group G, a "G-set" is a
    set equipped with an action of G. If G = Z (the additive group of integers),
    this amounts to a one-to-one and onto function from some set to itself.

    No matter what you call them, these are fundamental things. So, let's
    look at the *category* of Z-sets! Here the objects are Z-sets and the
    morphisms are functions that commute with time evolution.

    As explained near the end of "week216", we can define a kind of zeta
    function for a Z-set as follows:

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

    where |fix(f^n)| is the number of fixed points of f^n. Of course, this
    only makes sense if all these numbers are finite; henceforth I'll assume
    my Z-sets are "finite" in this special sense.

    It turns out that you know a finite Z-set up to isomorphism if you know its
    zeta function. So, a zeta function is just a sneaky way of talking about
    an ISOMORPHISM CLASS of finite Z-sets.

    This is a fancy example of something we all learn as kids: counting! When
    we "count" a finite set, assigning a natural number to it, we are really
    determining its isomorphism class. Two finite sets are isomorphic if and
    only if they have the same number of elements. Operations on finite sets,
    like disjoint union and Cartesian product, are what give rise to operations
    on natural numbers, like addition and multiplication.

    Summarizing this, we have the following motto, suitable for making into
    a bumper sticker:


    Similarly, this is what we're seeing now:


    Beware: here I'm only talking about zeta functions of the above form. There
    are lots of other things people call zeta functions. So, don't read too
    much into this statement. But don't read too little into it, either! With
    an extra twist we can get most of the zeta functions showing up in number
    theory. In number theory, we typically get a Z-set for each prime p,
    coming from the "Frobenius" for that prime. We thus get a bunch of "local"
    zeta functions Z_p(x), one for each prime. We then multiply these to get
    one big fat "global" zeta function:

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

    Each local zeta function is a formal power series, while this global zeta
    function is a Dirichlet series. As I mentioned in "week217", formal power
    series live in the monoid algebra of (N,+,0), while Dirichlet series live
    in the monoid algebra of (N,x,1). (N,+,0) is the free commutative monoid
    on one generator, while (N,x,1) is the free commutative monoid on countably
    many generators - the primes! Everything fits together sweetly.

    So, it's a good first step to think about the zeta function of a single

    Now, there's another motto along the lines of the above two, which I've
    talked about before:


    I explained this in "week185", "week190", and "week202". I've even taught a
    whole course on structure types (also known as "species") and the
    combinatorics of Feynman diagrams. The course notes by Derek Wise are
    available online:

    6) John Baez and Derek Wise, Quantization and Categorification, available at:

    So, I think this third example of decategorification is great. But, I'm not
    going to explain it in much detail here - just enough to say how it's related
    to zeta functions!

    A stucture type F is a gadget that gives a set F_n for each n = 0,1,2,....
    We think of the elements of F_n as "structures of type F" on an n-element
    set - for example, orderings, or cyclic orderings, or n-colorings, or
    whatever type of structure you like. We only require that permutations of
    the n-element set act on this set of structures. But for what we're doing
    now, let's also assume this set F_n is finite.

    Any structure type has a "generating function", which is a formal
    power series |F| given by

    |F|(x) = sum ------- x^n

    Isomorphic structure types have the same generating function. However,
    structure types with the same generating function can fail to be isomorphic.
    This is why I said generating functions are "a" decategorification of
    structure types, instead of "the" decategorification.

    Despite this defect, generating functions are still very useful in
    combinatorics. So, when we see a zeta function like

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

    as a trick for decategorifying Z-sets, we should instantly wonder if it's
    a generating function in disguise. And of course, it is!

    Actually it's easiest to leave out the exponential at first. This power

    sum_{n>0} |fix(f^n)| x^n / n

    is the generating function for the structure type "being cyclically
    ordered and equipped with a morphism to the Z-set X".


    We "cyclically order" a finite set by drawing it as a little circle of dots
    with arrows pointing clockwise from each dot to the next. A cyclically
    ordered set is automatically a Z-set in an obvious way. So, here's a type
    of structure you can put on a finite set: cyclically ordering it and
    equipping the resulting Z-set with a morphism to the Z-set X.

    And, if you work out the generating function of this structure type, you get

    sum_{n>0} |fix(f^n)| x^n / n

    Check it and see!

    What about the exponential? Luckily, there's a standard way to take the
    exponential of a structure type: to put an exp(F)-structure on a finite set
    S, we chop S into disjoint parts and put an F-structure on each part. So,
    the zeta function

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

    is the generating function for "being chopped up into cyclically ordered
    parts, each equipped with a morphism to the Z-set X".

    But this is just a long way of saying: "being made into a Z-set and equipped
    with a morphism to the Z-set X".

    Or, in category theory jargon, "being a Z-set over X".



    By the way, this is the kind of thing you could do with *any* structure type
    F. Given an F-structured set X, we get a new structure type "being an
    F-structured set equipped with a morphism to X". Or, in category theory
    jargon, "being an F-structured set over X". The generating function of this
    could be called the "zeta function" of our F-structured set X. I have no
    idea how important this is...

    ... but I want to keep gnawing away on the connection between zeta functions
    and the generating functions of combinatorics, because to understand number
    theory, I need all the "inside tracks" I can get!

    Quote of the week, brought to me by David Corfield:

    The scientific life of mathematicians can be pictured as a trip inside the
    geography of the "mathematical reality" which they unveil gradually in their
    own private mental frame.

    It often begins by an act of rebellion with respect to the existing dogmatic
    description of that reality that one will find in existing books. The young
    "to be mathematician" realize in their own mind that their perception of the
    mathematical world captures some features which do not fit with the existing
    dogma. This first act is often due in most cases to ignorance but it allows
    one to free oneself from the reverence to authority by relying on one's
    intuition provided it is backed by actual proofs. Once mathematicians get to
    really know, in an original and "personal" manner, a small part of the
    mathematical world, as esoteric as it can look at first, their trip can
    really start. It is of course vital not to break the "fil d'arianne" which
    allows one to constantly keep a fresh eye on whatever one will encounter
    along the way, and also to go back to the source if one feels lost at times...

    It is also vital to always keep moving. The risk otherwise is to confine
    oneself in a relatively small area of extreme technical specialization, thus
    shrinking one's perception of the mathematical world and its bewildering

    The really fundamental point in that respect is that while so many
    mathematicians have been spending their entire life exploring that world they
    all agree on its contours and on its connexity: whatever the origin of one's
    itinerary, one day or another if one walks long enough, one is bound to reach
    a well known town i.e. for instance to meet elliptic functions, modular
    forms, zeta functions. "All roads lead to Rome" and the mathematical world
    is "connected".

    In other words there is just "one" mathematical world, whose exploration is
    the task of all mathematicians, and they are all in the same boat somehow.

    - Alain Connes

    Previous issues of "This Week's Finds" and other expository articles on
    mathematics and physics, as well as some of my research papers, can be
    obtained at


    For a table of contents of all the issues of This Week's Finds, try


    A simple jumping-off point to the old issues is available at


    If you just want the latest issue, go to

  2. jcsd
  3. Oct 11, 2006 #2
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