Author: Kowalski

Should the proof of a theorem (taken in isolation) allow us to reconstitute precisely its statement? That seems like an interesting question, and I guess my personal answer would be that it should, more or less, given maybe enough context information (and with some restrictions on the length of the proof).

However, there might be other opinions. For instance, it is clear that if the proof does not make use, explicitly, of one of the assumptions, by hiding it in a computation or check left to the reader, then the reconstruction from the proof might miss it (I mentioned earlier reading a proof of the Banach-Alaoglu theorem where the important fact that one works with the weak-* topology is hidden from view).

In this spirit, today’s challenge is to find the theorem for which this short sentence is supposedly a proof:

The heat kernel defines a renormalization-group invariant plaquette action.

Back in the days when fractals where the most fashionable thing, I had heard of fractal cabbages, and seen pictures of them. However, they are typically not available in French or American stores, so I didn’t see a real one until noticing that they are very common in Swiss supermarkets. Here’s a picture of one, but I should say that pictures do not quite convey the actual feel of seeing and handling this vegetable (not to mention eating it — and it is indeed quite tasty). The diameter (of horizontal cross-sections) of the specimen displayed here is between 10 and 15 centimeters, and the height is slightly larger.

Every once in a while, “at an uncertain hour”, algebraic fever returns, and I look at inequalities with mistrust. (This is a dramatized introduction, to be sung to the tune of The Rime of the Ancient Mariner; as Wodehouse would say, the poet Coleridge puts these things well.)

Compared with an identity such as

$latex \sum_{n\geq 1}{\frac{1}{n^2}}=\frac{\pi^2}{6},$

which one can easily imagine occupying pride of place in a platonic heaven, what is one to make of an inequality like

$latex |K(p)|\leq 2\sqrt{p},$

where

$latex K(p)=\sum_{x=1}^{p}{\exp\left(2i\pi \frac{x+x^{-1}}{p}\right)},$

for every prime number p? (This is the famous Weil bound for Kloosterman sums). Or what should one think of a statement like: for every ε>0, there exists a constant C_ε such that

$latex d(n)\leq C_{\eps}n^{\eps}$

for all positive integers n, d(n) being the number of positive divisors of n? (See this post by T. Tao for an enlightening discussion of this well-known inequality, which was apparently first proved by Runge in 1885, in a paper in Acta Mathematica on solvable equations of the type x⁵+ux+v=0 — this slightly surprising reference is given in Montgomery and Vaughan’s Multiplicative Number Theory).

The suspicion that inequalities are not quite “right” may have led to a number of devices to transform them into equalities (or identities) as much as possible. For instance, one may say that any inequality of the type

$latex A\geq 0,$

where A is an arbitrarily complicated real-valued expression (notice that any inequality could be written in this way…) is a bad version of an identity

$latex A=B^2$

where B is some more intrinsic (and possibly even more complicated) expression. Note that this is in fact of some importance in logic (and in algebra, with the theory of real fields): extending slightly, it shows that the positive integers are definable existentially in the integers by the first order formula φ(n) given by

$latex \exist a\ \exist b\ \exist c\ \exist d,\ n=a^2+b^2+c^2+d^2$

in the language of rings (and let me recall the much subtler formula of J. Robinson that extracts the integers from the rationals, though not purely existentially — whether the latter is possible is still an open problem).

I have also heard it said that Selberg sometimes claimed that his whole career was built on the fact that the square of a real number is non-negative, but I have no idea if he actually did — at the very least, it seems to completely ignore the Trace Formula…

Turning to Kloosterman sums, algebraists certainly sleep better at night knowing that the “correct” statement is that

$latex K(p)=\sqrt{p}(\alpha_p+\bar{\alpha}_p)$

for some complex number α_p of modulus 1, which can be defined in some beautifully elegant manner: the Weil bound then becomes a simple matter of neglecting part of this interesting information, and only remembering the triangle inequality.

Another way of understanding certain inequalities is to rephrase them as instances of bounds arising from the norm of a linear operator between normed vector spaces: if T is such an operator, its norm is equal to the infimum of numbers C such that

$latex ||T(v)||\leq C||v||$

for all vectors in the source space. So, for instance, one of the large sieve inequalities, as a purely analytic statement, is that for any N and any complex numbers a_n we have

$latex \sum_{q\leq Q}{\ \ \ \ \ \sum_{1\leq a\leq q,\ (a,q)=1}{\left|\sum_{n=1}^N{a_n\exp(2i\pi an/q)}\right|^2}}\leq (N-1+Q^2)\sum_{n}{|a_n|^2}$

and one could say that N-1+Q² is here a placeholder for the “right” quantity which is simply

$latex \Delta_{N,Q}=||T_{N,Q}||^2,$

the norm of some (fairly obvious) linear map between two finite-dimensional Hilbert spaces. The implied criticism would be that it is only because we are not clever enough to find a formula for this norm that we have to do with disappointing inequalities. (Though Ramaré’s investigations of eigenvalues of the large sieve operator show that this operator is in fact quite mysterious: in the critical case where N and Q² are of comparable size, there seems ot be a limiting distribution for the eigenvalues, but it has a very strange look).

This method of introducing linear operators applies for many well-known inequalities, for instance the Cauchy-Schwarz and Hölder inequalities: they can be interpreted as giving the formula for the norms of linear forms

$latex f\mapsto \int_X{f(x)g(x)d\mu(x)}$

between suitable L_p spaces.

But now, this ghastly tale being told, I come back to my analytic senses: I am sure that there is much more to inequalities than being des identités manquées! But it’s not clear if, or how, this might be formalized. Maybe what is needed is a very elementary inequality where the best possible constant is known, but involves much more sophisticated notions than the statement of the inequality? Possibly, the Hilbert inequality

$latex \left|\sum_{1\leq n,m\leq N}{\ \ \ \frac{a_nb_m}{n+m}}\right|\leq \pi ||a|| ||b||$

(again with arbitrary complex coefficients a_n and b_m, and with arbitrary N) might be interpreted in this way. The constant π is here best possible, but since π occurs everywhere from the most elementary mathematics, it may not be “sophisticated” enough to carry conviction. Are there other known operators with similarly simple descriptions and norm known to be very complicated numbers (in some sense)?

Another possibility towards proving that inequalities are unavoidable would be to look at something like the function

$latex \epsilon\mapsto \max_{n\geq 1}\ \ \frac{d(n)}{n^{\epsilon}}$

and hope to show that, in some sense, it is “much more complicated” than the divisor function, and thus unlikely to be replaceable by something nicer — though in that specific case, it seems not clear that it can be true, because the divisor function itself is really quite a complicated object already (for instance, with respect to its algorithmic computability).

One situation in which, I think, there is a fairly general consensus that the inequality can not be replaced in a non-tautological manner even by an asymptotic formula, is that of class numbers of imaginary quadratic fields, a famous problem going back to Gauss. This can in fact be described very elementarily (though the algebraic interpretation is probably the only way to justify why one would look at this particular question): for a positive squarefree number d, let h(d) be the number of integral solutions (a,b,c) , with no common factor e>1, to the (in)equations

$latex -a<b\leq a\leq c,\ \text{with }\ b\geq 0\ if\ a=c,\ \ \ \ \ b^2-4ac=d.$

The question is then to know the size of h(d), and this is a notoriously difficult problem. It is relatively simple to show an inequality like

$latex h(d)<2\sqrt{d}\log d$

and it is further known that, for every ε>0, there is a constant C_ε>0 such that

$latex h(d)\geq C_{\epsilon}d^{1/2-\epsilon}$

(a theorem due to Siegel, which is much harder, and for which no one knows how to compute C_ε if ε is small enough; and here I can’t help quoting what may be the century’s greatest understatement, taken from MathWorld: “There are at least two Siegel’s theorems“).

Those two inequalities show that the class number is of size about d^1/2, in some sense, but after extensive work, I don’t think anyone who has looked at the problem in some depth would expect to get even an asymptotic formula

$latex h(d)=g(d)+\text{(smaller remainder)}$

where the function g(d) is “elementary” in a reasonable sense. Of course, this is not a formal statement, and it’s not clear if a precise version is possible (this may be another interesting somewhat meta-mathematical problem to consider…)

In that particular case, algebraists might exclaim that the Class Number Formula of Dirichlet provides the required “identity” version of h(d): up to minor (explicit) factors, we have

$latex h(d) \simeq \sqrt{d}L(1,\chi_{-d})$

relating the mysterious class number to a special value of a Dirichlet L-function associated with the underlying imaginary quadratic field. But this gives essentially no information on the size of h(d), so this transcription is not as convincing as what we saw in the case of Kloosterman sums. (Although this formula has been the basis of the deepest estimates for h(d), which have been deduced from bounds for L-functions.)

When I was young, I read most of the works of Jules Verne with enthusiasm. Thinking that my own children might one coming day also start doing this, I recently started looking again at some of the books I remember particularly, to check whether this infatuation was something to hide shamefully, or if instead I could only hope that my boys would develop a liking for them too. (As an aside, it should be said that in the Standard Model of French Literature, it is indeed a shameful thing to like Jules Verne; if I remember right, not a single line of his was included in the textbooks of French literature that I used in high-school — which, it must also be said tangentially, only had a single short excerpt from the terrible Vicomte de Bragelonne to represent the whole output of Alexandre Dumas père and his many “collaborators”, a book so bad that only a professor of literature with a spoilt and perverse mind would read it entirely to select parts of it instead of picking instead one of the many outstanding pyrotechnic displays of Les trois mousquetaires or some of the quite deep psychology of Le comte de Monte Cristo).

Thanks to the internet (and the fact that the works of Jules Verne, including many translations, belong to the public domain) it is fairly easy today to survey the whole Voyages extraordinaires without moving from one’s armchair. I recommend (for French-reading readers) to look at the versions in the French National Library archive since, being original versions, they contain the drawings which were part of the charm of the books for me (though I would typically read them in second-hand modern paperbacks, those always included the same drawings to illustrate the most suspenseful parts of the action).

The previous paragraph probably already reveals that my conclusion is that there is a lot to like in Jules Verne. The most obvious (and the most attractive force for me then) is that Jules Verne is, if not the only one, then certainly the best known writer in French literature of the 19th Century to show some interest in science. This must have been rather striking at that time (and must explain partly his success). It is not even simply that he has a positive attitude to science (which after all can be argued endlessly to be a misguided viewpoint), but the simple fact that science exists at all in his work isolates him from all of his equally-remembered contemporaries (I have no idea if there existed similar writers at the time who just never became successful enough to be remembered). He certainly conveys a strongly positive idea of science overall, and manages in his books to present aspects of it in very different styles: mineralogists, natural scientists, physicists, mechanicians, aeronauts, all appear somewhere. These scientists are not always mad or paranoiac or even absent-minded (though those also appear!), and can be very fascinating and attractive for children or teenagers interested in scientific matters. There are touches of humor which are also very pleasant (another quite un-French trait at that time), as for instance the deadpan description of the “Gun Club” and its artillerists disappointed by peace at the beginning of De la Terre à la Lune (“From earth to the moon”).

How far he was ready to include science in his plots can be seen from this equation appearing in the sequel to De la terre à la lune, where it plays the role of a thunderbolt showing to the three brave men in the cannonball that the amount of powder was miscalculated and would prevent them from reaching the moon. Even more impressive is the final chapter of Sans dessus dessous (“Topsy-turvy”) — with the charming subtitle Dont peu de personnes prendront connaissance (“That few people will acquaint themselves with”) — where M. Badoureau, Ingénieur des Mines, gives full and complete details of the computations explaining how a gigantic cannon, firing cleverly at the right point and time, might change the orientation of the earth so that its axis of rotation would become perpendicular to the ecliptic plane (i.e., bringing the axial tilt to zero). It is revealed in the book itself that the characters again miscalculated, underestimating the size of the earth by a factor of one thousand, so that their cannon shot barely moved the earth by the amount of three microns…

It has to be admitted that (as far as I remember) the scientists of Jules Verne are always male. He was not particularly progressist in many respects, as can also be seen from the many occasions when colonial questions and settings arise (in Africa and India in particular), but in one other aspect his world is strikingly different from the one that might be inferred from reading other 19th Century French novelists (beside containing those strange characters, scientists): it includes other countries besides France (!). In fact, his heros originate from a remarkable variety of places, and are portrayed with a remarkable display of sympathy. There are Frenchmen, of course, but also Englishmen (the heros of Trois semaines en ballon, for instance, though one is a Scotsman), Americans (from both North and South America, as in La Jangada), Russians (in Michel Strogoff), Germans (professor Lidenbrock in Voyage au centre de la terre, where the old Icelandic explorer Arne Saknussem also plays an important role…), Dutchmen, Turks (both in Kéraban le têtu, “Keraban the stubborn”), etc.

Here are a few of the books which I remember with pleasure, and which are not among his best known works:

(1) I already mentioned Kéraban le têtu; the plot device here is quite bizarre: the Turk merchant Keraban, who is visited by a Dutch friend, refuses to pay a new tax to cross the Bosphorus to go back to his house on the Oriental side of Istanbul; being moreover prone to sea-sickness, he chooses to go back home by going around the Black Sea on land. At the very end, finally arrived after much delay and subplotting, pressing circumstances require that he cross the Bosphorus in the opposite direction without losing a minute. How will he do it without paying the iniquitous tax?

(2) The little-known La Jangada is the story of a big log-raft going down the Orinoco river over 800 miles. I don’t remember much of the plot, except that there is a secret message that is only decrypted in the nick of time to save an innocent man at the end, and that a crucial part of the plot involves an amusing game of following an almost infinitely long liana in the surrounding jungle — of course, at its end is found a poor young fellow trying to use it to hang himself, and who, being saved, will find love during the remainder of the work…

(3) In Le testatement d’un excentrique, a rich millionaire leaves a last will where more or less randomly chosen strangers will play a gigantic Game of Goose with the States of the Union taking the place of the spaces of the board. (This type of idea has apparently been used in other books afterwards). Who will win?

(4) In Les aventures d’Hector Servadac, one of the strangest, a comet just barely touches the Earth tangentially around Algiers and takes with it a small part of land including atmosphere. After one orbit that takes it and its unfortunate inhabitants far from the Sun in the coldness of space, the comet comes back and brings the heros back in the exact same spot (though they do have to escape in a balloon; amusingly, the “vilain” in this tale is the French astronomer who tries to stay on the comet to continue exploring space…)