When, exactly two years ago, I published my earlier post containing the story of J. Ménard, I was apparently suspected by some people of being the author of that text. I tried for a long time to find the original Spanish version mentioned in the text, whose existence conclusively refutes this assertion (since my understanding of Spanish is, unfortunately, non-existent). After much effort, I have finally succeeded!
This afternoon, while chatting with Will Sawin, between addressing rather technical points of ongoing projects, we observed that although we’ve seen seminar talks shared between two speakers, it was never with simultaneous speakers. It was just a step to jump from there to the idea that someone should write an Opera about a mathematical seminar talk, which — as opera does — would allow a duet, or trio, or quartet, or quintet, or sextet, of simultaneous speakers, including maybe some from the audience, or the chairperson trying to control the situation.
Unfortunately, I don’t know music, but if I were twenty years old, I’d be very tempted to write “Seminar”, the definitive opera about a math talk. At least, I can think of a libretto and try to write it (which language? I think French is best here, although the title should then be “Séminaire” or “L’exposé” in that case; which topic? good question — of course it would have to be a real talk; which style?)
In any case, I can safely predict a triumph in enlightened circles.
Keen-memoried readers will remember the word appearing before on this blog. As one of the happy few who have read “The Unconsoled” twice, I applaud with pleasure the honor given to K. Ishiguro! (If my credentials are disputed, let me clearly state that I can answer the question: “Which spectacular goal scored by a Dutch player during the 1978 World Cup is described in the book?” — or rather, almost, since the description is ambiguous and could apply to two goals by the same player, during different games; I actually remember watching at least one of them).
As the title indicates, we are looking this time at the support of the limiting random Fourier series that arose in that first paper, namely
where is a sequence of independent Sato-Tate-distributed random variables. In a strict sense, this should be a very short paper, since the computation of the support is easily achieved using some basic probability and elementary properties of Fourier series: it is the set of continuous functions such that (1) the value of at is real and belongs to ; (2) the function has purely imaginary Fourier coefficients for ; (3) we have for all .
So why is the paper 26 pages long? The reason is that this support (call it ) is a rather interesting set of functions, and we spend the rest of the paper exploring some of its properties. Most importantly, the support is not all functions, so we can play the game of picking our favorite continuous function on (say ) and ask whether or not belongs to .
- Fixing a prime , and , invertible modulo , does the Kloosterman path itself belong to the support? Simple computations show that it depends on ! For instance, the path for the Kloosterman sum , shown below, does not belong to the support. (As we observe, it looks like a Shadok, whose mathematical abilities are well-known — sorry, the last link is only in French ; I suggest to every French-aware reader to watch the corresponding episode, since the voice of C. Piéplu achieves the seemingly impossible in making this hilarious text even funnier…)
- On the other hand, the path giving the graph of the Takagi function (namely ) belongs to the support.
- But maybe the most interesting problem from a mathematical point of view is one of pure analysis: when we see a Kloosterman path (such as the one above), we only see its image as a function from to , independently of the parameterization of the path. So we can take any shape in the plane that can be represented as the image of a function satisfying the conditions (1) and (2) above, and ask: is there a reparameterization of that belongs to the suppport? For instance, for the Kloosterman paths themselves (as in (1) above), it is not difficult to find one: instead of following each of the segments making the Kloosterman path in time , one can insert a “pause” of length at the beginning and end of the path, and then divide equally the remaining time for the segments. (The fact that this re-parameterized path, whose image is still the same Kloosterman path, belongs to the support is then an elementary consequence of the Weil bound for Kloosterman sums).
- In general, the question is whether a given has a reparameterization with Fourier coefficients (rather, those of ) are all smaller than . This is an intriguing problem, and looking into it brought us into contact with some very nice classical questions in Fourier analysis, that I discuss in this later post. We only succeeded in proving the existence of a suitable reparameterization for real-valued functions, for reasons explained in the aforementioned later post, and it is an interesting analysis problem whether the result holds for all functions. A positive answer would in particular settle another natural question that we haven’t been able to handle yet: is there a space-filling curve in the support of the Kloosterman paths?
All this is great analytic fun. But there are nice arithmetic consequences of our result. By the definition of the support, we know at least that any has the property that, with positive probability, the actual path of the partial sums of the Kloosterman sums will come as close as we want (uniformly on ) to , and this is an arithmetic statement. For instance, simply because the zero function belongs to the support, we deduce that, for a large prime , there is a positive proportion of such that all partial sums
for , have modulus .
In other words, there is a non-zero probability that all the normalized partial sums of the Kloosterman sums are very small. (It is interesting to note that this is emphetically not true for character sums… the point is that their Fourier expansion involves multiplicative coefficients, so they cannot become smaller than .)
At the very least, nobody from the University of Oxford (except if some of the Anonymous collaborators of some of the plays were professors there). Indeed, all the editors listed on the covers come from other institutions.
In comparison with the 1987 edition (called more modestly, if apparently inaccurately, “The complete works”), the new version identifies more works where Shakespeare was involved, and (taking from the other hand) finds also more plays where other writers participated. This is all explained in fair detail in a companion book full of statistical studies of proportions of rhymes or of feminine endings, or other fine points of prosody. Maybe most interesting (to me) is the play “Arden of Fevershame” that is now attributed in part to Shakespeare at the very beginning of his career, since its theme (the story of a then fairly recent murder most foul committed among England commoners) is rather far from the themes of most of his other plays.
The impressive volumes also make for excellent book-ends.
And there is still apparently a further volume (or two) to come, of “alternative versions” of those plays that are known in two or more substantially different early texts (e.g., “King Lear”).
I am eagerly anticipating a similarly ambitious scholarly N.O.W (New Oxford Wodehouse); in fact, I am happy to volunteer for the exacting role of editor of the Jeeves & Wooster canon. Or, if objections are raised against the attribution of such a crucial part of the oeuvre to a Frenchman of Polish and Breton origins, I will gladly take responsibility for the volumes encompassing the acts of the fifth Earl of Ickenham, fewer in number but by no means in importance.