Arts – E. Kowalski's blog

The Magic Mountain

Once more, I have yielded to the arch-Tempter, the Book-Buying demon.

This time, it started when I bought (second-hand — I actually think today that only second-hand books are really authentic, unless of course the book is brand new) the translation by J. E. Woods of the novel “Joseph and his Brothers” by Thomas Mann. I expected that (like Joyce’s “Finnegans wake” and Faulkner’s “A Fable”, which I both own and in one case read) this was only a gesture of respect for the work of a writer that I admire. To my astonishment, I read this four-part fifteen-hundred page book (“The stories of Jacob”, “The young Joseph”, “Joseph in Egypt”, “Joseph the Provider”) in a few weeks, and found it too short, and realized that it was a masterpiece. The story of Joseph and Mut-em-Ênet in the third book is, indeed, an extraordinary act of literary empathy. And this story was written in exile by a conservative sixty-year old german, when most of everyone and everything he loved was either utterly betraying his culture or was being destroyed.

Well, so when I learnt (from a blog post of the ETH Bibliothek) that — after who knows how many years of work from the editors — the commented edition of this book was appearing in April this year (the Grosse kommentierte Frankfurter Ausgabe announced it in 2008 as “in plan, 2012”), I couldn’t resist and ordered it. I actually had already bought a German version of the book (“Die Geschichten Jaakobs”, “Der Junge Joseph”, “Joseph in Ägypter”, “Joseph der Ernährer”, to use the original titles), and since the available room in my apartment doesn’t really allow for more than one copy of thousand pages long German books, I donated these to my colleague Ian Petrow who had told me of his liking for the “Magic Mountain”.

But then, could I really keep my paperback German copies of “Der Zauberberg” and of “Doktor Faustus”, when both existed in the same amply commented edition? I couldn’t, donated the old ones (to the same colleague), and bought both. So here I am:

(on the left, the older (in)complete works of Shakespeare for scale).

The empty slot in the middle is that of the “Zauberberg”, which I am now trying to read in German, with much help from online dictionaries. And it reminds me that I started reading “The Magic Mountain” in Rutgers (and in translation, of course), when a friend there recommended it to me, especially because of the character of Lodovico Settembrini:

Auf dem Wege von links kam ein Fremder daher, ein zierlicher brünetter Herr mit schön gedrehtem schwarzen Schnurrbart und in hellkariertem Beinkleid, der, herangekommen, mit Joachim einen Morgengruss tauschte – der seine war präzis und wohllautend – und mit gekreuzten Füssen, auf seinen Stock gestützt, in anmutiger Haltung vor ihm stehen blieb.
GFKA, p. 88

For the Yiddish version, translated by Isaac Bashevis Singer, see here.

Like Joyce, Thomas Mann died in Zürich, and his grave can be found there.

Where will the Tempter bring me next? I believe that, most likely, it will be the Opere of Primo Levi, or those of Niccolò Machiavelli, although my Italian is now rather worse than my German.

“Jacques Ménard, autor de Nicolás Bourbaki”

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!

“Seminar”, the opera

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.

Consoled

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).

The support of Kloosterman paths

Will Sawin and I just put up on arXiv a preprint that is the natural follow-up to our paper on those most alluring of shapes, the Kloosterman paths.

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
$K(t)=t\mathrm{ST}_0+\sum_{h\not=0}\mathrm{ST}_h\frac{e^{2i\pi ht }-1}{2i\pi h},$
where $(\mathrm{ST}_h)_{h\in\mathbf{Z}}$ 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 $f\colon [0,1]\to \mathbf{C}$ such that (1) the value of $f$ at $t=1$ is real and belongs to $[-2,2]$ ; (2) the function $g(t)=f(t)-tf(1)$ has purely imaginary Fourier coefficients $\hat{g}(h)$ for $h\not=0$ ; (3) we have $|\hat{g}(h)|\leq 1/(\pi |h|)$ for all $h\not=0$ .

So why is the paper 26 pages long? The reason is that this support (call it $\mathcal{S}$ ) 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 $[0,1]$ (say $f_0$ ) and ask whether or not $f_0$ belongs to $\mathcal{S}$ .

For instance:

Fixing a prime $p_0$ , and $a_0$ , $b_0$ invertible modulo $p_0$ , does the Kloosterman path $K_{p_0}(a_0,b_0)$ itself belong to the support? Simple computations show that it depends on $(p_0,a_0,b_0)$ ! For instance, the path for the Kloosterman sum $\mathrm{Kl}_2(8,1;9)$ , 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…)

Kl_2(8,1;19)
On the other hand, the path giving the graph of the Takagi function $T$ (namely $f(t)=t+iT(t)$ ) belongs to the support.

Takagi function
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 $[0,1]$ to $\mathbf{C}$ , 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 $f$ satisfying the conditions (1) and (2) above, and ask: is there a reparameterization of $f$ 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 $p_0-1$ segments making the Kloosterman path in time $1/(p_0-1)$ , one can insert a “pause” of length $1/(2p_0)$ at the beginning and end of the path, and then divide equally the remaining time for the $p_0-1$ segments. (The fact that this re-parameterized path, whose image is still the same Kloosterman path, belongs to the support $\mathcal{S}$ is then an elementary consequence of the Weil bound for Kloosterman sums).
In general, the question is whether a given $f$ has a reparameterization with Fourier coefficients (rather, those of $t\mapsto f(t)-tf(1)$ ) are all smaller than $1/(\pi |h|)$ . 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 $f\in\mathcal{S}$ 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 $[0,1]$ ) to $f$ , and this is an arithmetic statement. For instance, simply because the zero function belongs to the support, we deduce that, for a large prime $p$ , there is a positive proportion of $(a,b)\in \mathbf{F}_p^{\times}\times\mathbf{F}_p^{\times}$ such that all partial sums
$\frac{1}{\sqrt{p}}\sum_{1\leq x\leq j}\exp(2i\pi(ax+b\bar{x})/p),$
for $1\leq j\leq p-1$ , have modulus $<\varepsilon$ .
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 $1/(\pi |h|)$ .)