Author: Kowalski

The conference “Random Matrices, L-functions and primes” which A. Nikeghbali and myself are co-organizing with the support of the Forschungsinstitut für Mathematik started yesterday, and the second day just ended with a colloquium lecture by N. Katz about “Simple things we don’t know”. Before explaining (hopefully, tomorrow) what were some of these simple things he mentioned, here are a few words about yesterday’s lectures (the schedule is visible online).

Since a fair number of talks were given using various forms of beamer presentations, we have started gathering the corresponding files to make them available. There are currently three talks on the web page (from yesterday’s lectures):

(1) my own introductory lecture, which was intended to present the basic techniques and problems of analytic number theory to those members of the audience whose expertise lies in the direction of probability theory;

(2) Ashkan’s own not-so-introductory lecture, which is devoted to limit theorems in probability theory from the point of view of our recent joint work with J. Jacod about what we call mod-Gaussian convergence;

(3) J. Keating’s discussion of some classical and some new aspects of the connection between Random Matrix Theory and the Riemann zeta function. His emphasis was in large part on the issue of predicting full asymptotic formulas, with lower order terms, for many of the quantities for which Random Matrix Theory had first given predictions restricted to “main term” behavior (e.g., the density functions for the pair correlation of zeros, or the asymptotic of moments of the zeta function on the critical line). The point of these investigations (which lead to some quite complicated formulas which may be discouraging to look at!) is that the resulting conjectures become much more testable numerically; this is very clear in the pictures that Keating showed (based on joint works with a number of people, including prominently E. Bogomolny, B. Conrey, D. Farmer, M. Rubinstein and N. Snaith, in various combinations). The point is that, for instance, the graph of a polynomial

$latex p(x)=0.0001 x^2+0.1 x-12$

does not look even remotely like that of the main term

$latex q(x)=0.0001 x^2$

unless x is very large, and hence hypothetical experiments based on this simple main term would be very misleading for those values of x.

Keating’s discussion was continued partly in B. Conrey’s talk (the slides for which will be available soon), who explained briefly some of the methods used for these recent precise predictions (based frequently on the so-called “ratios” conjecture). He also mentioned a number of problems suggested by these results, one of which is the following particularly striking conjecture of M. Watkins: consider, among positive integers m, those which are not divisible by the cube of any prime and are congruent to 1 modulo 9, and let S be the set of those values which are sums of two rational cubes:

$latex x^3+y^3=m$

(for instance, m=1729). Fix a prime p>3, say p=7, and two integers x and y modulo p (say x=2, y=3). Then Watkins conjectures that the limit

$latex \lim_{T\rightarrow +\infty}\frac{|\{m\in S\,\mid\, m\equiv x\text{ mod } p,\ m\leq T\}|}{|\{m\in S\,\mid\, m\equiv y\text{ mod } p,\ m\leq T\}|}$

exists, and predicts its value; in particular, for the example of p=7, x=2, y=3, the limit should be 2. In other words, among cubefree integers, those which are congruent to 37 modulo 63 should be about twice as likely to be a sum of two cubes as those which are congruent to 10 modulo 63!

Conrey said he found this particularly beautiful, and it is hard to disagree! The statement is completely elementary, despite being a prediction that is completely dependent on quite deep mathematics, involving elliptic curves, their L-functions and the Birch and Swinnerton-Dyer Conjecture in particular, and ideas of Random Matrix Theory. The papers of Conrey, Keating, Rubinstein and Snaith and of Watkins on quadratic and cubic twists of L-functions explain how all this comes about.

Due to a deplorable oversight in the preparation of my Functional Analysis course, I had forgotten to present in the first chapter the elementary properties of finite-dimensional normed vector spaces. Since I didn’t want to stop the flow of the course to come back to this artificially (and since I didn’t really need those facts yet), I delayed until a better moment came. In the end, I am going to prove these things in the chapter containing the Banach Isomorphism Theorem, by an argument that may be called “cheating”, but which I find amusing.

There are three properties, which are closely related, which I wanted to state for finite-dimensional normed vector spaces (over C, although everything also holds for real vector spaces):

(Property 1) Any finite-dimensional normed vector space is a Banach space (i.e., is complete);

(Property 2) Any two norms on a finite-dimensional space are equivalent: there exist constants c>0, C such that

$latex c||v||_1\leq ||v||_2\leq C||v||_1$

for all vectors v;

(Property 3) Any finite-dimensional subspace of an arbitrary normed vector space is closed with the induced norm.

Since Cⁿ is a Banach space with the norm

$latex (*)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ||(x_1,\ldots,x_n)||=\max |x_i|,$

we see that Property (2) clearly implies Property (1). Moreover, the latter gives Property (3), since if W is finite-dimensional in V, the induced norm will make it a complete subset of V by Property (1), and so it must be closed in W by basic topology.

The second property is typically proved with a little compactness argument in the first chapter of textbooks in functional analysis, and so the others follow. Here is the alternate argument I will present.

We prove Property (1) first, by induction on the dimension of the finite dimensional normed vector space V. For dimension 1, with V spanned by some vector e, homogeneity of the norm gives

$latex ||te||=c|t|\text{ with } c=||e||$

which implies that V is homeomorphic to C, hence complete. Now if we assume that this Property (1) holds for spaces of dimension n-1, and V has dimension n, we consider a basis e₁,…,e_n of V, and for every i=1,…,n, look at the i-th coordinate functional

$latex \lambda_i : V\rightarrow \mathbf{C}.$

We do not know (in our setting!) that these are continuous, but the kernel V_i is of dimension n-1, so with the induced norm, it is a Banach space by the induction hypothesis, and so in particular is must be closed in V. Since the i-th basis vector is not in V_i, a corollary of the Hahn-Banach Theorem (that is in fact typically proved directly in that particular case) states that λ_i does have a continuous extension to V, with the property that

$latex \tilde{\lambda}_i(e_i)\not=0.$

But this extension must then be a non-zero multiple of λ_i, and so those coordinate functions themselves are continuous.

Now, having done this, the linear map

$latex T : V\rightarrow \mathbf{C}^n$

mapping v to

$latex T(v)=(\lambda_1(v),\ldots,\lambda_n(v))$

is then clearly bijective, and it is continuous (where the target has any of the standard norms, for instance the maximum norm (*) above). The inverse is also continuous since it is given by

$latex T^{-1}(\alpha_1,\ldots,\alpha_n)=\sum_{i=1}^n{\alpha_i e_i}$

and

$latex ||\sum_{i=1}^n{\alpha_i e_i}||\leq D \max|\alpha_i|,\ \text{ where }\ D=n\max_i ||e_i||.$

In other words, T is a homeomorphism, and hence V is complete since Cⁿ is. [Note : as was pointed out in a comment, general homeomorphisms do not preserve completeness, but here T and its inverse are also linear, hence both are Lipschitz, and then completeness is preserved.]

Having proved this first property of finite-dimensional normed vector spaces, we obtain (2) as a corollary of the Banach Isomorphism Theorem: given an arbitrary norm on a finite-dimensional vector space, we can compare it in one direction to the norm

$latex ||\sum_{i=1}^n{\alpha_i e_i}||_2=\max|\alpha_i|$

defined in terms of a basis, as above. This means the identity bijective linear map

$latex (V,||\cdot||_2)\rightarrow (V,||\cdot ||)$

is a continuous bijection between Banach spaces (by Property (1), both norms are Banach norms, which is required to apply the Isomorphism Theorem), and hence its inverse is continuous by the Banach Isomorphism Theorem, which implies the two norms are in fact equivalent.

And as we already described, the Property (3) follows directly from Property (1).

As the application period for postdocs approaches, I’d like to mention those proposed by ETH Zürich and encourage all candidates to apply here. Basic information may be found on this web page, and it is then possible to apply online very easily by following the link to this form, with a formal deadline of November 30. (There is also an open position at the Assistant Professor level in applied mathematics; the application procedure is different, but some of the information below may still be useful to motivate possible candidates.)

One thing I’d like to comment on is the “light teaching load” which is mentioned: this very often takes the form of courses on topics chosen by the postdoc himself or herself. As an example for this semester, Anne Moreau is teaching an introductory course on algebraic groups. Such teaching can be very good opportunities for a young researcher: if, for instance, the theory of expander graphs is an important tool that you’ve used in your work but did not yet have time to study in full depth; or if your probabilistic work seems to have applications to number theory, but you never had an occasion to learn analytic number theory from the ground up because there were no graduate courses on the subject in your institution… then teaching an introductory course would probably be the best way to reach a state of satisfactory osmosis with such a subject.

And now here are some additional good reasons to want to come to Zürich for a postdoc, which are maybe not so obvious to every reader, especially among mathematicians from outside Europe.

(1) ETH Zürich has a strong history as a world-class institution in mathematics: this is where Pólya discovered random walks, to give just one example. There is in particular a tradition of links with physics, still currently reflected in the mathematics and in the theoretical physics department. There is also a very strong computer science department, both theoretical (algorithms, etc) and practical (as much in the sense of computer languages, and of concrete applications).

(3) The scientific environment is extremely good; for instance, because of the presence of the Forschungsinstitut für Mathematik (FIM), which runs a very active visitor programme, it is almost too easy to invite people to come for a week or longer for discussions and joint projects. And other people’s visitors are of course excellent opportunities to talk about mathematics… In addition, the FIM sponsors various special activities and the Nachdiplom lectures, which are graduate-level lectures given during a semester by outstanding mathematicians, often on the most recent developments in their field (for instance, Simon Brendle, from Stanford University, is lecturing this semester on the Ricci flow with applications to geometry).

(2) The location of the mathematics department is hard to beat in terms of convenience; it is located in the “main” building

of ETH (click for a larger picture), which is found in the center of the town of Zürich, literally 5 minutes (walk) away from the main train station. As a proof, here is one side of the view from my office

with the Zürich lake, and the other side

with the train station. (I should say that the 5 minutes figure is mostly the downhill time from ETH to the train station; going back up on foot usually takes a bit more time, but there is a very convenient cable-train to do this; the entrance on the ETH side can be seen on the second picture above). Most other departments of ETH are now located in other buildings, some of which are also close to the center, and others are found in another campus (Hönggerberg).
From the train station, all corners of Switzerland are very easily reached, as well as France, Germany and Italy. Scientifically, this includes the EPF Lausanne, the universities of Basel, Neuchâtel, Geneva, and others in Switzerland; in France, this includes Strasbourg (only two hours away), and Paris is 4h30 by train: an excellent opportunity to visit the Bourbaki Seminar, for instance. And to go further (or faster), the international airport is only a 10 minutes train ride away.

(4) In addition to ETH, the University of Zürich (i.e., that of the Canton of Zürich, in contrast with ETH which is a federal institution of the Swiss Confederation) also has an excellent mathematics department, and there are many joint activities between the two, in particular the Colloquium and the Zürich Graduate School in Mathematics.

(5) The quality of life in Zürich is outstanding. As an example, all the (many) water fountains in the town offer drinkable water. As another, the public transport system is among the very best in the world — for this, the relatively small size of the town is of course an advantage (compared, e.g., with Paris). Living without a car is possible in very good conditions, and does not mean that skiing, hiking, and so on, must be put aside, since the train network can bring you efficiently to most ski stations and to many wonderful places for walks. Also, it is true that life in Zürich is quite expensive, but the salaries are commensurate and certainly competitive with the best offers in the US or elsewhere.

Finally, to balance the picture, here are some small potential drawbacks — to show that I am trying to be objective…

(i) The mathematics library is good, but most collections (especially journals) are within the main ETH library, which is extremely extensive (in all domains of natural sciences, architecture, biology, etc), but is not a walk-in-browse-pick-up-and-go library: the collection is typically searched online, where books can be reserved and then picked up at the central desk, while journal articles are typically found on the catalogue, and then scanned by the library on request and sent to you by email (admittedly, this is also very convenient, but you have to wait at least a little while for the PDF file to arrive). Depending on your reliance on finding interesting sources of information by random browsing, this may be a problem.

(ii) If you do not already speak Swiss German, you will be coming in a town where the native language is not yours. Again, this may be an issue, although one can argue that travelling broadens the mind, and that such an experience can be very interesting anyway. This is certainly not a problem in terms of being able to work and live here, since most people speak English, many speak French or Italian (or both), and of course standard German is universal. As mentioned on the web site, teaching can be done in English. Note that it is reasonable to try and expect to learn standard German (which is the written language for Swiss German speakers), but Swiss German itself is not mutually intelligible for a German-speaker, and is quite difficult to learn since there does not exist a written version. But as already said, it is at least a very interesting experience to observe the linguistic features of Switzerland, where there are four official languages.

(iii) The weather in Zürich is not that of California, of course; but if surfing is not an option, skiing becomes available, and one can at least swim in the river Limmat or in the Zürich lake (whenever the temperature is compatible with this activity).

Motivated by a recent post on God Plays Dice concerning the maximal order of elements in the Rubik’s group (the permutation group that describes the movements of the divine cube), I of course decided to try to compute how many conjugacy classes in it you typically need to know to generate the whole group — in other (self-coined) words, I decided to entrust to Magma the computation of the Chebotarev invariant of this group of order 43252003274489856000.

After two days of hard work on my desktop computer, Magma gave the following answer: the expectation of the waiting time is 5.6686453… and the expectation of the square is 36.7870098… (giving a standard deviation slightly larger than 2). This is actually the longest individual computation I’ve performed myself, and it took about 1 Gb memory; computing the 20 conjugacy classes of maximal subgroups was done in about 1.3 seconds, computing the 81120 conjugacy classes took 45 seconds more, and then precomputing the matrix indicating which subgroups intersected which conjugacy class was roughly one day long, the second day being devoted to the alternating sum over the 2²⁰-1 non-empty subsets of maximal subgroups (with the formula indicated at the end of my original post defining the Chebotarev invariant) — this is about one million steps.

The computation was done with floating point numbers with 30 digits precision, but I think the last 10 or so are not to be trusted (and I did not write them down), since they come from this alternating sum with a million terms (but I would trust the first 15, say, because the summands are rationals, hence the approximations used for individual terms are really correct to 30 digits).

As I continue teaching Functional Analysis, I am happily learning new things, which typically come out of trying to think how to present the material in a way that motivates it for my students. I try to insert, at least at particularly important places, some example or indication that will give some understanding of why various steps (or constructions, or definitions) can be seen as natural ones. Often, this leads to natural questions which are not always explicitly mentioned or highlighted.

Here is one such situation: since I have started by treating Hilbert spaces, I try to present analogies and contrasts when it comes to Banach spaces. One such contrast, of course, involves the fact that Banach spaces are not always reflexive. What this means is that the natural (bi)duality map

$latex D\ :\ V\rightarrow V\prime\prime$

is not always surjective, where D is the linear map between a Banach space V and the dual of its dual space V”, given by defining D(v) as the linear functional

$latex D(v)\ :\ \lambda\mapsto \lambda(v)$

for all λ in the dual space.

For a Hilbert space V, the (or one of the) Riesz Representation Theorem gives essentially an identification of V with its dual (or rather the “conjugate” Hilbert space in the case of complex scalars), from which the surjectivity follows because, properly interpreted, D is simply the identity.

To start explaining this, and why it may fail, and what “reflexivity” means, I follow the following path, which is (hopefully) quite natural here, since it is based on the Hilbert space proof, but which turned out to not be really emphasized in the textbooks I am using as references.

First, the Riesz representation theorem is typically proved (and I proved it) using the general result of existence of a projection of a vector on a closed convex set, in particular on a closed linear subspace: given W a closed subspace of V, and a vector v in V, there exists a unique w such that

$latex ||v-w||=\inf_{y\in W}{||v-y||}$

and in particular the infimum on the right is attained. For a general Banach space, this result is not always true, both in terms of unicity, and of existence. I will now concentrate on the latter property (see this earlier post for an example and another nice result involving this type of minimization problems).

I used the question of extending this type of results as a first motivating fact for the Hahn-Banach theorem: one of the first corollaries of it is that, for any vector v, we have

$latex (*)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ||v||=\max_{||\lambda||\leq 1}{|\lambda(v)|}$

where λ ranges over all elements of the dual space which have norm at most 1, i.e., such that

$latex |\lambda(x)|\leq ||x||$

for all vectors x. The content is first that there is equality with a sup on the right-hand side, but in addition that this supremum is attained.

It is now perfectly natural to wonder whether the “dual” identity

$latex (**)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ||\lambda||=\max_{||v||\leq 1}{|\lambda(v)|}$

holds, or in other words, whether a linear functional λ always attains its supremum on the closed unit ball of V (since the supremum is exactly ||λ|| by definition of the norm on the dual space).

That these are maxima instead of minima as in the previous case of Hilbert spaces is not particularly problematic, since one can always go from one to the other; in fact the counterexample mentioned earlier was exactly of this type: for a certain linear form, (**) was not correct, and a counterexample to the minimization problem followed straightforwardly.

However, it is very easy to see that (*) implies that (**) is true, for all linear forms on V, provided V is reflexive. So we get from the previous example a very straightforward proof, based on trying to understand the contrast between Hilbert and Banach spaces, that the space c₀ that occured is not reflexive.

But then one can ask (and I asked myself): is the converse to this implication true? In other words, assuming we have the nice property (**) for all continuous linear maps on V, does it follow that V is reflexive?

It turns out the answer is indeed Yes, but it seems to be a difficult result of Banach space theory. This is a theorem of R.C. James, and I found it mentioned (in a slightly disguised form) in Conway’s book on functional analysis, and following the reference, found the proof contained in this paper (Studia Mathematica, 1964). It is, indeed, by no means straightforward (though I am certainly not expert enough to really understand the proof after a first glance…).

None of the other texts I am looking at for my course mentions this (unless it escaped my notice, which is quite possible). Presumably, this is because it is not a particularly fruitful approach to further studies of Banach spaces (as the unbalance between both sides of the implication suggests), but it is definitely nice to know and to mention to students.

(One of my teachers once said — and I’m sure this has been said many times of many subjects — that “completely general facts in topology are either obvious or false”; this result indicates that the same clearly cannot be said of Banach spaces).