Random matrices, L-functions and primes: what was said on Wednesday

After Tuesday, naturally, came Wednesday. We had planned a half-day of talks only, with a free afternoon, and at some point it seemed like a rather poor choice since Wednesday morning saw the worse snowfall ever to happen in Zürich in October (at least since a certain time, I guess):

Snow in Zurich

(I did mention in my post-doc marketing post that the weather here is not always that of California…)

Although the skies cleared up a bit in early afternoon, it was again snowing quite strongly by the time of the conference dinner in the evening. But in the morning, we had three very interesting talks:

(1) D. Bump explained the method used by A. Gamburd and himself to prove the formulas for the average over unitary matrices of values (at a fixed point) of characteristic polynomials. This proof comes historically before the probabilistic arguments that C. Hughes had described on Tuesday, and contains also a lot of interesting features from the point of view of the representation theory of unitary groups and of symmetric groups, and their inter-relations. In this setting, the average over U(N) (equipped with Haar measure) of

 |\det(1-A)|^{2k}

(where k is an integer) appears as the dimension of a representation of the unitary group U(2k) (a representation which depends on N of course). This dimension is then computed using the Weyl dimension formula. As background for the type of structure that emerges with the variation in N, Bump suggested to read A. Zelevinsky’s short book “Representations of Finite Classical Groups: A Hopf Algebra Approach” (Springer Lecture Notes 869, 1981) — which I intend to (try to) do as soon as possible.

(2) The next lecture was given partly by Nina Snaith and partly by her student Duc-Khiem Huynh, and described their joint work in trying to understand some surprising features of the observed location of low-lying zeros of L-functions of elliptic curves over the rationals. More precisely, for such an L-function

L(E,s)=\sum_{n\geq 1}{a_E(n)n^{-s}}

with conductor N, it is natural from the point of view of Random Matrix Theory (to distinguish the possible symmetry types) to compute the normalized ordinate of the first zero

\tilde{\rho}_E=\frac{\log N}{2\pi}\gamma_E

where

L(E,1/2+i\gamma_E)=0,\ \gamma_E\geq 0

and γE is the first such ordinate of a zero. (Experimentally, of course, it is found that it is on the critical line). It turns out, experimentally, that the distribution of this low-lying zero does not at all look like what the Random Matrix model suggests, at least for currently available data (this was first described by S. J. Miller; the basic problem is that the histograms show a repulsion at the origin). The lecture was then devoted to explain how more refined models could lead to possible explanations of these features; in particular, it was suggested that the discretisation feature of the values of the L-function at 1/2 could be a source of this discrepancy.

(3) To conclude the morning, D. Farmer gave a very nice description of some issues surrounding one of the features of the correspondance between Random Matrices and L-functions that is often taken for granted: the link between the height T (for the Riemann zeta function on the critical line; it would be the conductor for other families) and the size N of matrices given by

N=\frac{\log T}{2\pi}.

This is usually justified very quickly as “equating the mean-density of zeros” (there are N zeros of the caracteristic polynomial on the unit circle of length , and about log T/2π zeros of ζ(1/2+it) in a vertical interval of length 1 around T), but D. Farmer showed that one can say much more than that, and that the connection is still somewhat mysterious.

Random matrices, L-functions and primes, III

I think most people who have organized a conference (even in such outstanding conditions as provided by the Forschungsinstitut für Mathematik) will not be surprised that my promise of posting daily updates was not fulfilled. However, the web page where we have collected the PDF files with the contents of the beamer talks has been updated, and not too many are missing — and those are mostly because the speakers wanted to have a look and make a few corrections before making them available. We do not have notes, however, for those talks that were given on the blackboard or with old-fashioned slides — but I will describe all of them briefly in this post and the next ones…

In keeping with a noticeable need for rest, I will only continue my survey of the conference up to Tuesday and therefore up to N. Katz’s colloquium lecture.

(1) We started with Z. Rudnick’s discussion of his recent works with Kurlberg and with Faifman concerning certain cases where it is possible to consider a limit of curves over finite fields with increasing genus without needing to first have the finite base field become larger and larger (the “honest” limit if one is most concerned with analogies with the number field case, and in particular with the Riemann zeta function). This concerns the family of hyperelliptic curves, and the reason it is easier to control is related to their interpretation as families of quadratic L-functions with increasing conductor — so the analogy pays off, in fact, partly because such families have already been extensively studied over number fields!

(2) The next talk was given by D. Ulmer, who explained how to produce many examples (even parameterized families) of elliptic curves over function fields over finite fields with explicit points of infinite order generating a finite index subgroup of the Mordell-Weil group, such examples having unbounded rank (a feat which no one knows how to do over number fields). This involved some discussion of rather lovely algebraic geometry, and leads to the hope that such examples will be useful for numerical experiments in testing conjectures of various types.

(3) Following this was a talk by A. Gamburd on his recent work with Bourgain and Sarnak concerning the use of expanding Cayley graphs in attacking sieve problems of “hyperbolic” nature, and also how they managed to obtain the necessary tremendous expansion (pun intended) of the number of situations where expanders arise from reductions modulo primes of a subgroup of SL(2,Z) satisfying the minimal assumption that it is Zariski dense in SL(2). This involves a lot of insights from additive combinatorics (sum-product estimates), and the breakthrough of Helfgott concerning growth of subsets of SL(2,Fp) under products.

(4) After a well-deserved lunch break, the lecture of P. Biane (“Brownian motion on matrix spaces and symmetric spaces, and some connections with Riemann zeta function”) gave a very nice introduction to an interpretation of the statistics of eigenvalues of random unitary matrices based on Brownian motion conditioned (in a suitable way) to remain within certain domains (Weyl chambers or alcoves). This was something I was not at all aware of (and I think the same was true for many of the number-theorists in the audience), and I think this was an excellent example of the type of “food for thought” we hoped that this conference could provide. We suspect that those eigenvalues are highly relevant in analytic number theory, but we do not know at the moment how this comes about, and any insight that may lead to a different way of thinking of the problem seems beneficial. [Update (11/11/08): P. Biane has just posted on arXiv a paper about the topic of his talk].

(5) There followed an equally insightful and enthusiastic lecture of C. Hughes, presenting the recent probabilistic interpretation of Haar measure on unitary matrices due to P. Bourgade, A. Nikeghbali, M. Yor and himself. This interpretation leads to a decomposition of the value of the characteristic polynomial (at a given point of the unit circle) as a product of fairly simple independent random variables, and from there it is easy to get either new proofs of known results concerning the distribution of those values (e.g., the moments of the values, first computed by Keating and Snaith), or new results which seem hard to derive from the other approaches.

(6) And the final lecture was N. Katz’s colloquium, “Some simple things we don’t known”. The type of question considered was the following: suppose you play a two person game where one person selects an integral polynomial f (of some odd degree 2g+1), without saying which polynomial it is, nor even what g is. Suppose further that this first player then gives to the second player two pieces of data: (1) an integer N such that all prime divisors of the discriminant of f divide N; (2) for every prime not dividing N, the number np of solutions (x,y) of

y^2=f(x)\text{ modulo } p,\text{ where } (x,y)\in \mathbf{F}_p^2.

Then the second player’s goal is to compute the integer g (which is the genus of the underlying hyperelliptic curve). As Katz explained, this should be possible, conjecturally, by computing

\limsup_{p\rightarrow+\infty}{\left|\frac{n_p-p}{\sqrt{p}}\right|},

because this quantity should be exactly 2g. However, although we know since A. Weil’s proof of the Riemann Hypothesis for curves that

\limsup_{p\rightarrow+\infty}{\left|\frac{n_p-p}{\sqrt{p}}\right|}\leq 2g,

obtaining equality is intimately related with particular cases of the generalized Sato-Tate conjecture, and this is unknown for any polynomial f with g>1 (except for very special exceptions, having to do with curves whose jacobians are isogenous to a product of the same elliptic curve)!

Random matrices, L-functions and primes: the conference, live

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

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

does not look even remotely like that of the main term

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:

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

\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.

Finite dimensional normed vector spaces

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

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

(*)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ||(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

||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,…,en of V, and for every i=1,…,n, look at the i-th coordinate functional

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

We do not know (in our setting!) that these are continuous, but the kernel Vi 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 Vi, a corollary of the Hahn-Banach Theorem (that is in fact typically proved directly in that particular case) states that λi does have a continuous extension to V, with the property that

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

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

Now, having done this, the linear map

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

mapping v to

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

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

and

||\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

 ||\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

(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).

Postdocs

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

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

View to the lake

with the Zürich lake, and the other side

View to the train station

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).