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):

(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

(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

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

where

and *γ _{E}* 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

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 *2π*, 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.