Deligne's "Weil II" paper includes a far-reaching conjecture to the
effect that for a smooth variety on a finite field of characteristic p,
for any prime l distinct from p, l-adic representations of the etale
fundamental group do not occur in isolation: they always exist in
compatible families that vary across l, including a somewhat more
mysterious counterpart for l=p (the "petit camarade cristallin"). We
explain in more detail what this all means, indicate some key
They actually satisfy a stochastic PDE with time-space white noise.
Can we say more using higher order cumulants?
How combinatorial techniques can help to analyze this departure from chaos?
The problem of finding metrics with constant Q-curvature in a prescribed conformal class is an important fourth-order cousin of the Yamabe problem. In this talk, I will explain how certain variational bifurcation techniques used to prove non-uniqueness of solutions to the Yamabe problem also yield non-uniqueness results for the constant Q-curvature problem. However, special emphasis will be given to the differences between multiplicity phenomena in these two variational problems. This is based on joint work with P. Piccione and Y. Sire.