The Langlands and Fontaine–Mazur conjectures in number theory describe when an automorphic representation f arises geometrically, meaning that there is a smooth projective variety X, or more generally a Chow motive M in the cohomology of X, such that there is an equality of L-functions L(M,s) = L(f,s). We explicitly describe how to produce such a variety X and Chow motive M in the case of powers of certain automorphic representations, called algebraic Hecke characters. This is joint work with J. Lang.
School of Mathematics
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?
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.
How combinatorial techniques can help to analyze this departure from chaos?
A Riemannian manifold is called Besse, if all of its geodesics are periodic. The goal of this talk is to study the energy functional on the free loop space of a Besse manifold. In particular, we show that this is a perfect Morse-Bott function for the rational, relative, S1-equivariant cohomology of the free loop space. We will show how this result is crucial in proving a conjecture of Berger for spheres of dimension at least 4, although it might be useful for proving the conjecture in full generality.