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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.

On an elliptic curve $y^2=x^3+ax+b$, the points with coordinates $(x,y)$ in a given number field form a finitely generated abelian group. One natural question is how the rank of this group changes when changing the number field.

An important question in hydrodynamic turbulence concerns the scaling proprties in the inertial range. Many years of experimental and computational work suggests---some would say, convincingly shows---that anomalous scaling prevails. If so, this rules out the standard paradigm proposed by Kolmogorov. The situation is not so obvious if one considers circulation around loops as the scaling objects instead of the traditional velocity increments.