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NRODUCTION LET y be a rectifiable Jordan curve in three-dimensional euclidean space. Answering an old question, whether y can bound a surface with minimal area, Douglas [l I] and Rad6 [45](independently) found a minimal surface spanning y which is parametrized by the disk. This minimal surface has minimal area among all Lipschitz maps from the disk into R3 which span y. The question whether this solution has branch points or not was finally settled by Osserman [42], who proved that there are no interior true branch points, and by Gulliver [lS], who proved that there are no interior false branch points.

It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Poisson varieties. For a large class of upper cluster algebras (injective-reachable ones with full rank coefficients), we describe all of its bases with these properties. Moreover, we show the existence of the generic basis for them. In addition, we prove that Bridgeland's representation theoretic formula is effective for their theta functions (weak genteelness). Our results apply to (almost) all well-known cluster algebras arising from representation theory or higher Teichmüller theory, including quantum affine algebras, unipotent cells, double Bruhat cells, skein algebras over surfaces, where we change the coefficients if necessary so that the full rank assumption holds.

We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.

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