We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M, g) that are C0-asymptotic to Schwarzschild of mass m > 0. Refining an argument due to H. Bray, we obtain an effective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in (M, g) for all sufficiently large volumes, and that they are close to centered coordinate spheres. This implies that the volume-preserving stable constant mean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume. This confirms a conjecture of H. Bray. Our results are consistent with the uniqueness results for volumepreserving stable constant mean curvature surfaces in initial data sets obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian. The additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work.

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We propose a dynamically bi-orthogonal method (DyBO) to solve time dependent stochastic partial differential equations (SPDEs). The objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi-orthogonal basis. It is well-known that the KarhunenLoeve expansion (KLE) minimizes the total mean squared error and gives the sparsest representation of stochastic solutions. However, the computation of the KL expansion could be quite expensive since we need to form a covariance matrix and solve a large-scale eigenvalue problem. The main contribution of this paper is that we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the KL expansion. Unlike other reduced model methods, our method constructs the reduced basis on-the-fly without the need to form the

We develop a theory to connect the following three areas: (a) the mean field equations on flat tori, (b) the classical Lame equations and (c) modular forms. A major theme in part I is a classification of developing maps f attached to solutions of the mean field equation according to the types of transformation laws (or monodromy) with respect to the period lattice satisfied by f.