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Peskin's Immersed Boundary (IB) method is one of the most popular numerical methods for many years and has been applied to problems in mathematical biology, fluid mechanics, material sciences, and many other areas. Peskiness IB method is associated with discrete delta functions. It is believed that the IB method is first order accurate in the $L^{\infty}$ norm. But almost no rigorous proof could be found in the literature until recently [Mori, Comm. Pure. Appl. Math: 61:2008] in which the author showed that the velocity is indeed first order accurate for the Stokes equations with a periodic boundary condition. In this paper, we show first order convergence with a $\log h$ factor of the IB method for elliptic interface problems with Dirichlet boundary conditions.

Let w be an Abelian differential on a compact Riemann surface of genus g ≥ 1. Then |w|2 defines a flat metric with conical singularities and trivial holonomy on the Riemann surface. We obtain an explicit holomorphic factorization formula for the ζ-regularized determinant of the Laplacian in the metric |w|2, generalizing the classical Ray-Singer result in g = 1.

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We exhibit the first examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies on hyperbolic geometry in an essential way; it combines an explicit upper bound for the first eigenvalue on coexact 1-forms λ_1^* on rational homology spheres which admit irreducible solutions together with a version of the Selberg trace formula relating the spectrum of the Laplacian on coexact 1-forms with the volume and complex length spectrum of a hyperbolic three-manifold. Using these relationships, we also provide precise numerical bounds on λ_1^* for several hyperbolic rational homology spheres.