http://hdl.handle.net/1721.1/7841 http://hdl.handle.net/1721.1/41793 Spaces of algebra structures and cohomology of operads Rezk, Charles W. (Charles Waldo) Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996. Includes bibliographical references (p. 85-86). http://hdl.handle.net/1721.1/41725 Viscous fluid sheets Savva, Nikos We present a general theory for the dynamics of thin viscous sheets. Employing concepts from differential geometry and tensor calculus we derive the governing equations in terms of a coordinate system that moves with the film. Special attention is given to incorporating inertia and the curvature forces that arise from the thickness variations along the film. Exploiting the slenderness of the film, we assume that the transverse fluid velocity is small compared to the longitudinal one and perform a perturbation expansion to obtain the leading order equations when the center-surface that defines the coordinate system is parametrized by lines of curvature. We then focus on the dynamics of flat film rupture, in an attempt to gain some insights into the sheet breakup and its fragmentation into droplets. By combining analytical and numerical methods, we extend the prior work on the subject and compare our numerical simulations with experimental work reported in the literature. Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007. Includes bibliographical references (leaves 108-117). http://hdl.handle.net/1721.1/41724 Extension of the Hodge theorem to certain non-compact manifolds Shapiro, Yakov (Yakov Mikhaylovich) We prove an analogue of the Hodge cohomology theorem for a certain class of non-compact manifolds. Specifically, let M be a compact manifold with boundary OM, and let g be a metric on Int(M). Assume that there exists a collar neighborhood of the boundary ... We then describe doubly weighted Sobolev spaces on M. For elements of these spaces the harmonic parts of w1 and w2 lie in one Sobolev space, while the non-harmonic parts of w1 and w2 lie in a differently defined Sobolev space. We prove that ... is Fredholm on almost all of these doubly weighted spaces, except for a finite number of values of w. This gives us an analogue of the Hodge decomposition theorem and leads to the result. This work generalizes earlier theorems of Atiyah, Patodi and Singer for b-metrics (case a = b = 0) and of Melrose for scattering metrics (case a = b = 1). Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007. Includes bibliographical references (p. 92). http://hdl.handle.net/1721.1/41723 Dirac operators and monopoles with singularities Yang, Fangyun, Ph. D. Massachusetts Institute of Technology This thesis consists of two parts. In the first part of the thesis, we prove an index theorem for Dirac operators of conic singularities with codimension 2. One immediate corollary is the generalized Rohklin congruence formula. The eta function for a twisted spin Dirac operator on a circle bundle over a even dimensional spin manifold is also derived along the way. In the second part, we study the moduli space of monopoles with singularities along an embedded surface. We prove that when the base manifold is Kahler, there is a holomorphic description of the singular monopoles. The compactness for this case is also proved. Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007. Includes bibliographical references (p. 75-77).