Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1989. / Includes bibliographical references (leaves 97-100) and index. / by Alan Edelman. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / Includes bibliographical references (p. 79-81). / The classical Brill-Noether theorems count the dimension of the family of maps from a general curve of genus g to non-degenerate curves of degree d in projective space Pr. These theorems can be extended to include ramification conditions at fixed general points. This thesis deals with the problem of imposing a ramification condition at an unspecified point. We solve the problem completely in dimension 1, prove a closed-form existence criterion and a finiteness result in dimension 2, and provide an existence test and bound the dimension of the family in the general case. / by Rebecca C. Lehman. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 189-203). / In this work, we study two different manipulations of electromagnetic waves governed by macroscopic Maxwell's equations. One is frequency conversion of such waves using small intrinsic material nonlinearities. We study conversion of an input signal at frequency w1 to frequency Wk due to second or third harmonic generation or four-wave mixing using coupled-mode theory. Using this framework, we show there is a critical input power at which maximum frequency conversion is possible. We study in depth the case of third harmonic generation, its solutions, and their stability analysis. Based on the dynamics of the system, we propose a regime of parameters that 100%- efficient frequency conversion is possible and propose a way of exciting this solution. We also look at same analysis for the case of degenerate four-wave mixing and come up with 2d and 3d designs of a device that exhibits high-efficiency second-harmonic generation. Second, we consider proposals for invisibility cloaks to change the path of electromagnetic waves in a certain way so that the object appears invisible at a certain frequency or a range of frequencies. Transformation-based invisibility cloaks make use of the coordinate invariance of Maxwell's Equations and require complex material configuration e and p in the cloak. We study the practical limitations of cloaking as a function of the size of the object being cloaked. Specifically, we study the bandwidth, loss, and scattering limitations of cloaking as the object gets larger and show that cloaking of objects many times larger than the wavelength in size becomes practically impossible. / by Hila Hashemi. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1969. / Vita. / Bibliography: leaves 137-139. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1990. / Includes bibliographical references (leaves 134-136). / by Matthew B. Stenzel. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996. / Includes bibliographical references (p. 140-141). / by Richard Stone. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007. / Includes bibliographical references (p. 92). / 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). / by Yakov Shapiro. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998. / Includes bibliographical references (p. 84-86). / by Radica Šipčić. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007. / Includes bibliographical references (p. 161-168). / The Donaldson invariants for CP2 were obtained as the u-plane integral from a N = 2 supersymmetric topological U(1)-gauge theory by Moore and Witten. We derive the generating function for the Donaldson invariants of CP2 as the stationary phase approximation of the low-energy effective U(I)-gauge theory on CP2 thus obtaining an interpretation of the u-plane integral in terms of determinant line bundles. For the product of the determinant line bundles, the local and global anomalies vanish. Moreover, the product has a canonical trivialization. We show that the u-plane integral also arises as the stationary phase approximation of a heterotic o-model on an elliptic curve at the boundary of the Coulomb branch with the target space CP1 x U(1). The semi-classical generating function is described in terms of determinant line bundles on the Coulomb branch. We show that in terms of the partition function on the elliptic curve, the blow-up function for the Donaldson invariants derived by Fintushel and Stern arises in a natural way. / by Andreas Malmendier. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2001. / Includes bibliographical references (p. 104-107). / The noncommutative symmetric functions Sym of Gelfand et al. give not only a lifting of the well-developed commutative theory of symmetric functions to the non-commutative level, but also relate the descent algebras of Solomon and the quasi-symmetric functions, where the latter are dual to the noncommutative symmetric functions equipped with the internal product, which are anti-isomorphic to the descent algebras. Using this anti-isomorphism, properties of both noncommutative symmetric functions and of descent algebras can be studied. Generalizations of the above theory are made in the present work. The starting point is the quasi-symmetric functions of type B, BQSym, which are shown to have an algebra, a comodule, and a coalgebra structures. The noncommutative symmetric functions BSym are then introduced as a module over Sym dual to the comodule structure of BQSym. It is then made into a coalgebra dual to the algebra structure of BQSym, and into an algebra dual to the coalgebra structure of BQSym. The latter duality defines the internal product *B on BSym, which makes (BSym, *B) anti-isomorphic to the descent algebra [Sigma]Bn of the hyperoctahedral groups Bn, studied by Bergeron and Bergeron. / (cont.) Lie idempotents of both BSym and [Sigma]Bn are then studied via the anti-isomorphism. In particular, a one-parameter family of Lie idempotents, which is a q-analog of a known idempotent, is found. A specialization of this family gives, in the descent algebra [Sigma]B, a Dynkin-like idempotent whose action on words is a signed left bracketing. Natural noncommutative generalizations of the Eulerian numbers and of the Euler numbers of type B are given. By a specialization, formulas for some refinements of the Euler numbers of type B are also derived. / by Chak-On Chow. / Ph.D.