Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1989. / Includes bibliographical references (leaves 28-29). / by Paul Bressler. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994. / Includes bibliographical references (p. 51-52). / by Alan Blair. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1984. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 132-133. / by Jin-Gen Yang. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (p. 155-163) and index. / Random matrix theory is a maturing discipline with decades of research in multiple fields now beginning to converge. Experience has shown that many exact formulas are available for certain matrices with real, complex, or quaternion entries. In random matrix jargon, these are the cases β = 1, 2 and 4 respectively. This thesis explores the general P > 0 case mathematically and with symbolic software. We focus on generalizations of the Hermite distributions originating in physics (the "Gaussian" ensembles) and the Laguerre distributions of statistics (the "Wishart" matrices). One of our main contributions is the construction of tridiagonal matrix models for the general (β > 0) 0 β-Hermite and (β > 0, a > β(m - 1)/2) β-Laguerre ensembles of parameter a and size m, and investigate applications of these new ensembles, particularly in the areas of eigenvalue statistics. The new models are symmetric tridiagonal, and with entries from real distributions, regardless of the value of β. The entry distributions are either normal or X, so "classical", and the independence pattern is maximal, in the sense that the only constraints arise from the symmetric/semi-definite condition. The β-ensemble distributions have been studied for the particular 1, 2, 4 values of p as joint eigenvalue densities for full random matrix ensembles (Gaussian, or Hermite, and Wishart, or Laguerre) with real, complex, and quaternion entries (for references, see [66] and [70]). In addition, general -ensembles were considered and studied as theoretical distributions ([8, 51, 50, 55, 56]), with applications in lattice gas theory and statistical mechanics (the parameter being interpreted as an arbitrary inverse temperature of a Coulomb gas with logarithmic potential). / (cont.) Certain eigenvalue statistics over these general β-ensembles, namely those expressible in terms of integrals of symmetric polynomials with corresponding Hermite or Laguerre weights, can be computed in terms of multivariate orthogonal polynomials (Hermite or Laguerre). We have written a Maple Library (MOPs: Multivariate Orthogonal Polynomials symbolically) which implements some new and some known algorithms for computing the Jack, Hermite, Laguerre, and Jacobi multivariate polynomials for arbitrary. This library can be used as a tool for conjecture-formulation and testing, for statistical computations, or simply for getting acquainted with the mathematical concepts. Some of the figures in this thesis have been obtained using MOPs. Using the new β-ensemble models, we have been able to provide a unified perspective of the previously isolated 1, 2, and 4 cases, and prove generalizations for some of the known eigenvalue statistics to arbitrary β. We have rediscovered (in the Hermite case) a strong version of the Wigner Law (semi-circle), and proved (in the Laguerre case) a strong version of the similar law (generalized quarter-circle). We have obtained first-order perturbation theory for the P large case, and we have reason to believe that the tridiagonal models in the large n (ensemble size) limit will also provide a link between the largest eigenvalue distributions for both Hermite and Laguerre for arbitrary P (for β = 1, 2, this link was proved to exist by Johannson [52] and Johnstone [53]) ... / by Ioana Dumitriu. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994. / Includes bibliographical references (p. 106). / by Andrew Hassell. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1996. / Includes bibliographical references (p. 81). / by Edith Mooers. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1990. / Includes bibliographical references (leaf 71). / by Patrick T. McDonald. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (p. 143-144). / We classify a certain class of vertex algebras, finitely generated by a Virasoro field and even (resp. odd) primary fields of conformal weight 1 (resp 3/2). This is the first interesting case to consider when looking at finitely generated vertex algebras containing a Virasoro field (the most interesting from the point of view of physics). By the axioms of vertex algebras it follows that the space g of fields with conformal weight 1 is a Lie algebra, and the space U of fields with conformal weight 3/2 is a g-module with a symmetric invariant bilinear form. One of the main observations is that, under the assumption of existence of a quasi- classical limit (which basically translates to the existence of a one parameter family of vertex algebras, the free parameter being the Kac-Moody level k), the complex connected algebraic group G corresponding to the Lie algebra 0 acts transitively on the quadric ... This generalizes a similar result of Kac in the case of conformal algebras. Using this observation, we will classify vertex algebras satisfying the above assumptions, by using the classification of connected compact subgroups of SON acting transitively on the unit sphere. The solution is given by the following list ... However, if one removes the assumption of existence of quasi-classical limit, the above argument fails and the problem of classification has to be studied using different techniques. In the case in which is a simple Lie algebra and U an irreducible g- module, we will prove, under some weak technical assumption, that no examples with "discrete" values of the Kac-Moody level appear. / by Alberto De Sole. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2013. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 123-125). / In this thesis, we extend Manolescus and Kronheimer-Manolescus construction of Floer homotopy type to general 3-manifolds. This Floer homotopy type is a candidate for an object whose suitable homology groups recover Floer homology. The main idea is to apply finite dimensional approximation technique and Conley index theory to Seiberg-Witten theory of 3-manifolds. Another part of the construction involves a concept of twisted parametrized spectra introduced by Douglas. We also provide explicit computation for the manifolds S 1 x S 2 and T 3 . / by Tirasan Khandhawit. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (leaves 59-61). / The thesis gives a comprehensive study of elastic instability in growing yeast colonies and thin sheets. The differential adhesion between cells is believed to be the major driving force behind the formation of tissues. The idea is that an aggregate of cells minimizes the overall adhesive energy between cell surfaces. We demonstrate in a model experimental system that there exist conditions where a slowly growing tissue does not minimize this adhesive energy. A mathematical model demonstrates that the instability of a spherical shape is caused by the competition between elastic and surface energies. The mechanism is similar to the Asaro-Tiller instability in prestressed solids. We also study the buckling of a highly constrained thin elastic plate under edge compression. The plate is clamped lengthwise on two edges and constrained by foam pieces along one of the shorter edges. The remaining edge is free. Applying uniform compression along the clamped edges generates a cascade of parabolic singularities. We apply the theories pioneered by Pogorelov, who showed that any zero gaussian curvature surfaces are solutions of the von Karman equations. When two such surfaces intersect, the adjoint surfaces remains a solution everywhere except at the boundary of intersection. However, for small plate thickness and the asymptotic limit, it is possible to construct a solution for the boundary. The total energy of the solution is then given as the sum of the energy of individual surfaces and the boundary energy. We demonstrate that by intersecting a cone and a cylinder the deformation of a parabolic singularity is entirely determined. / by Baochi Thai Nguyen. / Ph.D.