Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997. / Includes bibliographical references (p. 107-108). / by Michael G. Shefter. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. / Includes bibliographical references (p. 83-86). / This thesis begins with the study of a class of symmetric functions ... Which are generating functions for ribbon tableaux (hereon called ribbon functions), first defined by Lascoux, Leclerc and Thibon. Following work of Fomin and Greene, I introduce a set of operators called ribbon Schur operators on the space of partitions. I develop the theory of ribbon functions using these operators in an elementary manner. In particular, I deduce their symmetry and recover a theorem of Kashiwara, Miwa and Stern concerning the Fock space F of the quantum affine algebras ... Using these results, I study the functions ... in analogy with Schur functions, giving: * a Pieri and dual-Pieri formula for ribbon functions, * a ribbon Murnaghan-Nakayama formula, * ribbon Cauchy and dual Cauchy identities, * and a C-algebra isomorphism ... The study of the functions ... will be connected to the Fock space representation F of ...via a linear map [Iota]: F ... which sends the standard basis of F to the ribbon functions. Kashiwara, Miwa and Stern [29] have shown that a copy of the Heisenberg algebra H acts on F commuting with the action of ... Identifying the Fock Space of H with the ring of symmetric functions A(q) I will show that · is in fact a map of H-modules with remarkable properties. In the second part of the thesis, I give a combinatorial generalisation of the classical Boson-Fermion correspondence and explain how the map [phi] is an example of this more general phenomena. I show how certain properties of many families of symmetric functions arise naturally from representations of Heisenberg algebras. The main properties I consider are a tableaux-like definition, a Pieri-style rule and a Cauchy-style identity. / (cont.) Families of symmetric functions which can be viewed in this manner include Schur functions, Hall- Littlewood functions, Macdonald polynomials and the ribbon functions. Using work of Kashiwara, Miwa, Petersen and Yung, I define generalised ribbon functions for certain affine root systems 1 of classical type. I prove a theorem relating these generalised ribbon functions to a speculative global basis of level 1 q-deformed Fock spaces. / by Thomas F. Lam. / Ph.D.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 16). / Let G be a classical group and let g be its Lie algebra. For a nilpotent element X E g, the ring R(Ox) of regular functions on the nilpotent orbit Ox is a G-module. In this thesis, we will decompose it into irreducible representations of G for some spherical nilpotent orbits. / by Tonghoon Suk. / S.M.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / 169 / Cataloged from student-submitted PDF version of thesis. / Includes bibliographical references (pages 153-170). / Biological systems are extremely complex, and our ability to experimentally measure interactions in these systems is limited by inherent noise. Technological advances have allowed us to collect unprecedented amounts of raw data, increasing the need for computational methods to disentangle true interactions from noise. In this thesis, we focus on statistical methods to infer two classes of important biological interactions: protein-protein interactions and the link between genotypes and phenotypes. In the first part of the thesis, we introduce methods to infer protein-protein interactions from affinity purification mass spectrometry (AP-MS) and from luminescence-based mammalian interactome mapping (LUMIER). Our work reveals novel context dependent interactions in the MAPK signaling pathway and insights into the protein homeostasis machinery. In the second part, we focus on methods to understand the link between genotypes and phenotypes. First, we characterize the effects of related individuals on standard association statistics for genome-wide association studies (GWAS) and introduce a new statistic that corrects for relatedness. Then, we introduce a statistically powerful association testing framework that corrects for confounding from population structure in large scale GWAS. Lastly, we investigate regularized regression for phenotype prediction from genetic data. / by George Jay Tucker. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998. / Includes bibliographical references (p. 133-135). / by Radu Constantinescu. / Ph.D.

Thesis (Ph.D.) Massachusetts Institute of Technology. Dept. of Mathematics, 1947. / Vita. / Bibliography: leaves 267-269. / by J. Halcombe Laning, Jr. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (p. 81-84) and index. / This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. / It is well known that if a finite graded lattice of rank n is supersolvable, then it has an EL-labelling where the labels along any maximal chain form a permutation of [1, 2,..., n]. We call such a labelling an Sn EL-labelling and we show that a finite graded lattice of rank n is supersolvable if and only if it has such a labelling. This result can be used to show that a graded lattice is supersolvable if and only if it has a maximal chain of left modular elements. We next study finite graded bounded posets that have Sn EL-labellings and describe a type A 0-Hecke algebra action on their maximal chains. This action is local and the resulting representation of these Hecke algebras is closely related to the flag h-vector. We show that finite graded lattices of rank n, in particular, have such an action if and only if they have an Sn EL-labelling. Our next goal is to extend these equivalences to lattices that need not be graded and, furthermore, to bounded posets that need not be lattices. In joint work with Hugh Thomas, we define left modularity in this setting, as well as a natural extension of Sn EL-labellings, known as interpolating labellings. We also suitably extend the definition of lattice supersolvability to arbitrary bounded graded posets. We show that these extended definitions preserve the appropriate equivalences. Finally, we move to the study of P-partitions. Here, edges are labelled as either "strict" or "weak" depending on an underlying labelling of the elements of the poset. A well-known conjecture of R. Stanley states that the quasisymmetric generating function for P-partitions is symmetric if and only if P is isomorphic to a Schur labelled skew shape poset. / (cont.) In characterizing these skew shape posets in terms of their local structure, C. Malvenuto made significant progress on this conjecture. We generalize the definition of P-partitions by letting the set of strict edges be arbitrary. Using cylindric diagrams, we extend Stanley's conjecture and Malvenuto's characterization to this setting. We conclude by proving both conjectures for large classes of posets. / by Peter McNamara. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. / Includes bibliographical references (p. 30). / The category of Segal spaces was proposed by Charles Rezk in 2000 as a suitable candidate for a model category for homotopy theories. We show that Quillen functors induce morphisms in this category and that the morphisms induced by Quillen pairs are "adjoint" in a useful sense. Quillen's original total derived functors are then obtained as a suitable localization of these morphisms within the category of Segal spaces. As an application, we consider a construction of "homotopy fibres" within a homotopy theory modelled by a Segal space and show that the homotopy fibre of a map is preserved by a localization which remembers only the homotopy category plus the automorphism groups of objects. / by Hugh Michael Robinson. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006. / Includes bibliographical references (leaves 32-33). / The Scott rank of a countable structure A, denoted sr(A), was observed by Nadel to be at most wA + 1, where wA4 is the least ordinal not recursive in A. Let T be weakly scattered and L(a,T) be E2-admissible. We give a sufficient condition, the B,-hypothesis, under which T has model A with w4A = a and sr(A) = a + 1. Given the B,-hypothesis, an iterated forcing argument is used to obtain a generic Ta D T such that Th has a model with the desired properties. / by Alice Shih Ying Chan. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003. / Includes bibliographical references (p. 147-156). / We study the interaction between settling particles and a stratified ambient in a variety of contexts. We first study the generation of large scale fluid motions by the localised release of a finite mass of particles in the form of plumes or gravity currents. We present the results of a combined theoretical and experimental study describing the evolution of particle clouds formed by the release of heavy particles. In the early stages of motion, particle clouds behave as turbulent fluid thermals; however, their radial expansion eventually stops and particles settle from the base of the cloud at their individual settling speed. We focus on deducing a criterion for the various modes of particle deposition from particle clouds in a stratified ambient. We proceed to study the deposition patterns resulting from particle-laden gravity currents that spread horizontally when released in a particle-free ambient. Using a box-model, we focus on bidisperse gravity currents and examine the resulting particle distribution and maximal deposit length. We then turn to suspensions where particles are initially present throughout the fluid. The simultaneous presence of particles and of a stratified ambient may lead to behaviour analogous to double-diffusive systems, with particles playing the role of a diffusing component. We examine the linear stability of the settling of a particle concentration gradient in a stratified fluid. Numerical simulations allow us to determine the stability of the system for a broad range of particle settling speeds and diffusion coefficients. We then report on layering arising from sedimentation in a density stratified ambient beneath an inclined wall. / (cont.) From our experimental study, we describe the series of horizontal intrusions formed by particle-free fluid intruding at its level of neutral buoyancy. We present numerical models describing the time evolution of the concentration of particles and the layer formation. Finally, we present an experimental and theoretical study of the combined influence of hindered settling and settling speed variations due to an ambient stratification. We develop a criterion for the stability of a suspension settling in a stratified ambient and experimental observations allow us to qualify the main features of this instability. / by François Alain Blanchette. / Ph.D.