Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 129-130). / Let G be a connected complex reductive algebraic group with Lie algebra g. The Lusztig-Vogan bijection relates two bases for the bounded derived category of G-equivariant coherent sheaves on the nilpotent cone 11 of g. One basis is indexed by ..., the set of dominant weights of G, and the other by [Omega], the set of pairs ... consisting of a nilpotent orbit ... and an irreducible G-equivariant vector bundle ... The existence of the Lusztig-Vogan bijection ... was proven by Bezrukavnikov, and an algorithm computing [gamma] in type A was given by Achar. Herein we present a combinatorial description of [gamma] in type A that subsumes and dramatically simplifies Achar's algorithm. / by David B Rush. / Ph. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994. / Includes bibliographical references (leaf 23). / by Kian Boon Tay. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006. / Includes bibliographical references (p. 119-122). / Fully localized three-dimensional solitary waves, commonly referred to as 'lumps', have received far less attention than two-dimensional solitary waves in dispersive wave systems. Prior studies have focused in the long-wave limit, where lumps exist if the long-wave speed is a minimum of the phase speed and are described by the Kadomtsev-Petviashvili (KP) equation. In the water-wave problem, in particular, lumps of the KP type are possible only in the strong-surface-tension regime (Bond number, B > 1/3), a condition that limits the water depth to a few mm. In the present thesis, a new class of lumps is found that is possible under less restrictive physical conditions. Rather than long waves, these lumps bifurcate from infinitesimal sinusoidal waves of finite wavenumber at an extremum of the phase speed. As the group and phase velocities are equal there, small-amplitude lumps resemble fully localized wavepackets with envelope and crests moving at the same speed, and the wave envelope along with the induced mean-flow component are governed by a coupled Davey-Stewartson equation system of elliptic-elliptic type. The lump profiles feature algebraically decaying tails at infinity owing to this mean flow. In the case of water waves, lumps of the wavepacket type are possible when both gravity and surface tension are present on water of finite or infinite depth for B < 1/3. / (cont.) The asymptotic analysis of these lumps in the vicinity of their bifurcation point at the minimum gravity-capillary phase speed, is in agreement with recent fully numerical computations by Parau, Cooker & Vanden-Broeck (2005) as well as a formal existence proof by Groves & Sun (2005). A linear stability analysis of the gravity-capillary solitary waves that also bifurcate at the minimum gravity-capillary phase speed, reveals that they are always unstable to transverse perturbations, suggesting a mechanism for the generation of lumps. This generation mechanism is explored in the context of the two-dimensional Benjamin (2-DB) equation, a generalization to two horizontal spatial dimensions of the model equation derived by Benjamin (1992) for uni-directional, small-amplitude, long interfacial waves in a two-fluid system with strong interfacial tension. The 2-DB equation admits solitary waves and lumps of the wavepacket type analogous to those bifurcating at the minimum gravity-capillary phase speed in the water-wave problem. Based on unsteady numerical simulations, it is demonstrated that the transverse instability of solitary waves of the 2-DB equation results in the formation of lumps, which propagate stably and are thus expected to be the asymptotic states of the initial-value problem for fully localized initial conditions. / by Boguk Kim. / 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. 85-88). / In this thesis, we study ad-nilpotent ideals and its relations with nilpotent orbits, affine Weyl groups, sign types and hyperplane arrangements. This thesis is divided into three parts. The first and second parts deal with ad-nilpotent ideals for complex reductive Lie groups. In the first part, we study the left equivalence relation of ad-nilpotent ideals and relate it to some equivalence relation of affine Weyl groups and sign types. In the second part, we prove that for classical groups there always exist ideals of minimal dimension as conjectured by Sommers. In the third part, we define an analogous object for connected real reductive Lie groups, which is called 0-nilpotent subspaces. We relate 0-nilpotent subspaces to dominant regions of some real hyperplane arrangement and get the characteristic polynomials of the real hyperplane arrangement in the case of U(m, n) and Sp(m, n). We conjecture a general formula for other types. / by Chuying Fang. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 75). / Fix integers n and b with n =/> 3 and 1 =/< b < n - 1. Let k be an algebraically closed field. Consider the moduli space X of hypersurfaces in P" of fixed degree I whose singular locus is at least b-dimensional. We prove that for large 1, X has a unique irreducible component of maximal dimension, consisting of the hypersurfaces singular along a linear b-dimensional subspace of P". / by Kaloyan Slavov. / Ph.D.

Thesis: Sc. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 133-136). / This thesis studies the alternating sum of cohomology groups of a Springer fiber (in characteristic 0), called a total Springer representation, as a representation of both the Weyl group and the stabilizer of the corresponding nilpotent element. For classical types, we present explicit formulas for the decomposition of total Springer representations into irreducible ones of the corresponding Weyl group using Kostka-Foulkes polynomials. Also, the character value at any element contained in the maximal parabolic subgroup(s) of type A is explicitly given in terms of Green polynomials. As a result, closed formulas for the Euler characteristic of Springer fibers are deduced. Our proof relies on analysis of geometry of Springer fibers and combinatorics of symmetric functions. Moreover, we provide formulas for the character value of a total Springer representation at any element in the stabilizer of the corresponding nilpotent element. For exceptional types, the character values of total Springer representations are completely known. Here, we only describe the decomposition of such representations into irreducible ones of stabilizers of corresponding nilpotent elements. / by Dongkwan Kim. / Sc. D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1971. / Vita. / Bibliography: leaves 44-45. / by Kenneth Lange. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997. / Includes bibliographical references (p. 48-49 / by Mathieu Gagné. / Ph.D.

Massachusetts Institute of Technology. Dept. of Mathematics. Thesis. 1966. Ph.D. / Bibliography: leaf 153. / Ph.D.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 107-109). / Validation refers to the challenge of assessing how well a learning algorithm performs after it has been trained on a given data set. It forms an important step in machine learning, as such assessments are then used to compare and choose between algorithms and provide reasonable approximations of their accuracy. In this thesis, we provide new approaches for addressing two common problems with validation. In the first half, we assume a simple validation framework, the holdout set, and address an important question of how many algorithms can be accurately assessed using the same holdout set, in the particular case where these algorithms are chosen adaptively. We do so by first critiquing the initial approaches to building a theory of adaptivity, then offering an alternative approach and preliminary results within this approach, all geared towards characterizing the inherent challenge of adaptivity. In the second half, we address the validation framework itself. Most common practice does not just use a single holdout set, but averages results from several, a family of techniques known as cross-validation. In this work, we offer several new cross-validation techniques with the common theme of utilizing training sets of varying sizes. This culminates in hierarchical cross-validation, a meta-technique for using cross-validation to choose the best cross-validation method. / by Samuel Scott Elder. / Ph. D.