Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017 / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 121-125). / We study the asymptotic behavior of the representation theory of symmetric groups Sn, in positive characteristic as n grows to [infinity], with the goal of understanding and generalizing the Deligne categories Rep(St) as well as the theory of FI-modules and representation stability in the positive characteristic setting. We also give qanalogs of some of our results in the context of unipotent representations of finite general linear groups in non-defining characteristic. / by Nate Harman. / Ph. D. / Ph.D. Massachusetts Institute of Technology, Department of Mathematics

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017 / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 123-125). / This thesis consists of three parts. In the first chapter we discuss arrangements of equal minors of totally positive matrices. More precisely, we investigate the structure of equalities and inequalities between the minors. We show that arrangements of equal minors of largest value are in bijection with sorted sets, which earlier appeared in the context of alcoved polytopes and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the Eulerian number. On the other hand, we prove in many cases that arrangements of equal minors of smallest value are weakly separated sets. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the positive Grassmannian and the associated cluster algebra. However, we also construct examples of arrangements of smallest minors which are not weakly separated using chain reactions of mutations of plabic graphs. In the second chapter, we investigate arrangements of tth largest minors and their relations with alcoved triangulation of the hypersimplex. We show that second largest minors correspond to the facets of the simplices. We then introduce the notion of cubical distance on the dual graph of the triangulation, and study its relations with these arrangements. In addition, we show that arrangements of largest minors induce a structure of a partially ordered set on the entire collection of minors. We use this triangulation of the hypersimplex to describe a 2-dimensional grid structure on this poset. In the third chapter, we obtain new families of quadratic Schur function identities, via examination of several types of networks and the usage of Lindstrdm-Gessel- Viennot lemma. We generalize identities obtained by Kirillov, Fulmek and Kleber and also prove a conjecture suggested by Darij Grinberg. / by Miriam Farber. / Ph. D. / Ph.D. Massachusetts Institute of Technology, Department of Mathematics

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017 / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 201-245). / In this thesis, I introduce new methods for learning about diseases and traits from genetic data. First, I introduce a method for partitioning heritability by functional annotation from genome-wide association summary statistics, and I apply it to 17 diseases and traits and many different functional annotations. Next, I show how to apply this method to use gene expression data to identify diseaserelevant tissues and cell types. I next introduce a method for estimating genetic correlation from genome-wide association summary statistics and apply it to estimate genetic correlations between all pairs of 24 diseases and traits. Finally, I consider a model of disease subtypes and I show how to determine a lower bound on the sample size required to distinguish between two disease subtypes as a function of several parameters. / by Hilary Kiyo Finucane. / Ph. D. / Ph.D. Massachusetts Institute of Technology, Department of Mathematics

Throughout, R is a commutative ring with identity and X is an indeterminate over R. We consider R{X}, the polynomial ring in one indeterminate over R, and G(R), the group of R-automorphisms of R{X}. In particular, we consider the subring of R{X} left fixed by the group G(R), denoted by R{X}('G(R)). Let B(R) be the subgroup of G(R) such that (sigma) (ELEM) B(R) if and only if (sigma)(X) = a + bX, b a unit of R. If R is reduced, then G(R) = B(R); otherwise, B(R) (L-HOOK) G(R). We prove in Chapter I that R{X}('G(R)) = R{X}('B(R)). / In Chapter I we also prove that for R to be properly contained in R{X}('G(R)), it is necessary that R/M is a finite field for some maximal ideal M of R. Hence, if R is a quasi-local ring with maximal ideal M and R/M is infinite, then R{X}('G(R)) = R. / Let R be a quasi-local ring with maximal ideal M such that R/M is isomorphic to the Galois field with p('s) elements, where p is a prime integer and s (ELEM) Z('+). In Chapter II, we show that / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / where / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / In particular, we determine Z(,n){X}('G(Zn)) for n (ELEM) Z('+). Moreover, we prove that R{X}('G(R)) contains a nonconstant monic polynomial if and only if R is a 0-dimensional SFT-ring. / In Chapter III, we investigate R{X}('G(R)) for a von Neumann regular ring R. We obtain equivalent conditions for R{X}('G(R)) to contain a nonconstant monic polynomial; one of these is that {card(R/M)} is bounded for all maximal ideals M of R. Moreover, we prove that R is properly contained in R{X}('G(R)) if and only if R has a direct summand S such that S{X}('G(S)) contains a nonconstant monic polynomial. Finally, in Chapter III we construct a von Neumann regular ring B such that B/M is finite for infinitely many maximal ideals M of B, but B{X}('G(B)) = B. / In Chapter IV, we show that for any commutative ring R with identity, R{X}('G(R)) contains a nonconstant monic polynomial if and only if R is 0-dimensional, card(R/M) < N for some N (ELEM) Z('+) and for all maximal ideals M of R, and nilpotent elements have bounded order of nilpotency. / Source: Dissertation Abstracts International, Volume: 43-03, Section: B, page: 0747. / Thesis (Ph.D.)--The Florida State University, 1982.

The mechanism for cortical folding pattern formation is not fully understood. Current models represent scenarios that describe pattern formation through local interactions and one recent model is the intermediate progenitor model. The intermediate progenitor (IP) model describes a local chemically-driven scenario, where an increase in intermediate progenitor cells in the subventricular zone (an area surrounding the lateral ventricles) correlates to gyral formation. This dissertation presents the Global Intermediate Progenitor (GIP) model, a theoretical biological model that uses features of the IP model and further captures global characteristics of cortical pattern formation. To illustrate how global features can effect the development of certain patterns, a mathematical model that incorporates a Turing system is used to examine pattern formation on a prolate spheroidal surface. Pattern formation in a biological system can be studied with a Turing reaction-diffusion system which utilizes characteristics of domain size and shape to predict which pattern will form. The GIP model approximates the shape of the lateral ventricle with a prolate spheroid. This representation allows the capture of a key shape feature, lateral ventricular eccentricity, in terms of the focal distance of the prolate spheroid. A formula relating domain scale and focal distance of a prolate spheroidal surface to specific prolate spheroidal harmonics is developed. This formula allows the prediction of pattern formation with solutions in the form of prolate spheroidal harmonics based on the size and shape of the prolate spheroidal surface. By utilizing this formula a direct correlation between the size and shape of the lateral ventricle, which drives the shape of the ventricular zone, and cerebral cortical folding pattern formation is found. This correlation is illustrated in two different applications: (i) how the location and directionality of the initial cortical folds change with respect to evolutionary development and (ii) how the initial folds change with respect to certain diseases, such as Microcephalia Vera and Megalencephaly Polymicrogyria Polydactyly with Hydrocephalus. The significance of the model, presented in this dissertation, is that it elucidates the consistency of cortical patterns among healthy individuals within a species and addresses inter-species variability based on global characteristics. This model provides a critical piece to the puzzle of cortical pattern formation. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Degree Awarded: Fall Semester, 2009. / Date of Defense: October 8, 2009. / Cerebral Cortex, Prolate Spheroid, Turing System, Cortical Folding Pattern, Reaction-Diffusion / Includes bibliographical references. / Monica K. Hurdal, Professor Directing Dissertation; Oliver Steinbock, Outside Committee Member; Jack Quine, Committee Member; DeWitt Sumners, Committee Member; Richard Bertram, Committee Member.

We develop a spectral element method to price European options under the Black-Scholes model, Merton's jump diffusion model, and Heston's stochastic volatility model with one or two assets. The method uses piecewise high order Legendre polynomial expansions to approximate the option price represented pointwise on a Gauss-Lobatto mesh within each element. This piecewise polynomial approximation allows an exact representation of the non-smooth initial condition. For options with one asset under the jump diffusion model, the convolution integral is approximated by high order Gauss-Lobatto quadratures. A second order implicit/explicit (IMEX) approximation is used to integrate in time, with the convolution integral integrated explicitly. The use of the IMEX approximation in time means that only a block diagonal, rather than full, system of equations needs to be solved at each time step. For options with two variables, i.e., two assets under the Black-Scholes model or one asset under the stochastic volatility model, the domain is subdivided into quadrilateral elements. Within each element, the expansion basis functions are chosen to be tensor products of the Legendre polynomials. Three iterative methods are investigated to solve the system of equations at each time step with the corresponding second order time integration schemes, i.e., IMEX and Crank-Nicholson. Also, the boundary conditions are carefully studied for the stochastic volatility model. The method is spectrally accurate (exponentially convergent) in space and second order accurate in time for European options under all the three models. Spectral accuracy is observed in not only the solution, but also in the Greeks. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Degree Awarded: Spring Semester, 2008. / Date of Defense: November 15, 2007. / Rainbow Option, Basket Option, Jump Diffusion, Stochastic Volatility, Options, Convolution Integral, Spectral Element Method / Includes bibliographical references. / David A. Kopriva, Professor Directing Dissertation; Fred Huffer, Outside Committee Member; Bettye Anne Case, Committee Member; Alec N. Kercheval, Committee Member; Giray Okten, Committee Member; Xiaoming Wang, Committee Member.

Let M be a nonnegative measurable function on (0,$\infty)$ and let $\tilde{M}(x) = \vert x\vert\sp{{n\over p}-{n\over q}-n} M(\vert x\vert), x\in R\sp{n}.$ We can consider a convolution operator: for a suitable f, / (UNFORMATTED TABLE OR EQUATION FOLLOWS) / (a) Suppose $1\le s\le\infty.$ Then $M\in L\sp{t}({dr\over r})$ implies that $T\sb{M}:L\sp{p}(R\sp{n})\to L\sp{q}(R\sp{n})$ is bounded for all $({1\over p},{1\over q})$ in the type-diagram triangle with vertices $(1 - {1\over s},0),\ (1,{1\over s})\ {\rm and}\ (1 - {1\over(n+1)s},{1\over(n+1)s})$ if and only if s = t. / (b) Suppose $1<p<q<\infty.$ Let $s\sb0$ be the smallest value of $s\in\lbrack 1,\infty)$ such that ${1\over q}\ \ge\ {1\over n}({1\over p} - (1 - {1\over s}))$ and ${1\over q}\ \ge\ {n\over p} - n +\ {1\over s}$ Then $T\sb{M}:L\sp{p}(R\sp{n})\to L\sp{q}(R\sp{n})$ for all $M\in L\sp{t}({dr\over r})$ if and only if $s\sb0\le t\le\infty.$ / Results (a) and (b) solve the following problem: If $1<p<q<\infty$ find the ranges of s such that $M\in L\sp{s}({dr\over r})$ implies that $T\sb{M}$ is bounded from $L\sp{p}(R\sp{n})$ to $L\sp{q}(R\sp{n}).$ / Source: Dissertation Abstracts International, Volume: 54-07, Section: B, page: 3657. / Major Professor: Daniel M. Oberlin. / Thesis (Ph.D.)--The Florida State University, 1993.

In this dissertation, we focus on (1) improving the efficiency of some iterative domain decomposition methods, (2) proposing and developing a novel domain decomposition algorithm, (3) applying these algorithms to the efficient numerical solution of the finite element discretization of the shallow water equations on a 2-D limited area domain and (4) investigating parallel implementation issues. / We have closely examined the iterative Schur domain decomposition method. A modified version of the rowsum preserving interface probing preconditioner is proposed to accelerate the convergence on the interfaces. The algorithm has been successfully applied to the finite element shallow water flow modeling. / The modified interface matrix domain decomposition algorithm is proposed and developed to reduce computational complexity. The numerical results obtained by applying this algorithm to our problem improve upon those obtained by employing the traditional Schur domain decomposition algorithm. / We then investigate parallel block preconditioning techniques in the framework of three frequently used and competitive non-symmetric linear iterative solvers. Two types of existing domain decomposed (DD) preconditioners are employed and a novel one is proposed. The newly proposed third type of DD preconditioners turns out to be computationally the most efficient. / Parallel implementation issues of domain decomposition algorithms are then discussed. Typical parallelization results on the CRAY Y-MP are presented and discussed. / This dissertation also contains a relatively thorough review of two fast growing areas in computational sciences, namely, parallel scientific computing in general and iterative domain decomposition methods in particular as well as a discussion concerning possible future research directions. / Source: Dissertation Abstracts International, Volume: 55-07, Section: B, page: 2766. / Major Professor: I. Michael Navon. / Thesis (Ph.D.)--The Florida State University, 1994.

Motivated by questions in theoretical physics, there has been much interest in the problem of defining and calculating the determinants of differential operators, such as the Laplacian on manifolds. The most common method that makes sense out of the product of a discrete sequence, such as the spectrum of an operator, is the zeta function regularization, in which one defines the zeta regularized product by analytically continuing the zeta function to the origin. When the analytic continuation exists and how to implement the process are major questions in zeta regularization. / The purpose of this dissertation is to develop a systematic approach for operating with zeta regularized products which will make use of some basic properties to calculate zeta regularized products without finding the actual continuation. / First we establish a number of properties similar to that of ordinary products, second we use these properties to compute various zeta regularized products, in particular, the determinants of Laplacian on p-dimensional flat tori. Without involving the analytic continuation, our computation turns out to be much simpler and easier to understand. / Some analytic properties are also discussed, namely the Weierstrass factorization and the Laplace-Mellin transform. Relationships between gamma functions and zeta regularized products are established. It turns out that the gamma function as well as Barnes' multiple gamma functions can be represented as special zeta regularized products. Also some asymptotic expansions of zeta regularized products are obtained. As a result, the classic Stirling formula is generalized to a double Stirling formula. / Source: Dissertation Abstracts International, Volume: 54-12, Section: B, page: 6242. / Major Professor: John R. Quine. / Thesis (Ph.D.)--The Florida State University, 1993.

This thesis investigates the feasibility of the 4-D variational data assimilation (VDA) applied to realistic situations and improves existing large-scale unconstrained minimization algorithms. It first develops the second order adjoint (SOA) theory and applies it to a shallow-water equations (SWE) model on a limited-area domain to calculate the condition numbers of the Hessian. Then the Hessian/vector product obtained by the SOA approach is applied to one of the most efficient minimization algorithms, namely the truncated-Newton (TN) algorithm. The newly obtained algorithm is applied here to a limited-area SWE model with model generated data where the initial conditions serve as control variables. / Next, the thesis applies the VDA to an adiabatic version of the Florida State University Global Spectral Model (FSUGSM). The impact of observations distributed over the assimilation period is investigated. The efficiency of the 4-D VDA is demonstrated with different sets of observations. / In all of the previous experiments, it is assumed that the model is perfect, and so is the data. The solution of the problem will have a perfect fit to the data. This is of course unrealistic. / The nudging data assimilation (NDA) technique consists in achieving a compromise between the model and observations by relaxing the model state towards the observations during the assimilation period by adding a non-physical diffusion-type term to the model equations. Variational nudging data assimilation (VNDA) combines the VDA and NDA schemes in the most efficient way to determine optimally the best conditions and optimal nudging coefficients simultaneously. The humidity and other different parameterized physical processes are not included in the adjoint model integration. Thus the calculation of the gradients by the adjoint model is approximate since the forecast model is used in its full-physics (diabatic) operational form. / Source: Dissertation Abstracts International, Volume: 54-12, Section: B, page: 6242. / Major Professor: I. Michael Navon. / Thesis (Ph.D.)--The Florida State University, 1993.