Browsing Mathematics by Title
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On fvectors of polytopes and matroids
The fvector of a simplicial complex is a fundamental invariant that counts the number of faces in each dimension. A natural question in the theory of simplicial complexes is to understand the relationship between the ... 
On numerics and inverse problems
In this thesis, two projects in inverse problems are described. The first concerns a simple mathematical model of synthetic aperture radar with undirected beam, modeled as a 2D circular Radon transform with centers restricted ... 
On Particle Interaction Models
(20140224)This dissertation deals with three problems in Stochastic Analysis which broadly involve interactions, either between particles (Chapters 1 and 2), or between particles and the boundary of a C2 domain (Chapter 3). In Chapter ... 
On Selmer groups and factoring padic Lfunctions
Samit Dasgupta has proved a formula factoring a certain restriction of a 3variable RankinSelberg padic Lfunction as a product of a 2variable padic Lfunction related to the adjoint representation of a Hida family and ... 
On singularities of generic projection hypersurfaces
(2006)The present work studies singularities of hypersurfaces arising from generic projections of smooth projective varieties, in the context of Du Bois and semi log canonical singularities. It is demonstrated that Du Bois ... 
On special Lagrangian equations
(20140224)In this paper we study the special Lagrangian equation and related equations. Special Lagrangian equation originates in the special Lagrangian geometry by HarveyLawson [HL1]. In subcritical phases, we construct singular ... 
On TSemisimplicity of Iwasawa Modules and Some Computations with Z3Extensions
For certain Zpextensions of abelian number fields, we study the Iwasawa module associated to the ideal class groups. We show that generic Zpextensions of abelian number fields are Tsemisimple. We also construct the ... 
On the g2number of various classes of spheres and manifolds
For a $(d1)$dimensional simplicial complex $\Delta$, we let $f_i=f_i(\Delta)$ be the number of $i$dimensional faces of $\Delta$ for $1\leq i\leq d1$. One classic problem in geometric combinatorics is the following: ... 
On the Geometry of Rectifiable Sets with Carleson and Poincaretype inequlaities
A central question in geometric measure theory is whether geometric properties of a set translate into analytical ones. In 1960, E. R. Reifenberg proved that if an $n$dimensional subset $M$ of $\mathbb{R}^{n+d}$ is well ... 
On the mod 2 general linear group homology of totally real number rings
(1997)We study the mod 2 homology of the general linear group of rings of integers in totally real number fields. In particular, for certain such rings R, we construct a space JKR and show that the mod 2 homology of JKR is a ... 
The OrnsteinUhlenbeck Process In Neural DecisionMaking: Mathematical Foundations And Simulations Suggesting The Adaptiveness Of Robustly Integrating Stochastic Neural Evidence
(20130225)This master's thesis reviews the concepts behind a stochastic process known as the OrnsteinUhlenbeck Process, and then uses that process as a way to investigate neural decision making. In particular, MATLAB simulations ... 
Path algebras and monomial algebras of finite GKdimension as noncommutative homogeneous coordinate rings
This thesis sets out to understand the categories QGr A where A is either a monomial algebra or a path algebra of finite GelfandKirillov dimension. The principle questions are: \begin{enumerate} \item What is the structure ... 
Permutation diagrams in symmetric function theory and Schubert calculus
A fundamental invariant of a permutation is its inversion set, or diagram. Natural machinery in the representation theory of symmetric groups produces a symmetric function from any finite subset of <bold>N</bold><super>2</super>, ... 
Polynomials in Multiview Geometry
(20130417)We study multiview geometry and some of its applications through the use of polynomials. A threedimensional world point gives rise to n ≥ 2 twodimensional projections in n given cameras. The object of focus in this ... 
The Positive Semidefinite Rank of Matrices and Polytopes
The positive semidefinite (psd) rank of a nonnegative <italic>p</italic> × <italic>q</italic> matrix <italic>M</italic> is defined to be the smallest integer <italic>k</italic> such that there exist <italic>k</italic> × ... 
Problems in Algebraic Vision
This thesis studies several fundamental mathematical problems that arise from computer vision using techniques in algebraic geometry and optimization. Chapters 2 and 3 consider the fundamental question of the existence of ... 
Problems in computational algebra and integer programming
(2007)This thesis is a compendium of several projects that span the gap between commutative algebra and geometric combinatorics: one in tropical geometry, one in computational algebra, and two in discrete geometry and integer ... 
Qualitative stability properties of matrices
(1985)A matrix A is sign stable if some matrix with the same sign pattern of positive, negative, and zero entries has all eigenvalues with negative real parts. It is potentially stable if it is not sign unstable. The system A x ... 
The Radiative Transfer Equation in Photoacoustic Imaging
(20130725)Photoacoustic tomography is a rapidly developing medical imaging technique that combines optical and ultrasound imaging to exploit the high contrast and high resolution of the respective individual modalities. Mathematically, ... 
Random recursion
We study the limiting behavior of three stochastic processes. Two are interacting particle systems, the frog model and coalescing random walk. We work out transience and recurrence properties on various graphs. The last ...