Fall, 2007
The distributive lattices form the simplest nontrivial "variety" of lattice. They arise in many ways. For example, any collection of sets which is closed under pairwise intersection and union forms a distributive lattice under those operations. If we add complementation to the list of operations, we get boolean algebras. We will describe a fundamental theorem concerning the structure of finite distributive lattices, and then describe connections with familiar and less-familiar functions, such as the "volume functions," the "counting functions," and Möbius functions of partially ordered sets.
A number of meshless methods for solving PDEs such as the meshless local Petrov-Galerkin method have enjoyed success in applications, but lacked a mathematical theory providing error estimates and convergence. Schaback has recently provided such a theory, resulting in a framework which is applicable to a large class of methods for solving well-posed linear operator equations. I present Schaback's framework and specializations to methods involving meshless kernel-based trial spaces, and either weak or strong testing of residuals.
The weighted composition operator is the natural generalization of the composition and multiplication operators. Weighted composition operators appear in many areas of mathematics, such as Banach space theory and dynamical systems. In this talk, we present the foundations of the weighted composition operator acting on the Bloch space and little Bloch space of the unit disk. We also present necessary and sufficient conditions for the weighted composition operator to be bounded on the both spaces. Lastly, we will discuss my dissertation research goals in this area.
Deterministic differential equations and random processes combine unexpectedly with Dynkin's Formula. This formula provides the expected value of a stochastic process at a stopping time, but can also be used to solve a wide variety of deterministic partial differential equations such as Laplace's equation, the heat equation, and the Schrodinger equation. In this talk we will present the necessary concepts to understanding Dynkin's Formula to include Brownian Motion, stochastic processes, and stochastic differential equations; then we will apply the formula to solve Laplace's equation in an unconventional way.
In 1973, Fischer Black and Myron Scholes devised a model for pricing options under specific assumptions. In this talk, I will discuss the procedure used to price European options including Brownian Motion, Stochastic Integration, Ito's Formula, Martingales, and Filtrations. This construction is neccesary to get optimal results. Finally, I will talk about a few numerical procedures and results used in association with the Black-Scholes Model.
We examine the effects of gravity in a model of one-dimensional imbibition of an incompressible liquid into an initially dry and deformable porous material. We obtain analytic results for steady state positions of the wet porous material--dry porous material interface as well as liquid--wet material interface. The time-dependent free-boundary problem is solved numerically and the results compared to the steady state predictions. In the absence of gravity, the liquid rises to an infinite height and the porous material deforms to an infinite depth, following square-root in time scaling. In contrast, in the presence of gravity, the liquid rises to a finite height and porous material deforms to a finite depth. Dependence on model parameters such as the solid liquid density ratio is also explored.
Coin graphs are embeddings of graphs in the Euclidean plane such that each vertex corresponds to a non-overlapping closed disk. The problem (and solution) of determining the maximum number of edges of a unit coin graph on n vertices will be presented. There are several generalizations which lead to interesting open questions. My research generalizes the problem to coin graphs that satisfy certain conditions relating to the ratios of the possible radii of coins in the graph. Further, we will explore the algebraic equations describing wheel graphs as they relate to the maximum number of edges in certain coin graphs.
Given the formal complex power series ring on an infinite set of variables, a permutation acts on the elements of the ring by permuting the variables. A subset of the functions fixed by this action is the ring of symmetric functions. We will briefly discuss the structure of this ring. An important example of such functions are Schur polynomials, which will be presented from both a combinatorial and algebraic point of view. We will close with a brief application to graph colorings.
Van Douwen spaces are the exact spaces needed to produce a less than or equal to 2 to 1 mapping from the Čech-Stone remainder of the naturals onto a separable space. In my talk I will discuss the natural relationship between Van Douwen spaces and absolutes, as well as the construction of such spaces as dense subsets of absolutes. As an aside, a brief history and open problems will also be given.
Attempts to construct wavelet bases in higher dimensions with some of the properties of the Haar basis for L2(R) will be investigated. This leads to systems of sets which resemble iterated function systems. The main construction is based on the concept of a multiresolution analysis. We will see connections between self-similar tilings and compactly supported wavelet bases, as well as some coset counting arguments.
When G is a locally finite group, acting on a commutative unital ring R via ring automorphisms, then RG &sub R satisfies universally going-down. Through the investigation of the effect of easing the restriction of local fintiteness on G, Dobbs and Shapiro developed an example in which G acts on R such that the ring extension RG &sub R does not satisfy the going-down property. The construction of this group will be presented, along with the proof that the going-down property from the ring to the fixed subring fails.