The University of Iowa Mathematics Research Program
Summer 2011 Mathematics Faculty Research Projects
Prerequisites: Two semesters of Calculus and one semester of Linear Algebra
A "ring" is an algebraic system that has key properties of the integers: addition and multiplication operations, identity elements for both, distributive property, etc. Examples include Z_n (the integers mod n), the set of all polynomials in one or more variables, the set of all (3 x 3) matrices. We say a ring is 'commutative' if the multiplication operation is commutative, which eliminates most matrix rings. One of the fundamental questions in commutative ring theory is deciding when some element can be factored as a product of others. In this project, students will investigate questions about factorization and other problems in ring theory.
We will engage in a project in matrix theory. We have two ideas with some history in mind. This first is a counting problem regarding Fitting’s Lemma. This last says that every matrix is a sum of an invertible matrix and an idempotent matrix. How many ways can you do this? The second explores the theme of row-reduced matrices. It is not well known but products of row-reduced matrices are row reduced. Are there other collections of matrices that behave in the same way? There is some literature on this up to the 3x3 case. Can we do anything in the 4x4 case?
Title: Numerical Modeling and Simulation in Biology and Medicine
Description: In this project we will explore computational approaches to understanding problems in the life sciences using numerical methods for differential equations. Projects will be selected from several active research areas at Iowa such as modeling the heart, hearing, bones, etc. The projects will be implemented using MATLAB. (Introduction to MATLAB will be provided.)
Students in this group will study the mathematics of digital image processing. Sample topics: grey-scale matrices; multiple matrices for the basic colors; matrix operations that shift between resolution levels; and intermediate detail data. Resolution levels and details are represented in vector spaces, and the processes by transformations between them.
Students will get to experience how the mathematical theory can be applied to image processing and work on their own photos. The projects will be implemented using MATLAB. (Introduction to MATLAB will be provided.)
Cyclotomy comes from the Greek word for "circle division" and, in its original form, it was concerned with computing certain real and complex numbers that help you to draw polygons. Although this is the origin of cyclotomy, it turns out the numbers you compute are very useful in several parts of applied mathematics, including coding theory. The idea behind this project is that you can construct certain finite groups which have the property that their representations give you information about cyclotomic numbers. We will construct these groups and then use representation theory to compute these numbers and discover their properties.
In this project, we will study curves and surfaces in 3-space, in particular questions about symmetry and parameterizations. We all know how to parameterize a circle as [cos(t), sin(t)]. Similarly, we can parameterize space curves as [x(t), y(t), z(t)]; and surfaces as [x(s,t), y(s,t), z(s,t)]. We will start by considering a lot of questions, and then concentrate on the one(s) most interesting to the group: How can you tell from the equations whether a given space-curve is self-intersecting? How can you see from the equations whether a given curve or surface has some kind of symmetry. We can build a symmetric object by starting with a small piece and moving that around under some combination of rotations and reflections: How can we do this so as to obtain a smooth closed curve or surface? It is easy to parameterize a torus: can you parameterize a double torus? triple torus? Using TRIG functions as we can for a single torus? Students will use software such as Maple, Mathematica, or Matlab and introduction will be provided.
Exploration of Dynamical Systems:
From a certain point of view a dynamical system is a consistent collection of functions that gives you the future state of a physical system based on the present state. The time can be assumed to be discrete or continuous. In the discrete time situation, a good way of describing the system is the mapping that bridges the shortest time interval. In the continuous time situation, one often has a system of ordinary or partial differential equations governing the evolution. We will look at some such systems where it is possible to give a closed form solution; for others we will get an intuitive understanding of what is going on by means of a combination of numerical experiments and theoretical analysis.
Visual Depth Perception from Motion Parallax and Optic Flow