Summer 2008 Faculty Research Projects
Things that are almost Symmetric
Mentor: Prof. Jonathan Simon, Professor of Mathematics, The University of Iowa
This project involves a combination of geometry, topology, and algebra. We will study the basic idea of "symmetry" and try to find ways to make an "almost-symmetric" object into a really symmetric one. We will start working with simple objects such as triangles, squares, etc., and then move on to knots.
Numerical Methods for Solving Differential-integral Equations
Mentor: Prof. Weimin Han, Professor of Mathematics, The University of Iowa
We will introduce and analyze numerical methods for solving some differential-integral equations arising from transport equations. We will explore numerically convergence behavior of the numerical solutions. MATLAB codes will be developed to implement the numerical methods and to do numerical experiments. This project will give students experience with some of the important techniques in modern applied mathematics.Prerequisite: Knowledge of Calculus and Linear Algebra; some experience with MATLAB helpful.
Learning by Discovery: Fractal Geometry
Mentor: Prof. Yi Li, Professor of Mathematics, The University of Iowa
We will work on topics in fractal geometry arising from continuous or discrete dynamical systems/differential equations.
Subgroups of Direct Products of Groups
Mentor: Prof. Dan Anderson, Professor of Mathematics, The University of Iowa
Given two groups H and K we can form their direct product H x K. What are the subgroups of H x K? We first answer this question and try to extend this result to a finite direct product of more than two groups.
Drug Dynamics, Exponential Functions and Differential Equations
Mentor: Prof. Keith Stroyan, Professor of Mathematics, The University of Iowa
Differential equations are useful in studying the dynamics of a drug in the body. The study of such dynamics is called “pharmacokinetics.” Why should we care about such dynamics? Some drugs have undesirable, or even dangerous, side effects if their concentration is too high. At the same time those drugs must be above a certain concentration to be effective. As the drug is eliminated from the body, doses need to be given periodically in order to maintain the threshold level for effectiveness, yet doses cannot be too frequent or too large or the concentration will exceed a dangerous level. Students will have the opportunity to work on projects involving the main steps in mathematical modeling: setting up the differential equations, solving the differential equations, and testing the model against real data. Students may also work on projects using differential equations to model populations of competing species.
Find full project description HERE
Algebraic Aspects of Graph Theory
Mentor: Prof. Victor Camillo, Professor of Mathematics, The University of Iowa
This project will explore a connection between graphs and matrices. It will unite graph theory and abstract algebra. In particular, with each graph we will associate a matrix and we will be able to multiply these matrices to create an object called a ring. This ring is a different way to look at a ring we can create with our graph.
Matrix Groups
Mentor: Prof. Phil Kutzko, Professor of Mathematics, The University of Iowa
As you will learn this summer, the mathematical concept of a group is one of the most useful concepts in modern mathematics. Often, a collection of matrices forms a group using matrix multiplication. Such a group is called a matrix group. We are going to study matrix groups of 2x2 matrices and you will discover how you can describe these groups using generator and relations. Prerequisite: linear algebra.
What is the Shape of a Curve?
Mentor: Prof. Oguz Durumeric, Professor of Mathematics, The University of Iowa
The “shape” of a thing, whether the thing is a molecule, a car, or a person, is a fundamental part of its nature. But the idea of “shape” is not precise. When we restrict our attention to a particular set of mathematical objects, such as curves in 3-space, the question becomes more reasonable. For a curve in the plane, or in 3-space, we can talk about ideas like “length,” “curvature,” “thickness,” and various types of “energies.” In this project, we will learn about curvature, torsion, and other geometric properties of curves and how the properties are related. Afterwards, we will study some types of energies and thicknesses, possibly introduce new ones, and try to find some optimal shapes of smooth or polygonal curves in the plane and space with respect to these energies. The idea of a flexible object seeking some kind of energy-minimizing shape is fundamental to understanding how large molecules behave.
Connections Between SL(2,R) and Truncated Moment Problems for the Upper-half Plane
Mentor: Prof. Raul Curto, Professor of Mathematics, The University of Iowa
Given a list of numbers, {s1, s2, ..., sr}, the average is (1/r)(sum of the numbers).
If there are weights {r1, r2, ..., rr}, then the center of mass is (1/r)(sum of rh ´ sh ). The important quantity here is the so-called first moment 
. We also can consider higher-order moments such as the moment of inertia 
. Now extend these ideas to systems of points 
in the plane, and moments of all orders 
. If we are given the points and the masses, we certainly can calculate the moments. But what about the other way around: Given the moments, is there a system of points and masses having those moments?
Find project description HERE.