The University of Iowa Mathematics Research Program
Summer 2009 Mathematics Faculty Research Projects
Prerequisites: Two semesters of Calculus and one semester of Linear Algebra
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Computing Cyclotomy using Representation Theory
Mentor: Prof. Phil Kutzko, Professor of Mathematics, The University of Iowa
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.
Projects in Mathematical Modeling
Mentor: Prof. Keith Stroyan, Professor of Mathematics, The University of Iowa
Students in this project group will use mathematics to model problems in areas such as epidemics, drug metabolism, physical systems, or economics. Some sample projects are here
The goal is for students to do something themselves: formulate a problem, use software to explore it and make conjectures, prove (some of) the conjectures.
Numerical Methods for Differential Equations
Mentor: Prof. Laurent Jay, Professor of Mathematics, The University of Iowa
In this project, we will explore different methods to solve systems of differential equations numerically on computers using MATLAB. We will test some methods for some simple model problems in mechanics with kinematic constraints (e.g., a mobile robot, a sleigh, a skate, a "snakeboard", etc.) We will also develop and test some new methods. We will be particularly interested in numerical integration methods which can reproduce or mimic the properties of the original systems of equations well.
Group Rings and Representations
Mentor: Prof. Victor Camillo, Professor of Mathematics, The University of Iowa
As you will learn in the Groups and Characters workshop, studying the representations of a group can sometimes get a little complicated. One way to improve this situation is to view the representations of a group as modules over a ring associated to the group. This approach makes all the tools of module theory available. We will spend a little time learning about what a module is and how representations of a group can be thought of as modules. We will then apply this knowledge to some problems concerning representations.
A Topological Project Concerning Proteins
Mentor: Prof. Isabel Darcy, Professor of Mathematics, The University of Iowa
Proteins are long, linear polymer chains made up of combinations of twenty naturally occurring amino-acids. The 1-dimensional primary sequence of amino acids dictates the three-dimensional folded shape of a protein molecule and, ultimately, its biological function. Any sufficiently long polymer chain is expected to become knotted with high probability. For example, long extension cords frequently contain local knots. However proteins only rarely contain local knots. We will investigate knotted proteins. We will compare their functions and folding mechanisms to unknotted and nearly knotted proteins.
Shape and Symmetry
Mentor: Prof. Jonathan Simon, Professor of Mathematics, The University of Iowa
In this project, we will explore ideas of 'shape' and 'symmetry'. Given a set of points in space (or a curve or a surface), can we say one is "more spherical" than another; or one is "more symmetric" than another; or "these two shapes are very similar"? These questions arise, for example, in studying shapes of molecules or tumors.
Applications and Generalizations of Goursat's Lemma
Mentor: Prof. Dan Anderson, Professor of Mathematics, The University of Iowa
The direct product of two groups seems to be a simple idea. But finding all the subgroups of a direct product is not so simple. Goursat's Lemma tells how to find the subgroups of a direct product of two groups. We will explore Goursat's Lemma with an eye to applications and extending it to a direct product of more than two groups.
What is the Shape of a Curve?
Mentor: Prof. Oguz Durumeric, Professor of Mathematics, The University of Iowa
The shape of an object is one of its fundamental properties; shape affects how the object behaves. This is especially true for proteins and other large molecules. Many flexible objects find their optimal shapes by minimizing certain types of energies and thicknesses. In this project, we will learn about curvature, torsion, some classical theorems of curves in the plane or three dimensional space, and study some types of energies and thicknesses. We will try to find some optimal shapes of smooth or polygonal curves.
Hankel Matrices and the Vector Space of all Polynomials
Mentor: Prof. Raul Curto, Professor of Mathematics, The University of Iowa
Find a more precise project description HERE