University of Iowa REU Program

The University of Iowa program will accept 9 students total for projects in mathematics. Students at the University of Iowa will live together with the students in the VIGRE program and students in the Iowa Summer Institute in Biostatistics (ISIB) in the Mayflower Residence Hall. Alliance Scholars will share cultural and social activities with students in the VIGRE and ISIB programs. Dinner will be provided at Burge Residence Hall, and a resident assistant will be available in the evenings and on weekends for support.

Mathematics

The Department of Mathematics will offer 3 workshop courses that will feed into 9 mathematics research projects. The UI mathematics classes, as well as social and cultural events, will also be shared with the NSF VIGRE REU students who represent the Heartland Partnership schools. Prerequisites include two semesters of Calculus and one semester of Linear Algebra.

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Biostatistics

The University of Iowa Department of Biostatistics will offer a Biostatistics course and 8 research projects. Please visit the biostatistics REU homepage for more information on the Biostatistics summer program. Please note that you will need to fill out a separate application form for the biostatistics REU program. (You do not need to fill out the Alliance application for this program.)

Iterative Procedures in Differential Equations

Mentor: Prof. Juan Gatica, Professor of Mathematics, The University of Iowa

Problems in differential equations often are transformed into fixed point problems for operators in function spaces. We will study cases when the problem is amenable to use iteration of this operator and use this to produce numerical approximatyiosns to solutions, when they exist.

Computing Cyclotomy using Representation Theory

Mentor: Prof. Phil Kutzko, Professor of Mathematics, The University of Iowa
Mentor: Oscar Vega, Assistant Professor, California State University, Fresno

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.

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.

Shapes and Tangling of Filaments

Mentor: Prof. Jonathan Simon, Professor of Mathematics, The University of Iowa

In this project, we will explore ways of describing and measuring the "shapes" and the "tangling" of long filaments such as proteins or DNA. We have two goals: Devise mathematical definitions to capture the intuitive ideas of "shape" or "tangling", and try to find relations between different quantities. Think of words such as "length", "curvature", "knotting", "spatial packing". Students will [learn to] use computer software such as Maple (or Mathematica or Matlab) and KnotPlot.

Suggested background: Calculus II minimum, Calculus III or Linear Algebra desirable.

Decompositions of Linear Transformations

Mentor: Prof. Victor Camillo, Professor of Mathematics, The University of Iowa

We continue the project of finding decompositions of linear transformations as sums of invertible matrices and projections. This project is amenable to computer modeling. We also investigate and ongoing conjecture, not found in the literature about minimal ways to do this. We are motivated by the classic result: Fitting's Lemma.

Numerical Methods for Differential Equations

Mentor: Prof. Laurent Jay, Professor of Mathematics, The University of Iowa

In this project, we will explore different numerical methods to solve systems of differential equations on computers using MATLAB. We will test some recent methods for some problems having different time scales (e.g., having strong oscillations) and also for problems in mechanics (e.g., a mobile robot model, a skate model, etc.). We will be particularly interested in numerical integration methods able to reproduce or mimic the properties of the original systems of differential equations well and to interpret mathematically the numerical results obtained in the sense of backward error analysis.

Tangle analysis of protein-DNA complexes

Mentor: Prof. Isabel Darcy, Professor of Mathematics, The University of Iowa

Some proteins will cut DNA and change the DNA configuration before resealing the DNA. Thus, if the DNA is circular, the DNA can become knotted. When modeling protein-DNA reactions, one would like to know how to draw the DNA. For example, are there any crossings trapped by the protein complex? How do the DNA strands exit the complex? Is there significant bending? Topological analysis cannot determine the exact geometry of the protein-bound DNA, but it can determine the overall entanglement of this DNA, after which other techniques may be used to more precisely determine the geometry.

Numerical Solution of Differential Equations

Mentor: Prof. Weimin Han, Professor of Mathematics, The University of Iowa

This project is intended as an introduction to basic discretization techniques of differential equations. After a discussion of discretization methods for ordinary differential equations, we consider solving a few model partial differential equations. Essential features of numerical methods are presented, with emphasis on fundamental ideas of accuracy, stability, convergence, and numerical dissipation.

Finite Fields of Matrices and Relations to Finite Geometry

Mentor: Prof. Phil Kutzko, Professor of Mathematics, The University of Iowa
Mentor: Oscar Vega, Assistant Professor, California State University, Fresno

An affine plane is a just a generalization of the standard Euclidean plane; translation planes are affine planes in which lines have ''slopes'' that are represented by a suitable set of matrices. The best (in terms of geometric properties) finite translation planes can be obtained by constructing finite fields of matrices. In this project we will see how, using linear algebra, finite fields of matrices may be constructed. We will then look at the geometric properties of their associated translation planes (by studying the embedding of their corresponding field of matrices in the ring of all matrices). Some representation theory will be needed to investigate what type of ''symmetries'' these planes have.

Introduction to Knots

Mentor, William Schellhorn, Assistant Professor, Simpson College

http://www.simpson.edu/math/community/schellhorn.html If you tie a knot in a piece of rope and then connect the ends together, you have created a mathematical knot. This workshop is an introduction to the mathematical field of knot theory. Knot theory is a branch of three-dimensional topology, meaning that properties such as the length and thickness of the rope you used are traditionally disregarded. In recent years, researchers also have been studying how physical properties of the "rope" relate to knotting and tangling.

We will concentrate on the fundamental definitions in knot theory, ways that knots can be represented, how knots relate to surfaces, types of knots, and how to distinguish knots. We will also study other knotted objects that are closely related to knots, like tangles, braids, and chord diagrams. This workshop will include an introduction to the computer software KnotPlot.

Elements of Ring Theory

Oscar Vega, Assistant Professor, California State University, Fresno

http://zimmer.csufresno.edu/~ovega We are all very familiar with the set of integers, Z, due to all its wonderful algebraic (or number theoretical, if you prefer) properties and because it has been a commonplace in the history of mathematics. Rings, on the other hand, seem to be a bit more mysterious and difficult to understand. However, we can consider a ring to be just a generalization of Z; a set with two operations, usually denoted · and +, that satisfy a few simple properties that are related to each other only by the distributive law.

Rings are everywhere. The set of all n-by-n matrices with entries in the reals, and standard addition and multiplication, forms a ring. The set of all functions from C (the complex numbers) into itself, with composition as the multiplication and the obvious addition, is also a ring. The set of all polynomials with coefficients in Z is yet another example of a ring. Moreover, the study of rings and of the equations that can be solved in a particular ring is an area that has yielded many important results. For instance, polynomial equations over the integers with integral solutions, also called Diophantine equations, are related to a wide variety of topics ranging from Pythagorean triples, to arithmetic modulo an integer n, to important results in number theory such as Euler's theorem, to Fermat's Last Theorem, to secure ways to encrypt your credit card number for when you shop online.

In this workshop, students will start with learning the basics of ring theory, including results on homomorphisms, and products/sums of rings. Their study will continue with other important topics such as group rings, matrix rings and other non-commutative rings. Several examples will be considered.

More specific topics will be discussed towards the end of the workshop; these will be related to the projects students will get involved in the last four weeks of their REUs. For instance, a short introduction to representation theory will be given.

Scientific Computing and Numerical Methods

Matthew Rissler, Assistant Professor, Loras College

http://depts.loras.edu/academics/faculty/MatEngCompSciDivFaculty.html Modern science and engineering depend heavily on numerical analysis and computer simulations. This workshop introduces fundamental mathematical techniques and algorithms used in scientific computing.

Topics will include: function approximation, curve fitting, algorithms for solving linear and nonlinear systems of equations, matrix eigenvalue problems, optimization techniques, numerical solution of differential equations. Students will learn the high-level computer system MATLAB.

Prerequisites: Calculus I, II and a course in linear algebra.