The
Mathematics Departmentoffers undergraduate majors an opportunity to work closely with a professor on a project. The project might be publishable research, but it could also just be an interesting project that introduces students to some new mathematics. In any case the project would be worthy of a student presentation at a conference.

The duration of the project might be short or long: a student could work for more than one year on a long-term project, or less than a year on a short-term one.

This is not clerical work. The student will not receive payment or course credit for the project. This will be interesting mathematics that leads to something that a student can write or present. A student could end up with a finished project to hand to a prospective employer or graduate school admissions officer; or just to keep and say, "I did this."

If you are interested in working on a project, please contact the professor listed. You can also discuss project ideas you have with any faculty member, and that faculty member can suggest someone who will work with you.

## Potential Projects

In this project we will re-visit and expand upon a project of John Imbrie of the University of Virginia and his daughter Katherine matching periodic earth temperatures reflected in ice cores to when the earth axis tilt wobbled and the planets relative annual position to sun. We will investigate how key aspects of the ice-age record (such as shifts in dominant periodicities) follow from simple ordinary differential equations capturing the essential physics of the growth and decay of ice sheets.

**Prerequisite:** A student working on this project should have successfully completed MATH 241 Differential Equations.

### Magic Polygons, Dr. Shelly Bouchat

If you like number puzzles like Sudoku, you might be interested in “magic squares." A 3x3 magic square is a puzzle that has 3 rows, each of which contains 3 boxes. You must place numbers in each box so that the sum of each row, each column, and each of the two diagonals is the same number. This project will focus on expanding this classic puzzle to consider “magic polygons.” More details can be found on
Dr. Bouchat's web page.

**Prerequisite:** Just an interest in mathematics.

### ‘Area’ and ‘Length’ Application Using Green’s Theorem and More, Dr. John Chrispell

The idea of this project is to create an app for an Android or Apple device that allows the user to calculate the area of a selected region using a touch interface. For instance, an application user could calculate the approximate size of the state of New York. To calculate New York’s area a map interface would be used to collect user specified points around the boundary outlining the state and the application would compute the area using a line integral.

The application would be even more useful on a smaller scale. Computing areas of plots of land or towns by again selecting points around the region. As an added feature computing the length of a path selected on an image would also be a neat mathematical feature to add.

**Prerequisite:** The mathematics needed would be based on MATH 225 Calculus III with use of Green’s Theorem and arclength being the core concepts.

### The Gamma Function, Dr. Alfy Dahma

The theory of the gamma function was developed in connection with the problem of generalizing the factorial function of the natural numbers. The gamma function is defined as a definite, improper integral, and the notion of factorials is extended to complex and real arguments. This function crops up in many unexpected places in mathematical analysis, such as finding the volume of an n-dimensional “ball”. In this project we develop and explore the basic properties of this function.

**Prerequisite:** MATH 225 Calculus III preferably, but you can get by with a good understanding of infinite series and integration techniques used in MATH 126 Calculus II.

### Problems In Mathematical Analysis, Dr. Alfy Dahma

Are you interested in learning skills and techniques required to do research in mathematical analysis? If so, I have just the project for you. I’ve compiled a list of interesting problems in many areas of mathematics: calculus, set theory, linear algebra, topology, advanced calculus, general elementary analysis, etc. These problems range in difficulty level, and many of the solutions will require you do some of your own independent research. Solutions to some of these problems would be appropriate for presentation at an IUP colloquium or possibly a conference sponsored by a professional mathematical organization.

**Prerequisite:** Most problems require experience with writing proofs, but there’s no harm trying.

### Digital Image Processing, Dr. Ed Donley

In this project we will look at Digital Image Processing, studying processes whose inputs and outputs are images and that extract or enhance attributes from images.

**Prerequisite:** There are simple projects that require MATH 125 Calculus I and MATH 171 Introduction to Linear Algebra, and there are more advanced projects that require MATH 342 Advanced Mathematics for Applications and use Fourier Transforms and wavelets.

### Managing the Grading Process of Open-ended Questions on Nationwide Test, Dr. Yu-Ju Kuo

Have you ever taken a nationwide standardized test, such as an AP exam? Do you wonder how and where they are graded in a limited amount of time? This is a type of logistic problem occurring in business settings. Through this project, students will build a simplified *Arena* (simulation software) model for the above system, conduct sensitivity analysis for various resources and cost factors, propose reasonable ways to improve the system, and verify the proposed methods.

**Prerequisite**: No previous knowledge of *Arena* is required. Students should be comfortable playing with the software and will investigate the model in the directions of their choice.

### Least is the Best, Dr. Yu-Ju Kuo

A common concern in industry is optimization: minimizing the cost, maximizing the profit, optimizing resource utilization, and so on. Students learn basic optimization techniques in calculus courses. But to what is it applied? What if the objective function is non-differentiable? What if variables are discrete? In this project, students can choose their preferred "no-so-nice" application and explore heuristic approaches to estimate the optimum and the optimizer.

**Prerequisite**: A student working on this project should have successfully completed MATH 225 Calculus III, MATH 171 Linear Algebra, and COSC 110.

### Rafael Bombelli’s L’Algebra, Dr. Gary Stoudt

Rafael Bombelli’s **L’Algebra** is famous for being the first work to lay out the rules for working with what we now call complex numbers. It is also an excellent algebra book. While parts of the work have been translated into English, the entire work has not been. In this project we will translate portions of **L’Algebra** and see what gems we can find.

**Prerequisite:** A student taking on this project should know Italian or have a very good knowledge of some other romance language and a willingness to use it to "fake his/her way" in Italian.

### Conic Sections via Cones, Dr. Gary Stoudt

In this project we will work our way through **Conics** of Apollonius of Perga ca. 262 BC – ca. 190 BC. In this work Apollonius develops simple and not so simple properties of conic sections, many of which we now only know through calculus. We will also attempt to illustrate the propositions in **Conics** using the powerful mathematics software *Mathematica*.

**Prerequisite:** No previous knowledge of *Mathematica* is required. A student working on this project should have successfully completed MATH 225 Calculus III.

### The Method of Exhaustion for Areas and Volumes, Dr. Gary Stoudt

The method of exhaustion was developed by Eudoxus of Cnidus (ca. 410– ca. 355 BC) to prove results about areas and volumes using geometry (without limits). Archimedes (ca. 287 BC – ca. 212 BC) was its greatest proponent and used it to prove many results that we now demonstrate using calculus. In this project we will look at the works of Eudoxus and Archimedes and see first hand how the method of exhaustion was used.

**Prerequisite:** A student taking on this project should have a good knowledge of high school geometry.

### Euclid’s Elements and Geometer’s Sketchpad, Dr. Gary Stoudt

In this project we will go through Euclid’s **Elements** and illustrate his propositions using the geometry software *Geometer’s Sketchpad*. Euclid’s **Elements** is without a doubt the one of the most famous works of mathematics. In it Euclid lays the foundations for deductive geometry and goes on to prove very advanced theorems about geometry and number theory.

**Prerequisite:** No previous knowledge of *Geometer’s Sketchpad* is required. What is required is a good knowledge of high school geometry.

### Affine Transformations and Homogeneous Coordinates, Dr. Gary Stoudt

In this project we will look at geometric transformations using homogeneous coordinates and matrices. Affine transformations include translation, rotation, reflection, shear, expansion/contraction, and similarity transformations. This will show the student the relationship between high school geometry, linear algebra, and group theory. We will also illustrate properties in geometry and linear algebra using the powerful mathematics software *Mathematica* or *MATLAB*.

**Prerequisite:** No previous knowledge of *Mathematica*/*MATLAB* is required. A student taking on this project should have successfully completed MATH 171 Introduction to Linear Algebra.

### Approximation of Functions with Simpler Functions, Dr. Gary Stoudt

In this project we will use functions with "nice" properties to approximate other functions. An example that might be familiar to students is using polynomials to approximate certain functions via Taylor/Maclaurin series. Eventually we will look at families of so-called “orthogonal functions” and how they are used to approximate other functions. We will use the powerful mathematics software *Mathematica* to illustrate the approximations.

**Prerequisite:** No previous knowledge of *Mathematica* is required. Students will investigate theory only as their abilities dictate. A student taking on this project should have successfully completed MATH 171 Introduction to Linear Algebra and MATH 126 Calculus II.