# Pressing Toward the Prize

## Archive for the ‘Ideas’ Category

### Capstone Ideas

Posted on: November 5, 2010

The time has come to declare the topic for my capstone project, and I still find myself somewhat undecided. The two areas that interest me most are voting paradoxes and working with complex numbers. I like the practical application of studying voting processes, the problems that can arise when trying to determine the will of the people, and the manipulations (inadvertent or otherwise) that can greatly affect the results. Fairness in voting is a concept that should be of interest to us all, since the outcomes dictate many aspects of our lives.

A few weeks ago, my math professor suggested considering for our capstone project topics that may have been introduced in past math classes, but not studied in depth. My interest in complex numbers is a result of this suggestion. My exposure to the complex number system has mainly consisted of performing algebraic manipulations, and I would like to know more. Analysis of the complex numbers is such a broad topic that, should I decide to go this route, I will need some direction in choosing a focus. An intriguing application that my professor mentioned is one in which complex numbers are used to solve integration problems that would be quite difficult in the real number system. Both of these are worthy topics, but our formal proposal is due next week, so I will need to hop off the fence soon!

### Critical Points of Polynomials

Posted on: October 13, 2010

On October 6, we had a guest speaker in Capstone, Dr. Julie Eaton from the University of Puget Sound, who gave a presentation on locating critical points of polynomials. First, she presented the Lucas-Gauss Theorem: “The roots of the derivative of polynomials are contained in the convex hull of the roots of that polynomial.” For polynomials with real roots, it was easy to see graphically that the roots of the derivative, which are the critical points of the continuous polynomial curve, would be contained within the same interval as the roots of the polynomial. I had a little more difficulty following what was happening when some of the roots were complex numbers, however, because none of my classes to date have covered graphing in the complex number system. I know that when the root of a polynomial is a complex number, its conjugate is also a root. This made it easier to understand that the graph containing complex roots could be connected to form a region rather than an interval. It was interesting to see that the same principle in the real number system was true in the complex numbers. The region created by the roots of the derivatives was fully contained in the roots of the polynomial. For higher order polynomials, each time a derivative was taken, the shape of the region “decreased” and was still fully contained. For example, a fifth degree polynomial with one real and four complex roots created a pentagon, the first derivative created a quadrilateral, the second a triangle, and so on, each fully contained within the other. One of the things I love about math is finding patterns in “unexpected” places, so this was fascinating to me.

The second concept Dr. Eaton covered had to do with Newton polygons, which Newton created in 1671 along with an algorithm used to approximate the roots of polynomials as functions of their coefficients. I had no difficulty in actually using the technique to approximate the roots, but I didn’t fully understand the explanation of why it works. I guess that will wait for another day! I found the presentation quite interesting, but it went a little “fast” for me. I would like to have had more time to explore with Dr. Eaton a few of the concepts that were new to me, but she was under specific time constraints, and it was a lecture, after all. For me, the best part of the presentation is that, even though I didn’t fully understand everything she shared, it gave me ideas on new avenues of study.

### Meeting the Faculty

Posted on: September 23, 2010

In Capstone Seminar today we were introduced to the faculty members of the math department at PLU. Each professor gave a brief presentation of his or her areas of expertise, as well as some ideas for interesting Capstone projects. I was impressed by the wide array of mathematical fields represented by the nine professors, as well as the variation among the professors themselves. One works almost exclusively with probability and statistics, and another prefers topology and geometry, having declared statistics “boring.” There was so much information given that it was hard to take it all in, but many wonderful ideas were presented. With so much variety, I cannot imagine any of my fellow students not finding at least one topic that sparked his or her interest.

I am not very spatial in my thinking, so topology and geometry may not be the best for me, although I find them both quite interesting. I struggled a bit with multivariate calculus for that reason: I could do the math with no problem, but I sometimes had trouble “seeing” what was going on. At first blush, there were two topics that caught my attention. One had to do with educational assessments, involving both theory and development, and the other was number theory. I checked out some websites to investigate exactly what number theory entails, and I discovered that it is a very large branch of mathematics. Something that caught my eye, however, is the study of Diophantine equations, or equations that have only integer solutions, of which Fermat’s last theorem is one. I am not sure if either one of these topics will lead to my Capstone project, but they are possibilities to explore.