Pressing Toward the Prize

Archive for the ‘Speakers’ Category

The Final Round

Posted on: December 9, 2010

We had the final round of Capstone Proposal presentations today, and once again, a wide array of interests were represented. I am always humbled by others’ presentations, be it student or faculty, because they are a graphic reminder to me of how little I actually know! I listen and watch and try to understand… but mostly, I have no idea what they are talking about!! Some of the topics that totally lost me were Panel Data Modeling, the Riemann Hypothesis, and Applications of Fuzzy Set Theory and Finite State Automata (say that three times fast!!). I did a little better job of following the presentations about rankings and the Bowl Championship Series, the push for Proofs to be taught in high school, Digital Image Interpolation, and Error Detection and Correction in Data Transmissions. I was pretty excited that I understood some of the presentation on Markov Chains, since I can still recall a bit about stochastic processes from Linear Algebra. One of the presentations was about the Mozart Dice Game, which was apparently quite popular in Europe in the 18th Century. There is a 12 x 16 table, composition rules, and one uses the roll of the dice to randomly create a minuet. I don’t understand exactly how it works, but we were able to listen to a minuet that the student had created, and it was beautiful. She is working on a project that will connect art and math, and it should be quite interesting. I was also one of the presenters today, and all I can say about that is… I am glad it’s over!!

Topics Galore!

Posted on: December 4, 2010

In Capstone Seminar this week, we were treated to presentations in which nine different students shared the plans for his or her Capstone project. It was quite interesting to see the wide range of topics represented. One student will be programming virtual manipulatives (hands-on, interactive math learning tools) for mobile devices, one will be studying stocks, another dimension, and yet another will be evaluating Standards Based Grading in mathematics for secondary students. A couple of the students will be pursuing statistical topics, one involving Major League Baseball players and another looking at case control studies. In addition, modular origami and the Monte Carlo method will be explored. All of these are worthy topics, but my personal favorite is the study of traffic jams. Who has not been driving along and come upon a slow-down or back-up with no apparent cause? How frustrating! One student will be studying these “jamitons” in the hopes of using mathematics to gain insight into, and possible solutions for, such phenomenon. For all our sakes, I wish him well in his endeavor!

Random Fractals

Posted on: November 11, 2010

In Capstone Seminar today we had a guest speaker from Gonzaga University, Dr. Logan Axon, who gave a very interesting lecture on random fractals. According to Wikipedia, “A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole, a property called self-similarity.” Although mathematicians studied such continuous, non-differentiable functions as fractals beginning in the 19th century, the name fractal was given by Benoit Mandelbrot in 1975, derived from the Latin word fractus (broken or fractured), to describe sets that were too badly behaved for traditional geometry. A fractal is based on a recursive process, which may be determined or random.

The first example Dr. Axon presented was of the Cantor middle thirds set. You start with the unit interval [0,1]. You remove the middle third of the line segment, then the middle third of the two resulting line segments, then the middle third of the four left, and so on, continuing an infinite number of times. So what is the total length of all intervals removed in construction of the Cantor set C? Dr. Axon showed us that the sum of the lengths is 1, or the length of the interval we started with. The assumption would be that C is the empty set, since we removed the equivalent of the original length. But we could see that 0 is in every set, so is 1/3, and 1/9, and so on. In reality, there are as many points in the Cantor set as in the original set (interval), even though we removed all of them! I would have to say that this is pretty counter-intuitive!!

Next Dr. Axon listed the properties of C that all fractals should have:

• Self-similarity, scaled
• Detail to all scales
• Simple to define, but hard to describe geometrically
• Its “local geometry” is strange; every open interval containing one point of C must contain the other points of C as well as points not in C.

Dr. Axon described the von Koch curve, in which you begin with the unit line, replace the middle third with the top of an equilateral triangle, then replace the middle third of each of the resulting four line segments with the top of an equilateral triangle, and so on. The Dragon curve is a space-filling curve, and the Mandelbrot set is formed by iterations of a function on $\mathbb{C}$, the complex number system.

We were introduced to the dimension of a fractal, which is the main tool in fractal geometry measuring the “roughness,” or “texture” of the fractal, and how to find it using limits. Fractals also describe objects in nature, such as the Romanesco broccoli with its repeating spiral design. In addition, we saw that the fractal dimension of the Coast of Britain is 1.25, although I still don’t understand exactly how to interpret that! Random fractals, like the Random von Koch curve, can be used to create images that look less concocted, more natural, for things like computer graphics.

Before this lecture, I had no idea what a fractal was. I felt like I was understanding most of what was being presented, however, at least until a few of the professors started asking questions that I didn’t understand. That’s when I realized I may not be grasping it as well as I had thought! Needless to say, Dr. Axon’s responses to them were beyond my scope of understanding, as well. All in all, I found the entire topic to be quite fascinating, though, especially when we discovered that fractals often behave in very unexpected ways. Who knew you could remove all of a set, and still have all of it left?!?!

Knot Theory

Posted on: October 23, 2010

A few days ago, Dr. Daniel Heath, a math professor at Pacific Lutheran University, presented a lecture during Capstone Seminar offering an introduction to knot theory. He will be teaching a special topics class in the spring, and was hoping to pique our interest in the subject. Knot theory is a branch of topology in which a circle is embedded in 3-dimensional Euclidean space, $\mathbb{R}^3$. According to the Oracle ThinkQuest site, “a knot is a closed, one dimensional, and non-intersecting curve in three-dimensional space. From a more mathematical and set-theoretic standpoint, a knot is a homeomorphism that maps a circle into three-dimensional space and cannot be reduced to the unknot by an ambient isotopy.” So what you have is something like a knot in a string, but with the ends joined together so that it cannot come undone. There are various ways to describe a knot, so an important problem in knot theory is determining when two different representations are describing the same knot.  Wikipedia explains: “Two mathematical knots are equivalent if one can be transformed into the other via a deformation of $\mathbb{R}^3$ upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.”

Professor Heath showed us several knot diagrams, explained three legal “Reidemeister moves” that can be used to transform a knot, and then described how to determine if two knots are the same using “colorability” of the strands of a knot diagram as a classification criterion. Then he presented an important theorem: “If we have an unknot in a diagram with $n$ crossings, then it can be unknotted in less than $2^{n \times 10^{11}}$ Reideimeister moves.” This was apparently a huge breakthrough in knot theory, because it showed that there is an upper bound on the number of moves necessary to unknot a knot, but it is such a large number, that it is relatively useless from a practical standpoint.

I found the presentation interesting and informative, and was actually able to follow most of it. I found myself becoming a little lost when a theorem was presented about “n-colorability.” It is defined using modular arithmetic, which I understand, but the application to the knot diagrams went a bit “fast” for me. Even though there are important practical applications to knot theory, such as the properties of knotted vs. unknotted chemicals in chemistry, and string theory in physics, I find the whole concept of knot theory somewhat unusual. I know that we were exposed to only a small portion of what knot theory entails, but at this point in time, I don’t think it is a topic I am interested in pursuing. I meant to ask Professor Heath after the lecture why he became interested in knot theory initially, and also why he has continued to study the topic, because I am always interested in what motivates people to do what they do. I guess that will have to wait for another day.

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.