July 24, 2008
What a way to end my summer of research! While research has not exactly lived up to my expectations and it wasn’t as exciting as I would have hoped, this past week was actually fun for me and was definitely my favorite week. The reason for that, though, is that we weren’t actually doing much math. We were compiling our findings and our research into a presentation for a conference in Michigan that will occur early next week. I was finally back into my element. I was organizing and using multimedia and doing things that I really enjoy doing. While the rest of the group had written the “script” for the presentation, this was my task. I had a strict timetable and knew what had to be done and when. That was the structure that I had been missing all summer. So, while it may not sound like fun, I actually enjoyed the challenge of putting together our work. Even better, it solidified the knowledge in my mind and I actually have a very clear understanding of what we’ve accomplished the past ten weeks.
Us math researchers weren’t the only ones wrapping up our work. As a culmination of our summer research, all of the student researchers got together for a lunch and a presentation from each department (computer science, biology, chemistry, neuroscience, and math) on what they’ve been doing this summer. Somehow (I have yet to figure out how), I was selected to represent all of the math researchers. Needless to say, I was terrified because I would present my research (on which I felt my understanding was shaky at the time) in front of sixty other students (most of whom had far more experience and knowledge than I did). However, I put together what I felt was a good slideshow and presented it to the best of my abilities. I thought it went great. Plus, I had lots of people come up to me and say that they thought I did excellent and that I had the most easily understandable presentation by far. So, that made my day and gave me a lot of confidence. Now, I don’t need to fake confidence as much in my presentations. However, the one comment that was repeated several times in jest was that I needed to work on my accent. Apparently, my Northern accent comes shining through when I talk about the square “rut” of a function instead of the square “root” of a function.
To attempt to sum up my research is difficult, but I’ll try. We studied the historical figures of Gottfried Wilhelm von Liebniz and the Bernoulli brothers and what they stated regarding the optimization of travel between points. Then, we studied papers by Gary Lawlor and my adviser Michael Kerckhove about the shapes of time-optimization curves as well as the proofs by which they are proven to be time-minimizing. The foundation of these proofs rested on the “slicing method” in which we divided our paths into many “slices” using the extension of the normal vectors along the curve.
Our own extension of the research began as we attempted to not just prove a curve was time-minimizing, but to actually create a curve that was guaranteed to offer the least time of travel between points on the curve. All of these problems used a variety of speed functions. Of course it’s easy to know the fastest route with a constant speed (a straight line), but it gets more complicated when you have the speed function as “x^3-3xy^2″. Using osculating circles [Imagine driving along the road (curve) and having the steering wheel lock in place while turning a bend. The resulting circle you drive is the osculating circle] we approximated the time-minimizing path. Of course we ran into complications. For example, we had to begin using slicing circles instead of slicing lines. We also had to eliminate part of our proposed minimial-time paths because of slight accuracy errors. “Good regions” created by eigenvalues and eigenvectors, however, gave us the guaranteed-to-be-accurate regions. It is way more complex than this, but I’ve tried to “simplify” it for you.
I know this is all very complicated to you, so I will stop and say that it at least makes sense in my mind. I’m amazed right now how much I learned. I dealt with math that was way beyond the classes I had taken. Still, research isn’t something I expect to continue with. It is only for a specific type of person. I enjoy learning, but I need to actually apply it a lot more than simply investigating one area for a long period of time. I need variety. I think a lot of the other researchers also feel that way. While the summer revealed that research isn’t my main interest in life, it was beneficial in many other ways. I experienced a regular Monday to Friday 9-5 job. I also found out my comfort zone was in a workplace with a consistent routine and timetable as well as the ability for me to organize things. So, even though I thought I had ruled math out from my future earlier this summer, I may take a few more classes. There are some business routes that exist which would require several more math classes. However, if I were to take them, I have a very nice jump start due to this past summer. All told, I’m still very grateful for this experience.
Looking ahead, we have a conference in Michigan. It brings together the four colleges that participated in the LURE program. There are 30-minute presentations that eight groups will give (including ours) as well as poster presentations by the other groups. While it may be a long and tedious test of mental stamina to sit through two days of math talks, I’m sure I’ll make it out alive. However, actually getting home is my final test. One might think that there are plenty of flights from Lansing, Michigan to Minneapolis, Minnesota, but that would not be correct. In order to get home, I fly from Richmond to Lansing (for the conference) to Cincinnati to Atlanta to Minneapolis. Just for fun, I discovered that if someone left the Lansing airport at the same time as me and drove to the Minneapolis airport, we would arrive at the exact same time. However, I’m not going to complain since the LURE program reimbursed me for the equivalent of a roundtrip from Richmond. Since I didn’t want to waste time flying back to Richmond, I’ll just fly straight home. All told, I’m flying home for $2.50. That’s pretty sweet. I am so excited to get back.
I know this got really long and I don’t have any nice pictures to entertain you, so I’m sorry. I do feel as though I did not do a great job this summer adequately conveying what I’ve been up to with regards to research. It’s difficult to explain and I know that most of my posts didn’t live up to my standards. So for that, I apologize. My schedule this fall is teeming, so I’ll make it up to you then. Thanks for reading about my summer research. The next time I’ll check back in will be when I’m back home in Wisconsin!!!
Quote of the Week: “Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost.” -W.S. Anglin
Thought to Ponder: How is it possible for a Lansing to Cincinatti to Atlanta to Minneapolis flightplan to be the cheapest route? It’s 1,542 miles while it’s only 454 miles for a direct flight.
Shout Outs: The University of Richmond Class of 2012
YouTube Video of the Week:
This one has been around a while and has 37 million views
