My research for my dissertation focuses on groups (an algebraic structure) that have a property that my advisor, Dr. C, and I have named "the road trip property." Incidentally, he wanted to call it that instead of "the trip property" because trip made him think of acid. Anyway, we think of a space where we have shortest paths and defined distance - a sheet of paper for instance. We draw some hideous, crazy loop on it and measure its length and say that's our longest trip. Now, we promise not to drive more than a certain distance each day and put a bunch of destinations on our trip to ensure that. Then, it's possible for us to carefully shorten the trip in stages until it is a trip where we stay home.
This branch of geometric group theory is refreshing in the fact that we look at things from a distance, sometimes very far away. The idea is that if you look at certain spaces from far enough away, they look the same. It makes me smile to think that there is a loose equivalence between traveling across the country and back and never leaving home.
The rest of the research involves how this property relates to area bounded by the trips, which are labeled by algebraic words. Yes, there are words and alphabets in mathematical spaces. The nice thing about the property is that your space can be really terrible, like a wadded up sheet of paper, but you can map everything into a nice space and visualize it all through the use of van Kampen diagrams.
The upshot: I get to make dreamcatchers for my dissertation!
|Home is the bull's eye. Circles represent trips with dots for destinations. The blue lines are how the "trippers" are synchronized with each other.|
|Home is at the bottom. The big trip is the black ring. Other trips leave home on a blue curve, go out to a black dot, cross a red line through like-colored dots, hit another black dot, and take a blue curve home.|