Tuesday began with the algebra short course. We'd pretty much finished up with groups last Thursday, so we started in on rings. So in about a week, we'd skimmed through the material covered in my Abstract 1 class and were moving on to Abstract 2. It (There's no real comparison possible.)
Professor Aktar defined ring weirdly, in my opinion. Particularly, he required rings to have unity and be commutative. Rings without unity or non-commutative rings are both existent and important, but, under his definition, they must be specified as such. I much preferred the definition both my professor and textbook used last semester.
Right before stopping for a break, Dr. Aktar demonstrated how one could draw the zero-divisor graph of a ring. (Step one: make vertices out of all non-zero zero-divisors. Step two: Draw an edge between two vertices a and b if and only if a*b=0. There is no step three because drawing graphs is really easy.) During the break, I talked a little bit about the graphs of units and zero-divisors that I'd done while my classmates rolled their eyes.
Which means my classmates for this summer have realized the depth of my obsession with graphs much earlier than most people. I might have given it away with frequent comments about how I hoped we got to a project involving Cayley Graphs. (Or any kind of graphs.) Or maybe it was me mentioning that I'd managed to write a final paper on graph theory in three different classes. (Coincidentally, I've gotten to choose the subject for a final paper in three different math classes in college.) Or maybe it's the tattoo I was talking about getting on my forehead. OK, so it's not exactly a subtle interest...
Duing the research seminar, Dr. Farley acknowledged that he had not given us a terribly useful way of working with hyperbolic groups, and that he needed to give us more definitions or theorems. He didn't give us either, but he did acknowledge that he needed to give them to us. He did give us another graph that we could draw based off topological concepts. This time, the vertices are right cosets, and there is an arc between vertices as determined by the generators in the generating set. Once the proper 2-cells are attached, this becomes a covering space.
What's a covering space? That's a great questions... for another day.
Once the seminar ended, we moved into the break room for refeshments and the chance to meet our next colloquium speaker, Professor J.D. Phillips. He teaches at Northern Michigan University, and he came to Miami University to evaluate SUMSRI. While here, he might as well give a presentation on algebraic loops. (And hopefully not negatively evaluate the quality of external speakers brought in for colloquia...)
Loops are a type of algebraic structure. Like groups, they have a binary operation and a 2-sided identity. Unlike groups, they are not associative, which makes a lot of the beautiful proeperties that groups have go away never to be seen again. Other times, loop theorists can use their knowledge of groups to extend definitions and theorems to loops. And, most interesting, sometimes theorems that are really easy to prove for groups can be really hard when it comes to loops. Dr. Phillips talked about some specific examples of these kinds of problems, including some that are still unsolved.
One of the major techniques Dr. Phillips uses is proof by computer. It's faster and more accurate than human calculation would be. With proofs that can be over 10,000 pages long, it loses a lot of its elegance. However, a computer is not going to accidentally drop a set of parentheses midway through the proof. A human might, and, when you're working in a non-associative loop, that can really be a problem
It was the second presentation I'd heard on less-commonly-taught algebraic structures. (The other one had been a talk on bands by the professor who taught about half my Abstract 2 class. When I asked if Dr. Phillips recognzied the name, he said he did. Not any specific interactions, but just as two algebraists in the midwest who sometimes end up at the same conferences and such.) It's interesting to think of what a major difference the change of one or two requirements can make to the structure.
After the talk, several people went out to dinner with the speaker (it's pre-decided by Dr. Aktar. My turn will come next time) and the rest of us hung around, ate dinner, and then wandered around campus. We stumbled into a building that appeared to be a hall of fame/basketball court. During the summer, it is either preparing for a marathon retro-secretary competition or storing old technology as they renovated some part of campus that the monitors and keyboards and such were normally kept. (Probably the latter, though the former would be so much more exciting.) We ended up returning to the dorm just as the people who had gone to dinner were coming .So we all headed over to the Elliott for a very brief review session and homework set-up. Then Hannah left, and the girls returned to Stoddard to work through the homework. (None of the guys wanted to work on it then.)
At last, we were done, and I set out, intending to go for a walk. Except that it I was only a couple steps into it when I realized it was the perfect time of the day for catching fireflies. They were out and glowing, but it was still light enough you could see them flying if you knew where to look. So I went back in and got Maram. Griselda came with us, and we started running around outside trying to catch fireflies. Maram got over her fear that they'd bite and got rather good at catching them. Griselda mainly took pictures or videos but, a couple of times she did catch one. We stayed until the mosquitoes had an easier time finding us than we had finding fireflies, and then we went inside.
The day (like most others) consisted of sitting doing math and listening to lectures on subjects that would normally be taught in a graduate course And yet, in the evening, Maram and I were chasing around fireflies like we were 8. (The only thing we're missing is a Mason jar.) When I chase fireflies, it's with a fair amount of nostalgia, so I find Maram's enthusiasm energizing.
"This is the best part of the summer so far," she said, watching one light up as it left her hand. Here's hoping that it has so many other pleasant memories to compete with.
