In the algebra short course, we continued our discussion about modules before moving on to category theory. Along the way, Professsor Aktar took a detour to talk a bit about motivation for some of these ideas and definitions. This brought us to homology, confirming something that I'd begun to suspect on Monday when I'd checked a few books out of the library- I would not be cursing Dr. Jensen in grad school when I arrived there knew nothing about modules. I would be cursing her this summer because Galois Theory was helping me not at all, and strong knowledge of modules would have.
We got back on topic (unfortunately. Homology came closest to making sense when Dr. Aktar was explaining it) and spent about half the class proving certain parts of the snake lemma. To be honest, I wasn't paying that much attention during the proof. It was going to be really long day of math, and if we weren't even going to be able to prove the entire thing, (which we weren't, because it was too long and tedious) I didn't see that much value in taking copious notes about ideas that wouldn't maek sense until I reached graduate school and started diagram chasing for myself.
We went from in-depth technical explanations of aspects of modules back to the general description of categories. Which was in some ways a relief, and in some ways just irritating. Dr. Aktar told us that some people would describe it as “abstract nonsense,” then tried to defend why, although categoy theory was very abstract, it was not nonsense. This isn't a question that usually bothers me in mathematics, but I found myself wondering what value it had. But it's something new and challenging that is not going to be helping me after class on Thursday, so I had a really hard time caring.
Dr. Farley began the research seminar by asking what we'd worked on the night before, and if that had made things any clearer. We'd gotten stuck in trying to compute the homology groups of Thompson Group F, so in class we backed up anf worked through some examples leading up to it. Homology was making a bit more sense, but it still wasn't as nice as putting a discrete gradiant vector field on Thompson Groups. That made bizarre amounts of sense.
While I'm kind of on the topic, I might as well mention what the second research problem is: To compute the homology of Thompson Group T. The bad news (if you're us. Good news for the growing entity that is mathematics knowledge) is that this has already been computed. The good news (again, for us. Bad news for the average mathematician) is that the proof is so complicated, there are very few people who can understand it. Dr. Farley's hope is that we can stick a DGVF on Thompson Group T and calculate it from there. We shall see.
Realizing that time was running out, and we only had two projects, Dr. Farley let the example trail of so we could move on to a new topic- polygonal linkages. This involved another paper that Dr. Farley had read and thought looked interesting enough to explore with us. Along the way, we picked up new vocabulary, examples, and questions for homework.
Then it was time to go to the colloquium. This time, the speaker was David Friedenberg, a statistician and alumnus of Miami University, SUMSRI, and Carnegie Mellon. He is currently employed at Battelle, a think tank based out of Columbus, and has come to SUMSRI to talk for pretty much as long as anyone can remember. (Which makes him sound a lot older than he is. He graduated from Miami Univeristy in 2004, he's just been here every year since Hannah and Dr. Aktar were involved in the program.)
His question was what our dream project could be, if we could work on anything. Which is a difficult question for a bunch of undergraduates to answer. (It's probably a difficult question for everyone, but the other people in the room seemed to have an easier time and more specific, realistic, answers.) Our answers ranged from mathematical (“What we're doing right now, with algebraic topology”) to much less so (“I want to built a transporter.”) and from vague (“something combining math and foreign langauges and travel” to the very specific (“Solving Goldbach's Conjecture or the Riemann Hypothesis or something like that.)
I don't have much interest in statistics (that's a bit of an understatement. The closest I've gotten to larning anything related to statistics was a mandatory “Methods of Scientific Investigtation” class sophomore year of high school. I don't remember anything about that class, not even how to fold a paper airplane) bu the presentation was pretty intresting. There are a lot of different applications of statistics to everyday and not-so-everyday life, if that's the sort of thing you're interested in. Even if you're not, some of the projects are just plain amazing. Like the use of statistics in the software that attaches to the brain of a partially paralyzed human and allows him to pick up objects for the first time in years.
One of the things that I noticed was how different the style of each of the talks so far has been. The first talk, by the actuary, had been pretty strictly informational. (“Did you guys know that 'actuary' is a real job title? Here's the sorts of things we do, generally. Here's a specific example of the kind of calcultations we do. Don't you all want to become actuary's now?”) Every talk I've heard by an actuary has taken that form. By contrast, the talk by the program evaluator was very academic. It's the kind of talk I'd expect to hear at a conference, only explained from more basicconcepts than most presentations. (It didn't begin with history from the Book of Kells, so it doesn't win the award for “least technical introduction to a subject” from presentations I've seen.) And then today, the guy from Battelle gave his presentation like he was trying to recruit us. (He probably was. He'd been a SUMSRI student, and heard a presentation by an employee at Batelle. 8 years later... and they're currently in the process of trying to hire a newly-finished PhD student who was also a SUMSRI aluma.) Unlike the actuary, who seemed to just want people joining that field, Dr. Friedenberg wanted people working Battelle specifically. Other think tanks might do interesting things, but Battelle is really where it's at. Who knows? Maybe someone in the room will remember that in a number of years when they finish their undergrad, or Masters or PhD? (They hire at all levels. And have internships, if that's something we're interested in...)
After the colloquium, it was finally my turn to go to dinner! Dr. Dowling (real analysis professor) went as well, so you know our conversation was coherent and structured.
We ate at Fiesta Charro (a Mexican restaurant down High Street. I say “down High Street” like there are restaurants off-campus that are located anywhere else. Between conversations about research (Dr. Farley started to give us some idea of what the research schedule would look like for the next few weeks, but then we realized that we'd completely lost Dr. Dowling and our guest, so he had to drop the conversation while we were still confused. Just like in class...) and conversations about statistics, I asked a couple of general questions about math to everyone in the room., or to the people who had their PhD. This included my favoirte question to ask math people (“what was your favorite class that you took as an undergrad?”) and also the more practical “what's one thing you wished you'd known before starting your PhD?”
That question was initially just directed at Dr. Friedenberg, but when Dr. Dowling made a “good question, I have no idea how I'd answer that,” face, I decided to extend it ot the people who had gotten there PhDs less recently. Dr. Dowling took the cop-out answer of “what he said,” but between the three PhDs and one grad student, we had an intersting discussion about some of the aspects of PhD life that make things much easier and more feasible. (Like choosing a good advisor.) However, the general consensus of what people wished they'd known before they started was that they would in fact finish. And unfortunately, that's not a guarantee anyone can have.
Once dinner was over, it was time to go back for the review session, which lasted until 9. Making almost 12 solid hours of doing math and having mathematical conversations. I wish I knew for sure that it was all leading somewhere...
