On Wednesday, we got started an hour earlier than normal with a conversation with Dr. Phillips. He asked us a number of questions about our experiences so far at SUMSRI. Some of them we were able to answer easily and helpfully, (“What would you change about the living situation?” General consensus:“Have us all in the same building.”) some we were able to answer easily, but not terribly informatively (“Why did you choose to come to this program?” General response: “They accepted me.”) and others we couldn't answer. (“What's it going to be like in another few weeks when you move from research seminar to research?” “Umm....”)
Generally, although we had a lot of ideas of ways the program could be improved, none of our complaints were major enough to stop us from being very happy with the program. Dr. Phillips kept trying to get us to give more major criticisms, though he acknowledged that we might not feel comfortable complaining about parts of the program, or especially people, around the others.However, he didn't give us his e-mail or any other way to contact him privately if we did have more points we wanted to make. (If I were able to change anything about the external reviewer, I would have had that be an option.)
After that was over, we went to the research seminar. And life got confusing again.
The first thing to change was the seating arrangements. We were all sitting in the same seat that we'd chosen the first day. (There are basically two kinds of students: the kind who are find moving around from day to day [usually these are the same students who come in late and need to choose whatever seat is least obtrusive] and the kinds of students who will sit somewhere the first day and then not move. If someone sits in their seat, they will keep standing and hope the other student moves. If coffee and creamers are left over and taking up the space they would otherwise sit, they sit in that spot anyway and try and barely have enough table space.) And the first day, the guys had arrived first, followed by the girls. We'd sat down like that, so the first row was male and the second was female. So we claimed that the external reviewer had told us we should stop segregating ourselves by gender and finally mixed things up.
In the morning, I sat in the front row on the right and discoered I couldn't read the far left board. I moved in the afternoon to sit exactly one row in front of my usual spot, but I did miss a definition and an example. Which did not help the material make more sense.
We needed to begin with a more formal definition of what a covering map was. “More formal” being an exagerration, since we didn't have the background necessary for a truly formal definition. So we got a working definition just good enough to give us some idea of what was going on before getting the statement of theorems that were related to the project. At least, they probably will be some day.
So I was going along, trying to relate it back to class the last few day, feeling like we were approaching a new topic, and trying to follow. It was a little confusing at times, but I was mostly understanding the generalities of what was going on. And then out of nowhere we were suddenly back to talking about CAT(0) groups and non-positively curved spaces. Because there was a theorem that says the universal cover of a non-positively curved space is CAT(0). Under some situtations, it probably would have been a cool theorem related two things that I knew and cared about. Under the current situation, I ended up feeling betrayed.
After we got back to CAT(0) and NPC groups and spaces, we moved on to even more new definitions. Then he went back and formally defined CW complexes (again?) He'd given us a definition the first or second day of class before he decided that it didn't really matter, since all we cared about was 2-complexes. But now we cared about general CW-complexes and K(G,1) complexes, for reasons that would become clear. Eventually. Hopefully.
Analysis was, as always, a relief. (There are probably relatively few students who would agree with that statement.) We moved on from the undergraduate curriculum to measure theory, and from written notes to typset moves.
Dr. Dowling began by explaining some of the motivating reasons for Lebesgue and why anyone should would want to study it. Lebesgue wanted to be able to integrate more functions. (“A problem that I'm sure you've all had.”) It was all new material, but it was explained with the same detail of proofs and preciseness of definition I'd grown to know and love. Mmm. Analysis.
After class, we had a couple hours off, and then it was time to go to the review session. Hannah explained a couple more of the diagrams for automorphisms on the free group with three generators, and we each prepared notecards with one permutation on them. By combining the notecards together, we could test the links to see if they were CAT(0) or not. We concluded that none of the combinations that included an alpha (maping anelemnt to the concatenation of that and another generator) was not going to be CAT(0).
So... yay! Progress?
