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Problem One

USA | Friday, 12 June 2015 | Views [210]

Like Tuesday, today began with with the algebra short course. We got a new packet, though we spent the entire day finishing u pmaterial from the first packet. Which meant more groups, normal subgroups, quotient groups, and homomorphisms. So more Abstract 1 material (and moe time spent thinking about the final project for Abstract 2.)

The research seminar picked up about where the last one had left out.Unfortunately, since we don't have that many examples of groups we as a class have gotten comfortable working with, to come up with another, finite example, Dr. Farley went to the S_4.

S_3 and S_4 are both symmeric groups, or the groups of all possible permutations on 3 and 4 elements, respectively. The number of elements are six, which is not bad at all to graph, and 24, which is a bit worse. If we needed another example and went up to S_5, there would be 120 elements, which is not something I'd want to graph by hand.Fortunately, S_4 was seen as a sufficient example, and we moved on to infinite groups. (And I thought drawing 120 vertices and the edges between them was bad...)

The two infinite groups that we looked at (Z cross Z and the free group on two generators) are both pretty well-behaving groups, so it didn't takemany vertices or arcs to figure out what was going on. Which is good, since if we needed to draw the entire Cayley graph, we wouldn't be able to do anything else this summer (or for the foreseeable future.)

Then we left behind the beauty that was Cayley graphs to definte some more terms in topology and geometric group theory. Such as geodesics. And δ-hyperbolic. And geodesic metric spaces, and CAT(0) space, and non-positively curved geodesic metric spaces and mapping tori. And somewhere in the definitions and examples and theorems that we were supposed to believe were true was hidden the final key we needed to understand the first project.

Understand” might be a slight exagerration. There was a paper that Dr. Farley referred to several times that he hoped we'd be able to extend. Not generally, because it's not true generally, but for certain automorphisms of the free group it might work. He hopes so, and hopes that problem is accessible to undergraduates. Maybe we'll find out. It depends a bit on what the other problems are (there are two or three) and how much of a choice we have.

So, 10 hours of research seminar later, we know what one potential subject of resarch is. It's a start.

Today, we had a colloquia with an actuary. Before that started, we hung out in the mathematics break room for food, drink, and introductions. It was up to the speaker (Robert Dennison from Western and Southern Finanical Group) to come up with a question or two to pose to us. His were “what would you study if not math,” and “where would you want to be if you weren't here?” We went around the room introducing ourselvess and answering those questions.

The “what would you study if not math” struck me as an interesting question because, for some people, it seemed to be relatively easy to answer. I'd noticed previously that a number of natural science majors I've met have a back-up plan. It's to some degree joking, some degree genuine, but it's something that they can think longingly about while they're spending five hours on a single problem. And I'm not sure if that's as common in other fields (or even how common it is among math majors. I know I have a back up plan, and I've met another math major who does, so by induction, everyone who majors in math has a non-science subject to fall back on when classes get too hard.)

After the introduction (sample answers: foreign languages, music, English, linguistics, history; Australia, Canada, Austria, Saudi Arabia, University of Michigan, Hawaii) we went upstairs for the talk. It was probably the most interesting talk I've heard from an actuary, though that didn't take much.

It aslo confirmed once again that I have no interest in being an actuary. I still have a hard time imagining the kind of people who find actuarial work fun. I'm terrible with probability, and large amounts of money freak me out, so it's not something I'd ever want to do, so it's good for me that there are people who choose and enjoy a career as an actuary.

Robert Dennison talked a bit about work as an actuary in general, and his work in particular, and I came up with an idea for a thriller. Working title: Longevity. It's about an actuary serial killer who goes around murdering people in an attempt to lower the expected age of death. I predict it to be a best-seller which is subsequently made into a television show. You heard it here first.

We had a couple hours free (we being everyone except Dana, Griselda, and Sam, who were going to dinner with the professors, actuary, and Hannah) and then convened at 7 for the review session. The people who had gone out for dinner didn't return until 8, so the first hour of the review session was Angelo, Delaney, and I sharing stories from college and life. One of the nice parts of meeting new people is that you have all new stories to share.

The review session made the examples that Dr. Farley had given at the end of the class make a lot more sense. I still have no idea what the paper is saying, or how we're expected to be able to build on the results. But we have a project, or at least the rough outlines of one. In another two weeks, we might even know what exactly we're supposed to be doing. I'm not counting on it, though.

Tags: abstract algebra, graphs, math, research problem, topology

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