During dinner on Tuesday, Dr. Farley made a comment about how he wanted to resurrect the original problem. (You know, theo one about generalizing procedures from two generators to more which mostly consisteted of cutting up and rearranging squares.) No one except Sam had shown much enthusiasm for that project, so it ahd passed away quietly and unnoticed.
Until Wednesday when Dr. Farley began class by bringing it up again. He also brought up the project of putting a DGVF on Thompson Group T, and then went on to talk more about polygonal linkages. This was all in the first 90 minute session we had, and we emerged with three of four different questions to think about later that night.
I'd spent a little bit of time thinking about the DGVF for Thompsong Group T, and decided to put in a bit more work into gathering my thoughts. After all, I understood how it worked for Thompson Group F, so how much worse could it be to generalize that to T? (Imagine someone gave you a partially-drawn map and told you a process to draw in and label a new country. This country is chosen by beginning in the top left corner, ande then it goes left to right and top to bottom (in that order.) Now suppose someone gave you a globe, spun it, and told you they wanted you to come up with a process that would always insert a new country. [Country names are not a stable point of reference.] That's how much worse it could be.)
I had the beginnings of an idea that had occurred to me the night before, so during class I toyed around with it some more. And I discovered that I had a problem, in the form of the function not being injective. So I needed to play around with it much more.
No one else went to lunch in Armstrong, so I ate very quickly and headed to the library, mind still working at the problem. There was pretty much no one there, so I was free to pace around by the math books and mutter to myself. By the end, I thought I had discrete graidaint vector field, though I still needed to make sure it was had no closed nonstationary paths.
During the afternoon session, Dr. Farley decided we still didn't have enough projects and gave us a new set of definitions of explanations on Graph Braid Groups. I paid just enough attention to figure out thatt the title was a misnomer and it had very little to do with graphs. (He began by drawing a multigraph on the board as an example and, when I asked “so it doesn't need to be simple?” responded “our definition of graph is a one-dimensional CW complex.”) I wrote down the definitions and processess, realized that this didn't make half as much sense as the DGVF on Thompson Group T did, so I went back to thinking about it.
Unfortnately, I found out that the formula I'd come up with did not satisfy all of the necessary properties. I could come up with an infinite set of elements that would have non-stationary closed paths. Which was a problem.
Not nearly as much of a problem as not having any of the right terminology was. When I made comments to that effect, Dr. Farley and Hannah both responded that it wasn't important at this point, what was important was having the right ideas. It is important, though. It's important to me, because without the right language to express my ideas, I'm not sure I have the right idea. So I was trying to manage as best I could with as much rigor as I could manage.
During the evening study session, I had an opporuntiy to see if the ideas that I had made sense to other people or only to myself. So of course, the first thing I did was drop anything approaching a technical explanation to appeal to intuition.
The intuition was much easier to explain than the steps I'd spelled out that were lookng more like code than math. I was able to give a two-minute explanation of where I was and what knd of problems I faced, though it took away most of the details. Then we moved on, and I continued to work away at the problem.
After the study session was over, Maram asked for an explanation of what I was doing so that she could try and understand it better. So I went back to the board and wrote out more detail about everything I had so far for her and everyone else in the room. I was able to explain more detail about my algorithm (as a computer science double major, Maram's mind slipped into programming mode when I started describing the steps. I couldn't think of a better way to write it up then pseudocode.) They seemed to understand what was going on, which was nice, since at least a quarter of what I said aloud was muttering to myself about what I should try and consider.
It would be really nice to know the proper way to explain this. But until I have all that terminology, I guess I'll settle for just having the ideas. (Having ideas is very nice.)
