Over the past two weeks I’ve learned a significant amount I will get into more detail later, but perhaps one of the most significant events was getting a meeting with my mentor.
Report on Mentor
I visited my mentor’s dwelling over the weekend where we discussed the project. We had decided on where we wanted to take the project a couple days beforehand, so we planned on covering some bases and solving some canned problems. Unfortunately, my mentor has been incredibly busy within the last two weeks, so much so that it’d be unreasonable for me to ask him to prepare a lesson for this meeting. However, I had watched the first section of the lecture series by Matthew Macauley and had a substantial number of questions that we could work on.
The first question I presented him with was a question about Cayley graphs. They had been used extensively in the Visual Group Theory lectures, but it was never made clear how you went about making one. To tackle this question, we tried making the Cayley graph of an alternate presentation for the familiar D4. I had already given this a shot a couple days before, as it had been presented as a homework problem, but I hadn’t seen the proper solution yet.
For reference, figure 1.1 was my solution to the problem; it’s what I thought the Cayley graph should look like. Figure 1.2 is the alternate Cayley graph that my mentor and I made for the same group presentation. I think that the second major lobe in 1.2 is unnecessary, but the rest of the Cayley graph should still be valid. After having struggled with the confusion of two working Cayley graphs, we came to the consensus that Cayley graphs kind of suck. Multiple Cayley graphs can exist for the same group and even the same presentation, which makes them not very useful. I also learned that my mentor had never seen a Cayley graph until the lectures we watched for this project and, generally speaking, they aren’t included in standard courses on group theory. So we left that domain fairly quickly, and went to tackle a second (probably more important) question I had.
Next to the neat graphs, you’ll see some weird looking equation. That is the group presentation, and it’s one of the most common ways of representing groups. On the left of the vertical bar are the generators of the group, and on the right are the relationships that they need to satisfy. For the group shown (D4) applying s twice should equal the identity, and so should applying t twice or st four times. Changing the relationships will change the group, so if we had anything other than those three relations up top, the group would no longer be D4. What I was wondering is how you know that you have enough relationships to fully describe the group. If I only wrote s2,t2 how would I know that I was missing a relationship?
We talked on this problem a bit, and we landed on the rough hypothesis that the relationships act like independent linear systems. If you look at the top of figure 1.3, you’ll see the Cayley graph for D3. Essentially, the relationships in this group represent cycles in the Cayley graph. You can draw a lot of different cycles, an arguably unlimited amount of them, so we think that the relationships are the minimal set of cycles you need to generate every other cycle. The connection to linear systems being that the relations need to be completely different from each other (not generated by each other) in the same way that in a linear system, you can only solve for x and y when the equations are independent. You can see my mentor trying to explain this in figure 1.4.
What Have I Learned from My Mentor?
I think I got a neat look into what it’s like doing collaborative math. It was a very fun experience staring at our “whiteboard” trying to understand one topic or another. It was also neat to see the habits of someone who’s been in this sort of world for a while. For example, when working with group relationships, he wrote down a massively long chain of elements in the group from the Cayley graph and was able to simplify it with a relative ease that was very impressive. Also leafing through his bookshelf of thick-as-a-brick textbooks for the information we were looking for was very enjoyable.
What Do I Plan on Doing Next?
As said before, my mentor has been unexpectedly busy as of late, so we’ve set up a system that should be conducive to learning this subject in spite of that. We’ve largely chosen our topic (Galois theory) and our plan is to have me watch as many of the lectures as fits my pace, with him keeping up. Then, hopefully on time, we’ll have another meeting, and by this point I will have learned a lot more than I did last time and will hopefully have many more questions. We will then work through these questions in a similar manner as this meeting and go down interesting side-paths as we see fit.
I also have a separate set of bi-weekly blog posts that I haven’t been keeping up on. However, that will be where I relay progress on my “original result” which I have made incremental headway on.
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