Just a quick update on my PhD journey. I passed my compre last November. Barely. I presented my proposal last week. I need to revise it. Add a lot more of preliminary concepts. I don't even know where to start.
I also have a paper to write. And I hit a snag. The Lie algebra that I'm studying has a lot more elements than I thought. The results are still valid but describing them is a lot more difficult now. It's a nice problem to have though. It has become more interesting.
My adviser has asked me if I celebrated after my proposal. I actually didn't. I rested for a couple of days so I guess that counts as "celebration". And now I'm on my period so I can't work as much because of cramps. I haven't gotten out of the house either. I plan to after my period is over.
I don't know what to make of this discontent. I'm actually ahead of schedule. But the expectations are so high it feels like I'm lagging behind. Feeling grateful is not automatic.
I realize I'm learning a lot more now. Things that used to be difficult are easier now. But some things are still hard. I'll figure it out eventually.
I'm writing this post to remind myself that I'm lucky. Yes, I worked hard for them but there's still an element of luck in it. I actually manifested my research topic somewhat. It's crazy.
Words are indeed powerful.
The D-classes have a certain type of hierarchy. And to illustrate this, I'll give relevant properties of the rank of a transformation. The first property is that the rank of a product of two transformations is less than or equal to the minimum of the individual ranks. Let's flesh this out some more. Let's start with what I call the nobility in this hierarchy--the permutations. The noble class would like their wealth and power to stay within their class or family. How do they do that? The idea is to marry someone with the same social class as you are. It's a very common trope in movies and TV. So in the same way, for an "offspring" of a permutation to have the full rank still, both of its "parents" should be permutations. rank(ab)=n if and only if rank(a)=rank(b)=n As part of group theory, we study permutations. This is because a group is embedded in a permutation group by Cayley's theorem. In the finite case, every group of order n is isomorphi...
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