Mathematical Algebraic Structures

Algebraic structures, via Wikipedia
Algebraic structures, via Wikipedia

I

I’ve spent about 4-5 hours on my algebra homework. I still have another 27 problems to finish¹. Naturally, they get harder as you go along. Kind of annoying. I like writing easier ones first and then moving to harder ones a bit later. I like this to happen in each problem set. For some reason, I had trouble with cyclic groups and had to review the subject matter before completing two problems.

With these types of abstract math, it’s best to stop when you feel it slipping away or when you hit a problem that looks impossible to let it stew and come back to it. This has been my technique for the last few years and it works well. I have to be really careful with the solutions. I have all of the solutions of the problems that I’m doing in Hungerford’s Algebra².

Magma to group, via wikipedia
Magma to group, via Wikipedia

II

It’s getting chilly for the locals in Taiwan, which means that it’s really getting nice. When it isn’t raining, it’s not that humid and quite pleasant. Temperature is hovering around 20C, which is really nice.

III

I was preoccupied yesterday and fed Spike four times instead of two. Two times in the morning and two times at night. The last time, he gave me this pitiful look. I was wondering why he wasn’t eating, but it was his business. He used to be a picky eater when he was a puppy, but no more. I only realized later on how big his belly seemed, compared to normal. He had really eaten way too much. Naturally, what goes in must come out. In the lat 36 hours, he’s pooped 9 times. In order to even things out, I went on a long walk with him yesterday night. He enjoyed it since the dog park was empty because of the light drizzle.

For some reason, I go to the park at around 10:30PM. I realized that it’s too late for the dog park people. Chad was leaving when I arrived. I’ll have to try to head out at around 10PM.

Today, I had algebra on the brain. During the colloquium, there was a strange researcher from NTU who presented her findings on Clifford algebras. She went extremely fast, wrote really badly, and spoke as if we were all PhDs already. Only two or three people understood what she was saying³. In retrospect, after having talked to an algebra professor, who confessed that even he had trouble understanding parts, I think that colloquiums such as these are very good. They make you want to find out about things. I personally researched the main concepts to try and understand what was presented.

As for myself, since the presentation was in Mandarin, I didn’t get everything. The notes were in English and I managed to jot them all down pretty efficiently, but these types of algebras are somewhat esoteric to me. I researched them a bit after the talk, and I don’t really know how I feel about them⁵.

I knew of the underlying concepts, but her whole presentation was too fast and too complex for the target audience. Most of them abandoned trying to understand and worked on something else. This is not a bad thing for graduate students with an interest in algebra. It makes them curious about what is presented and lets them develop along their own path to a new understanding, something that wasn’t present before.

Clifford algebras are subject to the following condition:

v^2 = Q(v)\ \mbox{ for all } v\in V.

It can be rewritten this way:

uv + vu = 2\lang u, v\rang \mbox{ for all }u,v \in V,

She also had a lot to say about Azumaya algebras and central simple algebras.

IV

We always have a few Mr. Cats about in our apartment complex. They are usually community cats that are fed by the tenants. A while ago, we had a really nice Mr. Cat, who was very nice. We nearly adopted him/her, but she/he disappeared after a while. Since that Mr. Cat left, there have been new Mr. Cats to take up that space. A new feisty, hissing Mr. Cat, who looks pitiful enough, but seems to have quite a bit of fight left in him.

By the time that we started feeding him, a family on the 1st floor, with a door in the courtyard adopted him. I noticed today that they had 4 kittens as well. They were pretty cute. Having done the whole kitten thing once with Yoda, our kitty cat, I’m not anxious to do it again. I tend to say that I’m not a cat person, but a dog person, but I have both; a doggy named Spike⁴ and a cat named Yoda.

As I write this at 2:24AM, I hear two Mr. Cats duking it out. They make quite a bit of noise. I find it really funny how Mr. Cat keeps hissing at Spike. My wife told me that she had tried to pet him once, and he scratched her. I like to say that cats domesticated us, not the other way around. They tolerate us, and sometimes, still try to scratch our eyes out. That’s what Yoda does. Even though we saved her from the street when she was about 7 days old, she still has got a lot of ferocity in her. Not anymore, but she likes to grab things with her paws, and when she does, her claws come out.

* * * * *

[¹]: By my count, this means that I’ve got at least 10 more hours of homework to finish. I hope to finish at least half of the problem set by tomorrow. The prof gave them out last Thursday, but I was a bit lax on the weekend.
[²]: Springer Verlag, vol 125 in Graduate Texts in Mathematics.
[³]: This was especially true for the math education people. I saw one colleague who understood most of it, but even then, it was pretty advanced.
[⁴]: 5-year old French Bulldog from North Bay, Ontario.
[⁵]: I find them interesting, as I have an interest in algebraic structures.

Author: range

I'm mathematician/IT strategist/blogger from Canada living in Taipei.

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