Abstract Algebra Quirks

One of the things that has always driven my interest in Abstract Algebra as a field is the perceived multitude of quirks. It’s hard to speak of what I mean from a meta perspective, so I’ll just give an example.

Let k be a field. Recall that a ring R is called a k-algebra if k\subset Z(R) and if 1_k=1_R, where Z(R) is the center of R, that is, the elements x\in R for which xy=yx for all y\in R. From an algebra perspective, any ring R which is a k-algebra is, necessarily, both a ring and a vector space over k. For this reason, when speaking of R being generated by a subset of elements of R, it’s necessary to indicate with which regard the subset generates R.

One cool example of this necessity – an example which I find quirky, in some regards – comes in the form of the polynomial ring R=k[x] in one variable over k. Certainly with this construction, R is a k-algebra, and so R is both a ring and a vector space over k. Note, then, that as a ring, R is a a finite-dimensional k-algebra since x is a ring generator for R – that is, R=k[x] is the smallest ring over k containing x. On the other hand, k[x] has basis 1,x,x^2,\ldots as a vector space over k and hence is infinite dimensional as a k-vector space.

Long story short: Two different structural existences for a single object, and the two are, to some extent, polar opposites as one another.

Things like this always make me realize why algebra is such a necessary and beautiful field.

The End of Summer, or “Orientation”

My intention was to make my next post (i.e., this post) be some cool tidbits I’ve been pondering concerning the Gamma function (see here: http://en.wikipedia.org/wiki/Gamma_function), but alas, I’m simply too tired to squeeze that one in for today. I’ll try to do some of that later today – you know, when I’m not galavanting around Tallahassee, attempting to establish residency status. But that’s neither here nor there.

Instead, I’ll update about the most recent departmental things that have gone on, most notable of which was the first day of orientation.

The first day of orientation happened yesterday (as it’s 2:41 am now) from 10:45am until…. I wasn’t really sure what to expect, and as is usual in such situations, I was a little uneasy about being around a whole slew of new people, jumping through hoops, etc. etc. I can honestly say, though, that I was pleasantly surprised.

I woke up and saw that it was raining. Surprisingly, rain has been the ongoing theme since arriving here in Tallahassee, but we were fortunate to wake up early, leave early, and arrive with a few minutes to spare. At 10:45, we met Torrece – the Graduate Admissions Coordinator – to receive info packets. Not long after (about 11-ish), we went to “the big classroom” (LOV 101, which seats 120 students) for some general information, etc. During this, we got to “meet” (in actuality, we got to see from afar) some of the faculty who were “important for us to know,” and after much ado about nothing (or so it seemed), we went upstairs for lunch.

Lunch began at noon. It was a quaint setup with a light meal, and by way of how the setup was designed (or so I suspect), we were put right in the middle of forced socialization. As is often the case in such situations, the information I heard was largely forgotten afterwards, but what I did take away from the session is that most of the new admits were pretty nice folks. That’s always a plus.

After lunch, we dealt with a small scheduling mishap, after which we split up by area of study (pure, applied, bio and financial math are the options here) to register for our first semester of classes. During the pure math registration, I got to meet the advisor (Dr. van Hoeij) for the very first time; I also got the first WTF? You’re a Ph.D. student in what now, trying to take which classes for what? Say huh? moment of the semester when I asked Dr. van Hoeij about a scheduling quirk stemming from my having registered for two early qualifiers.

Those are always pretty funny.

Apparently, because I’m signed up for two qualifiers, I have to sign up for two additional “backup” classes in addition to the mandatory three everyone has to take. That means that for the first week or two of the semester, I’ll be devoting myself to five graduate math courses (for which I may or may not be qualified), in addition to teaching and lesson planning and whatever else may be required. I guess there’s no easing back into academia for me, eh?

Anyway, this is my current schedule as it stands now. Changes will be made later this semster:

Measure and Integration I, MWF, 11:15am – 12:05pm

Topology I, MWF, 12:20pm – 1:10pm

Abstract Algebra I, MWF, 1:25pm – 2:15pm

Complex Variables I, MWF, 2:30pm – 3:20pm

Groups, Rings, and Vector Spaces I, TR, 11:00pm – 12:15pm

I know you’re jealous of my Mondays/Wednesdays/Fridays, right? 😛

Anyway…after registering for classes, we were ordered to go to some building or other (I still don’t remember the name of it) to do “other stuff.” The other stuff was us getting assigned twelve different identifying pieces of information, having our pictures taken for the website, setting up email forwarding, etc; of course, all this stuff (it took about an hour, an hour fifteen total) happened after wandering around – still in the rain – trying to find some building I’d never been to before. The good news was that I finished everything (at 3:15pm) before I was even scheduled to start “other stuff” training (3:30pm), so I guess it wasn’t all bad.

Honestly, it was a little more underwhelming than I’d expected it to be. As I said previously, later today is going to involve me trying to remedy some domicile information, and then the rest of the week is eaten up with more registration stuff. The bad news about that is that my Saturday Sunday qualifier in Complex Analysis isn’t going to get nearly the last minute cramming it ought to; the good news? I’m not really sure yet. 😛

Stay tuned for some Gamma function nonsense, and maybe some other interesting tidbits as the journey towards academic greatness continues.

New To Town

Well, we’ve been in town for about a week now and we’re finally starting to get settled in. In the midst of settling down, I have the overwhelming burden that is studying for qualifying exams.

Each qualifying exam is a four hour test meant to evaluate whether a graduate student has the required knowledge to officially move into Ph.D. candidacy or not. In my case, I’m signed up to take two such qualifiers – one in Complex Analysis and one in Groups, Rings and Vector Spaces.

I can say, honestly, that I have been studying for these exams off and on throughout the summer; if I’m still honest, however, I’d have to say that my studying has been more off because of all the other random chores and responsibilities that have been on for what seems to have been nonstop. I’d managed to get by with that under the pretense of, Eh: I don’t even know when the exams are going to be given so I’m sure I have plenty of time. After all, ignorance is bliss.

As of a couple days ago, though, my ignorance (and, hence, my bliss) was erased.

I’m scheduled to take my Complex Analysis qualifier on Sunday, August 26 (from 1pm to 5pm) and the other on Saturday, September 1 (also from 1pm to 5pm). That means that I have no more than 18 days – indeed, two and a half weeks – until I’m put in a very stressful, very uncomfortable situation for which I should be prepared.

should be prepared, but alas, I’m fearful that I won’t be.

My game plan is to update this thing with my journey through mathematics: It began about a week ago, when we pulled into town, and it’ll be never-ending – exhilarating, refreshing, and frightening – for the next few years. I look forward to having folks along for the ride.