So, to summarize the direction of my most recent mathematical endeavors: I woke up and decided that part of my aspiration was to become a geometric topologist, and I did that despite the fact that topology is (far and away) my worst subject.

That sounds precisely as terrible as it probably is.

# And, it’s Sunday….

Tomorrow is Day 1 of Week 7.

You thought I’d stopped counting, didn’t you?

Today, I posted a couple more solutions from Hatcher. I also got the okay from Dr. Petersen on my poster for Math Fun Day. I’ll probably post a .pdf image of the poster as the big day gets closer.

For now, I’m going to come study some Riemannian Geometry: I have to (very soon) pick a topic for a presentation in that class, and so it’s getting more and more necessary that make sure I know what’s going on now. Maybe I’ll surprise myself and know a lot.

I don’t expect to surprise myself. 😛

# Week 3, Day 1 or Properties of Lie Brackets

Today is the first day of the third week of the semester. I know that counting down like this is going to make it seem longer than it already seems, but it seems so long that I can’t seem to help remaining conscious of the precise time frame I’m dealing with.

Such is life, I suppose.

I’ve noticed an amusing trend in my page views involving solutions from Dr. Hatcher’s book, namely that I’ve been receiving an abnormally-high level of page views lately, almost all of which seem to center on those solutions. I guess that means that the semester has started elsewhere too and that people find topology as difficult and frustrating as I do.

For those of you who fit this bill and who are reading this right now: My plan is to start doing more problems ASAP, so that page might get its first update in quite a while.

Today, though, I want (read: need) to talk about differential geometry. In particular, we spent some time in class last week discussing the Lie bracket and its properties, and because we have a derivation of one particular property, I wanted to take the time to put that here for my own benefit.

# One week in

So, I’ve officially survived the first week of my (second) second year as a Ph.D. student. It’s hard to imagine sometimes that I’ve been a grad student for over three years.

I’m such an old man.

It’s been even less easy than I’d anticipated, too, which is nicht spaß; sadly, too, this is the first week where teaching responsibilities are actually a thing (we get week 1 off from such things), so now it’s difficult and time-consuming. Jajajajaja.

Rather than making this entry all about `Yours Truly`, I figured I’d pit stop in and use some of my (*gasp*) unoccupied time write up a little spiel I saw for the first time in my Riemannian geometry class last week.

In topology, there’s an idea called invariance of dimension which can be stated in many different contexts, situations, etc. This can be modified in the case of manifolds with differential structures, and because the idea of the proof seemed a bit cool, I decided to throw it up here for you guys. Throughout, the notation $M^m$ denotes a manifold $M$ of dimension $m$ with an associated differential structure.

So since coming back around here last week, I’ve been working on an update.

Of course, I’d be lying if I said I’d been working non-stop on an update, but I have, in fact, been working on one. I’d say I’m a solid 75% finished with it now, even though it’s (a) not going as quickly as I’d expected and (b) probably not going to be written the way I’d anticipated. Oh well; such is life, I guess.

I’m down to my last seven days of Wolfram employment, and to say I’m a sad robot is understatement of the year. I’m hoping the hustle and bustle of a new school term with new responsibilities and opportunities and excitements will curb that somewhat, but at this point, I’m not 100% convinced.

Among new things that have happened in the last week:

• FSU made office assignments for the new year. Apparently I’m staying put. As much as I’d have liked a new office (you know, since my CEILING COLLAPSED AND DESTROYED ALMOST EVERYTHING I HAD THERE(!!!)), I don’t like the hassle that comes with doing something new. I’ll already be doing enough new things; figuring out a new office situation isn’t something I want to add to that list.
• Fall schedules have been entered. I’m officially taking the third semesters of abstract algbera (field theory + categories, I think) and topology (advanced algebraic topology), as well as a course on complex manifolds (taught from an algebraic geometry perspective, I’d guess) and Riemannian manifolds. I’m excited. Sincerely.
• I’ve gotten my change of residency file 95% compiled. Monday will be the day to finish it off and submit it.
• I’ve finally started training for my new gig with Pearson. I’m less than thrilled with the progress so far. We’ll see.

Otherwise, things have been kinda the same: I’ve been doing a lot of Wolfram stuff, I’ve been taking more time away to hang with my family, and I’ve been somehow managing to not think about the nervousness I’ll invariably feel when new TA responsibilities, etc., pick up.

Things are good, I’d say, even though I’ve got a lot of things I need to start doing otherwise. I need to start reading the books potential advisers have suggested; I need to start doing more independent research; I need to start getting back in school mode.

For the first time in a really really long time, I’m enjoying being in not-school mode. I wonder if this is that changing tide I always heard so much about.

Anyway, expect a new content entry soon enough. And maybe some to follow that one. We’ll see.

Peace.

# Update

I just wanted to drop in and update here. I haven’t been posting much in the last day or two, but not because I haven’t been workin’ it!

Here’s what’s been going on.

• Wednesday, I stayed home and had a Clifford Analysis day. I read a solid three or four pages of my professor’s paper before calling it a day.
• Because I felt like I hadn’t done enough on the Clifford front, I went to my office Thursday armed with new writing supplies and spent a solid few hours verifying the claims made in the aforementioned three or four pages I’d read. That was a good feeling.
• Friday was (differential) geometry day, and I started the day working some “trivial” problems from Spivak’s little book. In the middle of the day, I had a phone interview with Pearson for a potential part-time job; that interview went well and I’m moving on to the second stage of the employment process. I spent some more time in Spivak’s little book before spending the remainder of my evening working problems from Volume One of Spivak’s magnum opus. Those problems are also “elementary” but they’re a bit harder. The challenge was good.
• Today is supposed to be algebra day. Because we only recently were in a position to remedy some previously-existing financial woes, however, we spent most of the day split between running errands and spending time out and about with our son. I did take both Eisenbud/Harris and Perrin with me, along with my trusted G2 and Composition Book; very little progress was made, however.

I’m actually about to dip out for the evening here in a few minutes, but depending on how much energy I have tonight, I might buckle in and try to figure out some of this sheaf theory stuff. If I had a fourth Algebraic Geometry Observation published, it would be that transferring between theory and problems which apply said theory is very VERY difficult.

Okay, I’m out. Later guys!

# Study Plan, tentatively, + Algebraic Geometry Exercises

So I think it’s probably best to have a rotating study plan schedule that allows me to do certain topics on certain days. So far, I’m thinking of having a rotation that looks something like:

Differential Geometry -> Algebra -> Clifford Stuff -> Algebraic Topology (optional),

and since yesterday was (unofficially) differential geometry day, I’m going to spend today doing algebra.

First order of business: Eisenbud and Harris. And, since I’ve been meaning to write down some of the solutions to exercises I’ve passed, I guess I’ll do that here.