Welp, I haven’t been around these parts in a bit. That’s not to say I haven’t been working, though; I’ve just been working through things that don’t necessarily make good reading…

…like math….

Okay, so maybe that’s not entirely true. I have, however, been spending a lot of time on just a couple projects for my advisors and so there hasn’t been an abundance of topics to post about here. I think I’m getting to a point, though, where I can start posting some things here and maybe use that to reconcile the fact that I can’t understand most things I read, etc. We’ll see.

So yea…the semester’s over. It turned out not to be a terrible one for me in the end despite being pretty terrible throughout. The upshot: I managed a 4.0 that semester and ended up with advisors. That’s a victory for sure.

I’ll do my best to come back around these parts in the next day or two and post something of substance. I’d like to try to do some sort of expositing on geometric topology things (foliations, laminations, universal circles) and maybe some Clifford things too; I’d also like to attempt to reconcile my previous goal of learning how to do things from Hatcher.

Maybe I can do all of the above. 🙂

Until next time….

# Pre-Week 5, Post 1

This is a short little thing I’m throwing up here just to say a couple things:

1. I exist.
2. It’s been busy around this place.
3. It’s been busy around my life.
4. I owe you guys some stuff.

Re: The Last Item: I’ve got some moderating to do here. I’ve gotten a few comments on my Hatcher solutions pointing out that I’m really embarrassingly bad at topology. Rest assured: I’ve received your comments, I know how terrible I am, and I’ll be working to rectify the situation as soon as time permits.

Maybe I’ll have a follow-up to this later this evening.

Peace.

# Frustration, or Somebody’s got a case of the Tuesdays

This is going to be an entry about my algebraic topology class.

My previous topology classes were taught by someone who’s amazing <i>as a mathematician</i>. Most people in the class would agree, however, that this person was someone who was terrible as an instructor: Somehow, I made it through a graduate sequence of coursework despite receiving terrible grades throughout. This happened despite my spending 50+ hours a week on every homework assignment and slaving until I was on the verge of breakdown week in and week out. Somehow, I got a B.

My topology experience thus far is certainly not one of my finest achievements.

Fast forward to now and I’m in an algebraic topology class taught by someone who’s amazing. Amazing. Not amazing <i>insert quantifier here</i>, no – this person is simply amazing. And this topic is beautiful. And this class is hard.

This class is hard, too, despite the fact that we have no continual responsibilities. Indeed, we have zero homework whatsoever: Not required problems to turn in, not required problems to keep, not even suggested problems for our benefit. We simply have <i>zero homework</i> in this class. That’s a huge relief after last semester.

What we <i>do have</i>, though, are exams. We have three of them, and I have zero doubt now (nor have I had doubt at any point this semester) that my ass will be kicked by each and every one.

As a result, I’m working hard.

A week-ish ago, I spent some time going through the preliminary parts of the stuff we’re talking about (homology theory). I did examples, I spent lots of time drawing pictures, and I didn’t stop until I got it.

That’s right: A week-ish ago, I <i>got it</i>.

Today, however, I’m sitting in my office, frustrated and almost-defeated, blogging to you all and mourning the fact that a lot has apparently changed in the last week-ish.

Today, I just don’t get it.

If I were to make a list here cataloging the number of screw-ups I made trying to solve one problem over the course of about 20 hours, I’d be (a) making a really long list and (b) really <i>really</i> embarrassed.

I’m really <i>really</i> embarrassed right now.

Finally, after re-reading and re-re-reading Hatcher, I found source 1 of my confusion. Later, after consulting the online resources of mathematicians greater than myself (case in point here), I found the remaining sources of my confusion.

The upside is that now I’m no longer confused. On the other hand, the fact that I was as confused as I was (and about such basic material as that happened to be) makes me really <i>really</i> uneasy moving forward.

I need an intervention.

In the meantime, I’m going to try to dust myself off, hit the salt mines yet again, and lose my frustrations in the never-ending cycle of Lana del Rey that’s been permeating through my office for the past couple hours.

3 weeks, 2 days.

# Sunday Summary

My mathematizing wasn’t very impressive today. I:

1. Read some pages on Gröbner bases in Dummit and Foote.
2. Did some tutoring / tutor-related things.
3. Spent some time figuring out some solutions from Hatcher.

Despite it being 3:30am, my game plan is to be up around 8am to make a trip to campus. While there, I plan to do the usual errand things, and to then lock myself in my office for 5-or-so hours and do some legit math things.

That means I need to take good resources there with me.

Peace.

# Working leisurely or Doing nothing?

So here’s the thing: I haven’t really done anything today. What I mean is that I haven’t constructed anything new (a page, a list of definitions, a solution) that didn’t exist yesterday, and so – for all intents and purposes – I haven’t done anything.

But somehow, I haven’t done nothing either.

Some days, I make a plan to do something (“do” something), and I set out on that path. Sometimes, the path I reach has a bunch of hurdles that I’m not prepared to conquer, and so I set out on a side journey to obtain the skills necessary to progress down my original path. Sometimes – on days that are particularly unkind – the side paths have hurdles requiring sidepaths and the side-side-paths have hurdles requiring side-side-paths and so the whole journey gets twisted into some amalgamated blob of non-progress that somehow still manages to accomplish something.

That, ladies and gentlemen, was a metaphor. It’s a metaphor that fits my day rather well.

So as I mentioned earlier, algebraic topology was a bust. I decided, then, to finally take the plunge and to read something on $D$-modules via Google. My professor had suggested this as a nice algebraic way to derive lots of the differential geometry results by way of learning really difficult algebra stuff like categories and stacks and sheaves and schemes and what not. That, of course, got me interested. I did a little digging and found an online resource from Harvard and decided to take a stab. I made it through about a page before I realized I was missing stuff on stuff.

I freshened up on stuff about Lie groups and took a gander at what Wikipedia had to say about Universal Enveloping Algebras. Of course, Universal Enveloping Algebras required me to know things about Tensor Algebras, and when I decided to look up something more foundational like “rings of differential operators”, I decided that I should probably concurrently try to parse through some literature regarding Differential Algebra as well. That chase has brought me to where I am now and has sustained me for the better part of three hours.

In that three hours, I’ve found lots of good resources (including an online .pdf of Ritt’s text Differential Algebra) and have done quite a bit of reading, but if I were to die today and pass the totality of today’s efforts off to someone else, their inheritance would consist of precisely zero tangible work.

So yea: Not doing anything while not doing nothing is a thing and it’s called “research mathematics”. Such is life, I suppose.

I think I’m going to end today’s part of my quest on the differential algebra / $D$-modules front here: I’ve got some stuff to do and what not, yadda yadda yadda, etc. etc. I plan on working some more on Kobayashi and Nomizu before bed, though.

All I do is math math math no matter what….

# CW-complexes

So as I spend my days progressing through the very dense, very slow-moving genius that is Hatcher’s book Algebraic Topology, I’m constantly reminded of things I’m not very good at.

And believe me: There are lots of things in that book I”m not very good at.

One thing I’ve always struggled with were the technical details of CW-complexes. I’ve mentioned that before. As a result, I’ve spent the better part of an afternoon gathering online resources, etc., that would shed insight onto the things I’m not clear about. While I wouldn’t bet money on this, I feel confident that I’m now better-equipped to recognize and understand the theoretical construction and properties of CW-complexes…

…and yet, I still find that I can’t do many (as in, I can do almost none) of the problems in Hatcher.

I’m pretty sure I understand the gist for problem 0.14 (for example), which asks you to put a CW-cell structure onto $S^2$ with $v$ 0-cells, $e$ 1-cells and $f$ 2-cells, where $v,e,f$ are integers for which $v-e+f=2$. The gist seems simple: Add on a cell in a precise, regimented way, and define a characteristic map which correctly “incorporates” the additional construction into the given construction. I get that. But where do I start?

I just don’t seem to know enough to transition from sticking my toes in the water to jumping in and taking a swim. I’m pretty sure jumping in at this juncture means drowning.

As I’m sometimes inclined to do, I dug up some other solutions to see if they could shed insight. I feel like Tarun’s solution is overly complicated, while I feel like Dr. Robbin’s solution assumes way way more knowledge than I have at my disposal.

This leaves me feeling a little lost on the algebraic topology front, meaning I’m going to have to dig through some of the resources I have at home and try to figure something out.

Gah.

For completeness, it should be noted that I’ve scrounged up a couple of resources online: Find them here and here.

Maybe in the meantime, I’ll dig up a diversion or two: I’m thinking some differential geometry or maybe even some $D$-module theory. Maybe I’ll be adding some more around these parts later.

In the mean time, check out these easy-to-read, casual expositions on the proofs of the ABC conjecture and the Bounded Gap Conjecture for Primes. I find the first story particularly intriguing.

# Hump Day Math Day

Well, it’s Wednesday.

Generally, Wednesday is the cause for much joy because it means the terrible Mondays and Tuesdays are things of the past and that the future – those Fridays and Saturdays – are right around the corner. It makes sense, then, that Wednesdays would be good work days too: It should intuitively mean that the drag-assery of Mondays and Tuesdays has subsided and that the “time flies when you’re having fun” of Fridays and Saturdays has caught hold.

That hasn’t been the case here on the math front, today.

I’ve spent the better part of the morning and afternoon battling a migraine, which means I’ve held out on the “read new books” front (so long, Kobayashi and Nomizu…) and have instead been leisurely contemplating some of the unfinished problems from algebraic topology (…and hellllllllllo, Dr. Hatcher!). I’ve made some progress (read: I’ve done two problems), but it’s definitely not something to write home about.

But hey: Some days you’re the dog and some days you’re the hydrant.

I’ve finally reached the cluster of Hatcher problems dealing with CW complexes, and because I’m so genuinely awful at them, I’m going to take some time to read around and try to get a better grasp before moving forward. I have a good general idea and I can usually do some problems / work with the things themselves, but I always have a hard time processing the technical details thereof. Something about that whole connecting $n-1$ cells to $n$ cells by way of continuous maps that do some stuff on the boundary and yadda yadda yadda just never has settled with my brain cells.

And that, friends, is the point of summer: To force unsettled things to settle.