Update, briefly

Posted by Chas

I’m about to have to get ready to meet some friends, but just FYI: I’ve added a couple new Hatcher solutions from section 2.1. I’ve got several more written down, but this surely isn’t a speedy venture. Just FYI.

Woot, progress!

Another Sunday, or Awaiting Week 4

Posted by Chas

3 weeks.

I’ve officially survived the first three weeks of my second year of grad school (twice, actually). Again, I know keeping count of the days is a terrible thing to do to myself, particularly when there’s been a very small amount of good to come from my weeks thus far, but at this point, I’m sort of using that countdown as some sort of badge of accomplishment. Or something.

The coming weeks are going to be very very stressful and busy and stressful. Besides my usual load of stuff (I’m enrolled in 6 classes, I have a reading class in algebraic geometry starting up on Tuesday, I’m TAing for 1 lecture and 7 labs, and I’m trying to pick advisors / plan presentations I’ll need to give some timem soon), I also thought it was a good idea (remind me why?!) to make a poster to present at FSU’s upcoming Math Fun Day. That particular endeavor shouldn’t be especially difficult, but it requires time and time, ladies and gentlemen, is precisely what I have zero of.

Daunting is the adjective that comes to mind.

Also daunting is / was / has been the thought of continuing my goal to do all the problems in Hatcher. As you may recall, I spent the first half of summer slaving to acquire the information needed for the Chapter 0 exercises, only to have my plan for Chapter 1 totality derailed by that little piece of awesome that was my Wolfram internship. Long story short: The obsessive-compulsive part of me wants to not move forward until I hash out a Chapter 1 plan, but the This will benefit me in the class I’m taking now which, subsequently, hinges on my ability to understand Chapters 2 and 3 of Hatcher part wants to press forward.

I’m pleased to announce that the second guy won out.

In particular, my Hatcher Solutions page is showing signs of progress. It didn’t take as long as I’d predicted it to take to build that framework, and due to a random, unforeseen bout of sleeplessness at 3am this morning, I had precisely the opportunity needed to seize the moment. Right now, all those are empty pages, but I’m pleased to report that I seem to have accumulated approximately six solutions; if everything goes as planned, I’ll be taking time to update by including those as soon as possible.

In the meantime, I’m going to continue to hash out what to do about this paper. And what to do about the professors I’m emailing regarding potential advisor-hood. And what to do about the fact that I severely cut my weekend work time by spending yesterday ballin’ out of control in celebration of my wife’s birth. And what to do about….

Au revoir, internet. I bid thee well.

Oh, I just remembered: I have my first exam of the semester Friday. It’s on field theory. I’m less than pleased. $:\$

Half-June

Posted by Chas

So today wasn’t really my day, overall. Generally speaking, I woke up feeling congested and nasty, I spent the whole day with a migraine, and I was only not-lethargic for about four hours total overall.

Unsurprisingly, then, I realllly couldn’t force my brain to do any real math. For that reason, I completely avoided reading new things and instead typed up the expository analysis entry during the middle part of the afternoon. I ended the night doing some solutions for Hatcher – Chapter 0 of which I’m hoping to knock out soon to begin Chapter 1 – and drawing (really really) poor diagrams in MSPaint. I’ve emailed Dr. Sjamaar from Cornell to ask how he gets his diagrams drawn, but thus far have heard nothing back.

Pleeeeeeease don’t leave me hangin’, Dr. Sjamaar: My blog is evidence that I’m in desperate need of your resource knowledge!

Anyway, it’s almost 2am and I’m about to call it a night.

Auf Wiedersehen.

Some Exercises from Lee’s Introduction to Smooth Manifolds

Posted by Chas

These are exercises from the first few pages of Lee’s Introduction to Smooth Manifolds. These are pretty elementary from a topology perspective – maybe the middle-to-late part of a semester on point-set topology – but I decided to put them here to (a) remember some of that stuff, and (b) force myself to not delay doing them any longer. As it turns out, I struggled more with these than I should have and even had to consult the internet for a couple thing; in instances of internet borrowing, I provide links.

Note, finally, that if you’re trying to hunt these particular exercises down within Dr. Lee’s book, they’re actually dispersed throughout the text; suffice it to say, I’m not a fan of that particular method of delivery. Nevertheless….

Continue reading →

Sunday Summary

Posted by Chas

My mathematizing wasn’t very impressive today. I:

Read some pages on Gröbner bases in Dummit and Foote.
Did some tutoring / tutor-related things.
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.

Hump Day Math Day

Posted by Chas

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.

What dreams are made of

Posted by Chas

So for those of you who haven’t noticed, I’ve been doing algebraic topology a lot lately.

Pretty much any time I’m not doing life stuff or helping parent my son, I’m on my laptop with a black Pilot 1mm G2 (aka the greatest pen of all time) and a legal pad and I’m frantically trying to figure out things…or I’m trying to convince myself that I finally understand the differences between the definitions of retract and weak deformation retract and strong deformation retract and that the differences aren’t just subtle nuances but are actually deep and significant and deep.

I’ve been doing algebraic topology a lot lately.

Whenever I spend lots of time doing the same thing in succession, I find my brain existing in a very unilaterally-focused state where nothing really registers outside the thing I’m doing lots of. In such moments, I find that I’m constantly working on said thing when I’m walking in the park or when I’m grocery shopping or when I’m sleeping…

…I’ve been doing algebraic topology while sleeping….

Pretty much every night this week, I’ve snapped awake only to realize that the thoughts pre-snap consisted of homotopies or of commutative diagrams or of compositions of compositions of compositions of….

…I’ve been doing algebraic topology a lot lately.

Presently, it’s 6am. I’ve been awake for 3 hours because – let’s face it – I almost never sleep anymore. The first thing I did when I woke up at 3am? Log in to WordPress and break out the legal pad.

The good news: I’m making progress. The bad news: I’m starting to fear I won’t be able to make progress on any of the other topics I’d set aside for summer studies.

I think I’m going to be more pro-active in remedying the bad news there, though. As soon as I get done transcribing the solution (I’m sure you guys have noticed my solutions page, yes?) I just figured out, I’m going to break out a different book (perhaps even on a different subject?) and try my hands at that.

Ideally, I’d like to expand my solutions page (singular) to solution pages (plural) as I work through other meaningful texts in my life. I’d really like to take a stab at doing a large portion of Munkres’ problems, though I’m about 101% sure I’ll never be able to do all of them; I’d also like to take some time to add some solutions to differential something because – truth be told – I’m a differential something-er at heart. This is inescapable.

On the other hand, I really don’t want to ignore my original passion – algebra – though posting solutions there would be more challenging. Why, you ask? Well, working through Dummit and Foote has always been a goal of mine, but Nathan Bloomfield already has an absolutely tremendous WordPress blog set up for precisely such a work-through. So, long story short: Even if/when I do work through those problems, posting them seems completely pointless.

Now, however, I fear I’m just rambling.

Good night, world.

Study Plan + Hatcher Section 0 solutions

Posted by Chas

So going into the summer, I had a pretty concise game plan of how I wanted things to work. I wanted (a) to exempt all my qualifying exams so that I’d have fewer have to’s on my agenda, and (b) to spend all the time I’d save through completing (a) split between (i) studies to help me with my required coursework next semester and (ii) studies to help me towards being able to research things I want to research.

Worst case scenario, of course, was that I didn’t exempt qualifying exams and had to incorporate preparation for those into the mix. As it stood, though, I had a clear idea of what my summer was going to consist of, academia-wise. I planned on taking the first week (i.e., last week) to casually do some independent research without a rigid schedule in place, which would turn into a regimented plan – beginning today and lasting until the beginning of fall – consisting of one hour of studying per day in each of the following fields:

Clifford Analysis.
Differential Topology/Geometry.
Algebraic Topology.
Abstract Algebra.

Among the four fields listed above, the first two were for me and me alone, whereas the last two were to help me for next semester.

It’s now 4:30am Tuesday. I didn’t spend Monday adhering to that regimen and, for all intents and purposes, I probably won’t today. There’s a long list of excuses I could give about how I had to go to the ER, about how my infant son is teething and angry, etc. etc., but the story is simply this: I made a regimen that I don’t seem to be able to stick to. Bing, bang, boom.

I’m going to do my best to get something constructive out of this summer, but I’m facing the sad reality that it probably won’t be what I’d planned. Boo hoo.

To aid my attempts, I’m going to try to put some of my solutions here as a way of forcing myself to stay dedicated. Somewhat. For now, here are some solutions for the Chapter 0 material from Hatcher.

Hatcher, Algebraic Topology, Chapter 0
1. Construct an explicit deformation retraction of the torus with one point deleted
onto a graph consisting of two circles intersecting in a point, namely, longitude and
meridian circles of the torus.

Proof. Let $T$ be the torus and let $X=T\setminus\{\text{pt}\}$ be the space in question. By considering the square gluing diagram of the torus $T$ sans a point, a diagrammatic representation for $X$ can be given as shown in Figure 1 below.

Figure 1

Note that the gray part of Figure 1 represents the fact that the torus $T$ is filled in and that one can visualize the deformation retract in question by grabbing hold of the hole in $X$ and stretching it out so that the diagram in Figure 1 is hollow (i.e., not filled in, i.e. all white, etc.). Thus, the deformation retract in question amounts to the projection of the interior of a square (the square representing the gluing diagram of $T$ ) onto its boundary, the formula for which can be attained by presupposing that the gluing diagram for $T$ is placed in $\mathbb{R}^2$ as $J\times J$ where $J=[-1,1]$ . Basic arithmetic shows that, for $t\in I$ , the family $f_t:I\times I\to\mathbb{R}^2$ given by

$f_t(x,y)=(1-t)(x,y)+t\left(\dfrac{(x,y)}{\max\{|x|,|y|\}}\right)$

“does the trick.” Indeed, note that $f_0(x,y)=(x,y)$ , that $f_1(x,y)=(x,y)/\max\{|x|,|y|\}$ is an element of $\partial(J\times J)$ , and that $f_t(x,y)|_{\partial(J\times J)}=(x,y)$ due to the fact that $\max\{|x|,|y|\}=1$ on $\partial(J\times J)$ . Finally, note that continuity of the family $f_t$ is given due to the fact that $f_t$ is the composition of continuous functions of $x,y,t$ for all $t\in I$ . $\,\,\square$

2. Construct an explicit deformation retraction of $\mathbb{R}^n\setminus\{0\}$ onto $S^{n-1}$ .
Proof. This problem is essentially a problem from Calculus III. Note that for a vector $\mathbf{x}$ in $\mathbb{R}^n\setminus\{0\}$ , the normalized vector $\mathbf{x}/\|\mathbf{x}\|$ lies on $S^{n-1}$ . It suffices, then, to do the normalization process in a way that’s continuous for a time parameter $t\in I$ , and one way to accomplish this is to define a family $f_t:\mathbb{R}^n\setminus\{0\}\to S^{n-1}$ so that, for each $\mathbf{x}=(x_1,x_2,\ldots,x_n)$ in the domain,

$f_t(x_1,x_2,\ldots,x_n)=\left(\dfrac{x_1}{t\|\mathbf{x}\|+(1-t)},\cdots,\dfrac{x_n}{t\|\mathbf{x}\|+(1-t)}\right).$

As noted in problem 1, the function $f_t$ is continuous for each $t\in I$ . Moreover, $f_0(\mathbf{x})=\mathbf{x}$ , $f_1(\mathbf{x})=\mathbf{x}/\|\mathbf{x}\|$ , and $f_t(\mathbf{x})|_{S^{n-1}}=\mathbf{x}$ due to the fact that $\|\mathbf{x}\|=1$ for all $\mathbf{x}\in S^{n-1}$ . $\,\,\square$

riemannian hunger

A rambling narrative of one man's journey through mathematics

solutions

Update, briefly

Another Sunday, or Awaiting Week 4

Half-June

Some Exercises from Lee’s Introduction to Smooth Manifolds

Sunday Summary

Hump Day Math Day

What dreams are made of

Study Plan + Hatcher Section 0 solutions

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