Update

Despite my hope to the contrary, it would appear that the math I’ve done while here so far as not parlayed into me blogging super-frequently. For what it’s worth: Life is busy. Just in case you were wondering. ^_^

Lately, I’ve been working from home more than I’ve been going to Princeton/IAS. My goal is to change that soon and I actually had a wonderful day at IAS today. I’d like to go tomorrow but I have a work meeting at the least convenient time one can imagine; there’s also no topology seminar at the University tomorrow, so I suppose I’ll be staying in and working again. No harm no foul, I suppose.

So what have I been working on? Well:

• Universal Circles for Depth-One Foliations of 3-Manifolds. The gist here is: If you have a taut (e.g.) foliation on a 3-manifold, a theorem of Candel says we can find a metric on all the leaves so that they’re hyperbolic. Moreover, by tautness, you can lift to a foliation of the universal cover which is then a foliation whose leaves are hyperbolic discs. A ridiculously deep idea of Thurston was to look at the infinite circle boundaries of these disk leaves and maybe…glue them together? Canonically? And see if that gives insight about things?

  You probably already know how this ends: It’s doable (because he’s Thurston) and it does provide deep insight about the downstairs manifold (see, e.g., the articles by Calegari & Dunfield and/or Fenley, or Calegari’s book…)

  Now, let’s say we do this for certain classes of kind-of-understood-but-still-unknown-enough-to-be-interesting foliations like those of finite depth. Can we get cool manifold stuff by doing this process? I dunno, but maybe.

• Homologies. My ATE was about Gabai’s work on foliating sutured manifolds, so studying sutured manifolds is something I’m still interested in. One way of doing that nowadays is with this colossal, ridiculously-powerful tool called Sutured Floer homology. So…you know…homology…but when talking with other grad students about the millions of homologies out there and about how nobody really understands what motivates discovers of them, I realized that there was a lot I needed to know before focusing on one homology foreverever. So I’m working on learning stuff about homologies.
• Geometric Group Theory. Ian Agol is at IAS this year as the distinguished visitor and a lot of his work is on relationships between GGT and 3-manifolds. If you listen to any talk relating those two things, you realize there’s this whole dictionary of words and acronyms like QCERF and LERF and RAAG and Virtually Special, Residually Finite, etc. etc. I think in order to someday bridge the gap towards doing work like those guys do, I need to know what all these words mean, and what better time to figure that out than right now?! So yea…I’m doing that some, too.
• Dirac Operators, Spin manifolds,…. At some point soon, I’m going to start working on hypercomplex geometry again, and part of that will be the study of Dirac operators. So far, there are lots of perspectives on those, so we’re going to try to first establish the explicit connections between them and then maybe…do some stuff? I dunno. I also have stuff on Clifford analysis / geometry I want to look at, as well as some more things involving generalized geometries. Lots here.
• Topological Quantum Computing. This is a pipe dream until I’m able to feed my family and progress on my dissertation. It’s on the radar, though.

Okay, so this was an update! I’ve also been bookmarking some interesting proofs I’ve run across so I’ll know where to look when I decide to expand things here, and…yea.

Oh! And my professional webpage finally exited alpha and went into beta! http://www.math.fsu.edu/~cstover.

And now, Morrrr…se homology. Morse homology. That’s what I’m looking at as a segue into Floer. Another late night ftw!

Later.

Reading, and reading, and teaching, and reading, and reading, and…

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.

What’s been goin’ on…

So, I’ve been doing a piss-poor job of keeping this part of the internet pruned and tended to, etc. I’ve decided to stop in and give this thing a good once-over with how the semester’s been going now that the semester is (finally) nearing its end.

• My teaching assignment this semester was awful. I’ve been unimpressed mostly throughout.
• I gave two seminar talks at FSU’s complex analysis seminar: Complex Structures on Manifolds and Constructing Complex Manifolds Using Lie Groups. The first went pretty okay; the second was very spur of moment and came when I was in the middle of battling the flu and was unsurprisingly less-good.
• I’ve had two bouts of exams so far this semester and have managed to escape both with A averages.
• I recently concluded the two mandatory class-related presentations I had for the semester: I talked about Frobenius’ Theorem on the integrability of -plane distributions for my Riemannian Geometry class, and about Hyperkähler manifolds for my class on Complex Manifolds. Like above, the first of these was pretty okay and the second was kinda “meh”.
• I picked doctoral advisors.

That last point is one I’m particularly happy about.

As I tend to do, I managed to pick a path that’s not the standard among students (from what I can tell) in that I picked two advisors who work in two totally unrelated fields. Be that as it may, however, I’ll officially be under the tutelage of Drs. Sergio Fenley and Craig Nolder who – respectively – study geometric topology and hypercomplex analysis/geometry. For Dr. Fenley, I’m going to be studying various aspects of foliation theory; for Dr. Nolder, I think I’m going to be studying various aspects of lots of different things.

To say I’m excited would be an understatement.

Currently, then, I’m in the process of balancing end-of-semester duties and candidacy prep duties, which means I basically haul giant stacks of books around with me 24/7 and try to read any time my eyes/brain aren’t needed for something else. It’s exhausting and nerve-wracking and brain-intensive and amazing and surreal. I literally can’t express how excited I am.

When classes start back on Monday, there will be one week of non-finals classes followed by one week of finals; over the course of those two weeks, I’ll have lots of TAing to do and lots of exams to take. When those weeks are over, though, I’ll be enveloping myself in reading roughly 20 hours a day.

Or thereabouts.

I think that’s about all I’ve got presently. I’ve been on the look-out for various fellowship/scholarship opportunities, as well as various summer programs and internships, etc. I’ll try to post progress on those fronts (and others, too) here as I remember. Between all that, I think it’s safe to say that my updating of Hatcher solutions is on the (very very far) back burner for a bit, but if I’m able, I plan to spend time going through, correcting the screw-ups that exist (believe me, there are many) and trying to get generally better-familiarized with the techniques necessary to master that material.

Maybe Dr. Fenley will help. 🙂

Until next time….

Wonderment, or The Passage of Time.

Today is the first day of the eighth week of the semester.

The middle of the semester was officially last Friday.

Some people may read that and deduce that it’s all downhill from here. Any time I hear that phrase to describe midterm, I’m always a bit blown away. Really, it makes me wonder: Is this what downhill feels like?

I really hope there’s more to it than that. Heh.

Anyway, so here’s the usual rundown:

• FSU’s Math Fun Day 2013 was Saturday. It was pretty okay. Here’s a .pdf copy of my poster, for anyone interested.
• Apparently, I’m the speaker at Wednesday’s Complex Analysis seminar. Abstract and other info can be found here.
• Our next topology exam is scheduled for next Friday. I’m also anticipating an exam in Galois theory around the same time.
• I’m on a short deadline for picking a presentation topic for my Riemannian Geometry presentation.

So, I said all that to say: (a) I still exist. (b) Life is hectic. (c) I’m not sure when I’ll get around to posting more of Hatcher 2.1, but I’ll probably be moving on to Hatcher 2.2 here in about…30 seconds. Also, (d) I really need some down-time. And a haircut. And a drink.

I really need a drink.

Goodnight, Monday.

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. 😛

Update, briefly

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!

Quick in-between-TAing-assignments Update

I have another TAing duty starting up in about 20 minutes, but I decided to spend the rest of my time putting something here.

(I spent the time before this doing differential geometry computations on the blackboard)

Things are going. I made it through exam week(s) volume 1 without too much pain (other than the preparation therefor), and though I’m still waiting to hear back on how I performed on my homology exam, I think I managed to do pretty well overall. That’s a plus.

I also managed to finish the rough draft of my poster for FSU’s math fun day! That’s also a plus.

Now, I’ve begun looking to the (near) future. Indeed, I have some presentations coming up and I’m in the process of learning material and picking topics, etc. etc., to try not to put that stuff off until the last minute. Overall, I’m pretty excited: At the end of this semester, I’ll have grown quite a bit as a mathematician and will have done some of the “most meaningful work” of my career in the direction of being a professional mathematizer.

I like it.

I plan on spending some time this weekend putting Hatcher solutions up, as well as trying to finish catching up in do Carmo so I can be prepared for my Riemannian Geometry presentation.

Things right now are pretty not-terrible, though, despite my ceasing to exist here every now and again.

I hope to have some big things to roll out soon. Keep me honest, interwebz!

Peace.

Edit: In all this talk about my career, I forgot the most exciting news to happen in recent weeks! On September 23rd, my son turned the big zero-one! Yep, that means my wife and I managed to raise an amazingly awesome bag of awesomeness for an entire year without killing or seriously maiming him! We’re both as excited as that last sentence seems to convey we would be!

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.

Another Sunday, or Awaiting Week 4

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.

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.