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 SpecialResidually 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.

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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!

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. :\