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.

Update

I’ve fallen into a bit of mathematical stagnancy since the first week or so of living here but after much ado, I’ve finally become regimented enough to start doing work and doing math and juggling other obligations, etc. etc.

What can I say? Moving is hard business!

Since falling off the mathematical (and career) wagon, I have managed to buy some new math books (uber sale; it’s my weakness) and to completely build a 95%-ish complete version of a new professional homepage which I hope to deploy within a week or so. As of a few days ago, I also managed to climb back on to the career (sans math) wagon, and as of today (well, yesterday; it’s 4:30am “tomorrow” for me right now), I also managed to do some low-key math with my BFF L. Hoping that pans out.

Later today, I’m going to head to IAS and spend the day doing math things and listening to postdocs talk about stuff I’ll likely never be mature enough to comprehend. Hoping this is day 1 of a lot of consecutive days of doing that and/or things like it. We’ll see.

Mathly yours…

Travel Update Finale, or The Times They Are A-Changin’

Travel Update 1

Currently sitting in a hotel room in Virginia.

I’ve spent the last few days enjoying some travel time with the family – so far, we’ve driven through Georgia, South Carolina, and North Carolina – and things are great. Today, we took Aleks to the EdVenture children’s museum in Columbia, SC, and that place was killer! I’ve got a bajillion or so pictures, so I’ll probably come back and upload some of them soon.

Mathematically, things have been pretty stale (because, you know, ROAD TRIPPING). I have been pondering the possibility of adopting a Gödel-like approach to symbolic first-order logic using graph theory instead of numbers, etc. This is something proposed by a friend of mine and it’s opening me up to hopefully learn some stuff I ought to know already but don’t. We’ll see.

That’s all for now. I’m going to try to dredge up some old ideas for expository math things into some new entries not about my family and me. Maybe that’ll work out.

Talk soon, everyone.