Today, I spent the day at IAS, listening to Alex Eskin talk about Teichmuller dynamics.
I don’t know why, but I somehow struggle on some deeper level when it comes to that topic. These talks always start relatively similarly with billiards and the (non-)existence(?) of periodic orbits thereof before providing a dictionary between billiards and Riemann surface theory, an introduction to basic notions in ergodic theory (Ergodic, Uniquely Ergodic…), and then – apparently at some point when my brain shuts down – there’s really deep stuff including conjectures by Fields medalists, etc. etc. Somehow, I understand all the pieces before brain shut-down, but even so, the shut down always seems to happen and leave me scratching my head and wondering wtf happened during.
Maybe it’s a tumor.
I’ve been focusing more on stuff about universal circles. In particular, I’ve found some other documents online that summarize the Calegari-Dunfield paper a bit, and I’ve been using Calegari’s wonderful book to help get new views on things. It’s slow, but it’s progressing way better than it ever has.
Last week, there were three Minerva lectures at Princeton University by Maryam Mirzakhani. The creative ways in which she applies and broadens the scope of hyperbolic geometry is staggering, and as much as I’d like to say I understood a lot of things, I understood very small fragments of a handful of things. It was an amazing experience that I’ll cherish for a long time, but man – I was so tangibly outclassed during that it was almost embarrassing. Wonderful, but (almost) embarrassing.
Besides that, I’ve been working: Mostly boring monotonous things for Wolfram with the exception of breaking Wolfram|Alpha today, and then finally some progress on fixing the very badly-done FSU Financial Math pages. It’s a lot happening, but it’s all mostly enjoyable and I like being kept busy, etc. Always good.
Unfortunately (or perhaps fortunately for my progress on things that matter), I haven’t typed up any more interesting proofs or anything. At some point, I hope I can blog regularly without feeling like I’m missing out on more important things but honestly? Now is not that time.
I hope this finds everyone well, and if I don’t see you again first: Happy holidays!
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.
Had a great first week at IAS. Their math library is fucking unreal and it gave me a chance to read about tons of stuff I should have already read about but haven’t.
The end result is that I did very little in terms of wage earning, and in particular that our savings is down to approximately $0 and if I don’t start earning pay soon we’re going to starve. Even so, the math library here…?
Tomorrow is the first day of the year’s first directed workshop-thing on 3-manifolds (http://www.math.ias.edu/wgso3m/agenda) and I’m indescribably excited about that. I’ve also gotten to a point where I have a schedule in place to earn a livable wage between all that (yay no starvation!) and will hopefully be able to parlay some of the awesome math I’ve been absorbing into things to post here…
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.
So a while ago, I was reading Hatcher’s notes on 3-manifolds. In there, he defines what it means for a manifold to be prime and states, casually, that the 3-sphere is prime. He later says that it follows immediately from Alexander’s Theorem as, and I quote: Every 2-sphere in bounds a 3-ball. And that’s it. Done.
Elsewhere, Hatcher expands his above statement: …every 2-sphere in bounds a ball on each side…[and h]ence is prime. Again, though, it isn’t accompanied by anything, and while this is clearly a trivial result, I just couldn’t see it for the longest time…I knew that it followed from a number of things, e.g. the fact that is the identity of the connected sum operation, that is irreducible (and that every irreducible manifold is prime), that one gets the trivial sum by splitting along a 2-sphere in which bounds a 3-ball in , etc. Even so, I didn’t want to leverage some enormous machinery to deduce the smallest of results and what I really wanted was for someone to tell me what I was missing. So I never stopped thinking about this, even after moving forward, until finally – it just clicked!
I figure other people who are as visualization-impaired as I may benefit from seeing this explained in greater depth, so in lieu of typing a blog post containing something new and attention-worthy, I figure I’d share this instead. Details after the break.
Okay, so previously, I blogged about potentially implementing a series on interesting proofs. Unsurprisingly, nobody read that post and/or cared, so I decided to go ahead with it anyway because I’m a loner, Dottie, a real rebel…
…anyway, the framework for that is now in place. It’s a barren landscape presently but I have the content necessary to add one proof with (hopefully) more to come!
"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." - Paul Halmos