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.
Last week was the first of the big 3-manifolds events at IAS and overall, it was spectacular. The highlight, without a doubt, was Dave Gabai being amazing during the last talk of the week, but there were some other great moments too…
…and some not-so-great ones, including some woman whom I don’t know interrupting Genevieve Walsh‘s talk no fewer than 10 times to say random rude things about how it was not-good (which was untrue), unoriginal (only true in the sense that Dr. Walsh spent some time talking about general background that she didn’t claim to have invented), and a waste of time. I was pretty blown away that such things happened at pure math talks, but I guess pure math people are people too and – at the end of the day – people just look for a way to disappoint and/or bring down other people.
I learned a lot, though, and I came away with a new direction for my own research, so that’s going to be the goal moving forward: To balance the somewhat-regular yearly 3-manifolds talks at IAS with the stuff I need to figure out to get my own stuff knocked out.
Oh, and plus side: I actually got a full week of salaried work done! YAY FOOD! But the downside is that I’m having to drop $2k on random car things (making our tires able to withstand rain and snow and making it so that our heat keeps hypothermia at bay), so…YAY CREDIT CARD DEBT! ::wink::
Alright, well I’m awake for some dumb reason so I guess I’ll…try to do something…constructive. Or something. Hah.
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…
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.
Today is the fifth full day at our new place and things are finally starting to settle in. Until today, we’d been sleeping/sitting/otherwise living on the floor, for the most part. In particular:
A couple days ago, we got our Wifi connected so our internet access went from patchy and occasional to great and full-time.
After spending the first few days sleeping on the floor, we got a couple air mattresses on Monday. That came with some slight added comfort.
Today, our new couch came in. I can’t overstate how amazingly comfortable this fucking thing is, and believe me when I say: It’s completely changed my whole attitude to have a comfortable place to sit!
As a result of the added couch-induced comfort, I’m letting today be my first day transitioning to The Princeton Schedule of mathing all day and working (for a wage) at night. So far today, it’s been all 3-manifolds and foliations, particularly getting things I ought to already know typed into Mnemosyne so that I can make sure I know know them moving forward.
There’s so much math I should be better at; I’m really looking forward to using this year to bridge the gap from where I am to where I ought to be.
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!
The last time I posted something meaningful here (not counting the 2014 year-in-review and the most recent claim of attempting necromancy), it was June 2014 and I was about to embark on a summer of traveling. Around that same time, my son was 21 months old, I was working part-time at Wolfram, and I was a pre-doctoral candidate whose academic situation had gone (apparently without being blogged about) from two doctoral advisors with two separate projects to a single advisor plus a second non-advisor faculty colleague.
Typing that out makes me realize how much has changed.
For those of you keeping score, it’s now August 2015, and 13 months after the last update, lots and lots of things have changed. For example, my son is now one month away from being three years old. There’s also a lot of professional stuff, too. Let’s go somewhat chronologically.
I passed my advanced topics exam (ATE) and became a doctoral candidate. My work was on Gabai’s colossal (first) work on Reebless foliations in 3-manifolds, and while I definitely learned more significant math than I’ve ever learned, I feel like there’s so much in that paper than I’m years away from understanding.
I flew up to Baltimore to interview for an NSA gig. I didn’t get chosen.
I went to the 40th annual spring lecture series at the University of Arkansas and had a complete blast. I ended up slipping on ice, busting my ankle up pretty badly, and having some travel woes near the end but when all was said and done, I met some cool people (Benson Farb, Allen Hatcher) and saw some really great talks. Oh, and great coffee!
I went to Rhode Island College and gave an invited lecture on limit sets and computer visualization. It was an honor and I couldn’t have hoped for a better first invited lecture experience.
I finished a pretty uneventful spring semester at FSU. Lots of work. Lots and lots of work.
Once summer (2015) rolled around, I got accepted to some pretty great things:
I was fortunate enough to be awarded a pair of scholarships from the FSU math department.
And now, here we are! It’s officially September 1 (1:07am now): That means Fall semester has started at FSU (which means I’m now a fourth year doctoral student; eek) and things are back in full swing. It never gets familiar, really, no matter how many times it happens. C’est la vie, I guess.
I’ve got a bunch of stuff going on, professionally:
I’m still trying to make progress on my dissertation research (3-manifolds and, eventually, foliations).
I’m studying Dirac operators / spin manifolds / hypercomplex structures / supermanifolds / miscellaneous things that seem to get more and more into the realm of theoretical physics as we progress. This is with my non-advisor faculty colleague.
I’m trying to get a small research project going with an undergraduate at FSU on topological quantum computing (maybe Microsoft will take interest?).
Non-professionally, things have also happened. I got pretty serious into working out for a bit; later, I lost track due to travels, though I’ve since made some pretty considerable body transformations due to a healthier diet. I’ve also tuned back my Wolfram hours to give me more time to do student things; I’ve upgraded my workstations (desktop and mobile); I’ve made the switch from Windows to Linux (full-time rather than as a hobby)…
…that may actually be about it!
So there! Now we’re caught up! That means that I can pick up next time with an actual update / piece of newness / whatever. And who knows – maybe there will even be some math thrown in here! gasp
Observation I. Every article in this field is remarkably long.
I’ve currently got articles whose lengths are 60, 58, 50, 39, 59, 56, 58, 76, and 50 pages long in my “reading queue”. Of the others, there are none that are shorter than 25 pages with the exception of Gabai’s second article (18 pages, though sandwiched between a 60 page original and a 58 page triquel).
One would think that longer automatically implies “more detailed” (i.e., less terse, less difficult to read, etc.), but this isn’t necessarily the case; in particular, Gabai’s articles are ridiculously complex and brilliant and amazing, and even legitimate 3-manifold topologists specializing in foliation theory confess that it takes forever (literally not literally) to make it through even one of them.
My prediction is that when my son’s in college, the average math Ph.D. will take 10 years. Give or take.
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.
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.
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.
"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