Fields Medalists and Topology and Thesis Research and…

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!

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 since the update

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 spent summer 2014 traveling.
  • I went to an absolutely amazing topology conference at Cornell (and met a Fields medalist).
  • I spent a month learning how to use Mathematica better at a Wolfram summer school.
  • I went to an okay topology conference at CUNY Staten Island.
• Afterwards, I was offered a full-time position at Wolfram as Math Content Developer. I accepted and took the year off from teaching.
• I landed a lead role in a really awesome math-related project at Wolfram.
• I went to a great conference at Yale and really enjoyed New England. New Haven is absolutely incredible.
• 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 went to the Tech Topology Conference soon after becoming a candidate.
• Not long after, FSU had a pretty gnarly conference on Clifford analysis.
• 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:
  • An MSRI program on Geometric Group Theory. I eventually had to decline this one.
  • Moduli Crossroads Retreat; also had to decline this guy.
  • A Berkeley summer school in diffeomorphism groups. I went there and had an absolute blast.
  • The first annual Chicago summer school in geometry and topology. I can honestly say that this was the best mathematical experience I’ve ever had, and man: What a beautiful city Chicago is!
• 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

Good night, everyone.

PS: Oh! I was also introduced to Mnemosyne by a mathematician considerably better than myself! So far, I’m a pretty big fan.

Foliation Theory Observation I

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.