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.
-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”.
being monogenic: My original claim was that 
for monogenic functions, but today, I miscalculated the partials for an example that led me to believe that 
was actually the criteria. That’s not correct at all.
and 
, respectively, to be a single sum ranging from 
instead of a term for 
and 
, respectively, plus a sum for 
. Later, when there were two parameters 
in the sum, I claimed that 
implies that 
; this, of course, is false, since 
is identified with 1 so that 
. On the other hand, with a proper bit of rigor, the proof is still essentially correct. 
, an equivalent representation of which is 
where 
and where 
represent the scalar and vector parts of 
to be the derivative of ![D_n[\mathbf{sc}(f)]=0](https://s0.wp.com/latex.php?latex=D_n%5B%5Cmathbf%7Bsc%7D%28f%29%5D%3D0&bg=ffffff&fg=111111&s=0&c=20201002)
and ![D_n[\mathbf{vec}(f)]=0](https://s0.wp.com/latex.php?latex=D_n%5B%5Cmathbf%7Bvec%7D%28f%29%5D%3D0&bg=ffffff&fg=111111&s=0&c=20201002)
. With regards to the scalar part of 
.
, and so each component must be zero. Hence, 
for 
.![\begin{array}{rcl}0 & = & \displaystyle\left(\frac{\partial}{\partial x_0}+\sum_{l=1}^n e_l\frac{\partial}{\partial x_l} \right)\circ\left(\sum_{k=1}^n u_k(x)e_k\right) \\[2em] & = & \displaystyle\sum_{k=1}^n \frac{\partial u_k}{\partial x_0}e_k + \sum_{k,l=1}^n e_l\,e_k\frac{\partial u_k}{\partial x_l}\,\,\,\,\,\,\,\,\,\,(1)\end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Brcl%7D0+%26+%3D+%26+%5Cdisplaystyle%5Cleft%28%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x_0%7D%2B%5Csum_%7Bl%3D1%7D%5En+e_l%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x_l%7D+%5Cright%29%5Ccirc%5Cleft%28%5Csum_%7Bk%3D1%7D%5En+u_k%28x%29e_k%5Cright%29+%5C%5C%5B2em%5D+%26+%3D+%26+%5Cdisplaystyle%5Csum_%7Bk%3D1%7D%5En+%5Cfrac%7B%5Cpartial+u_k%7D%7B%5Cpartial+x_0%7De_k+%2B+%5Csum_%7Bk%2Cl%3D1%7D%5En+e_l%5C%2Ce_k%5Cfrac%7B%5Cpartial+u_k%7D%7B%5Cpartial+x_l%7D%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%281%29%5Cend%7Barray%7D&bg=ffffff&fg=111111&s=0&c=20201002)
.
sum to zero precisely when each sum itself is equal to zero due to the linear independence of the basis elements 
, 
. In particular, then, the second sum in 
cited there should be 
matrices instead of 
. Recall also that the first equation in the system of equations shown in the original entry – the equation 
– is achieved by combining the equation of “mixed partials” of the form 
from the second sum in 
from above.
-module theory. Until next time…. 
riemannian hunger 