Implementation for Interesting Proofs (Framework)

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!

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.
• 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:
• 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.

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.

What’s been goin’ on…

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 $k$-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”.

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.

Maybe Dr. Fenley will help. 🙂

Until next time….

Being (re)born(-again)

I wasn’t around these parts much yesterday for a number of reasons: I spent the better part of the midday on campus, splitting time between campus errands, meeting with one professor about the stuff he’s working on, and parsing through another professor‘s latest research paper; the times around that were being occupied by birthday things because yesterday was the twenty-seventh annual celebration of my being birthed.

That’s right: I’m 27 now, which means – among other things – that I have fewer than 365 days (I’m not sure exactly how many) to supplant Jean-Pierre Serre as the youngest Fields Medal winner. Not a good feeling for a just-blossoming mathematician. Le sigh.

I guess the good news is that I have 13 years to maybe squeeze one in there. Thirteen years…that’s 4,747 days from today. That makes it seem like I have plenty of time to work!

</forced optimism>

In any case…

The paper I’m reading is on the behavior of M-conformal mappings taking values in a Clifford algebra $\mathcal{C}\ell_{m,n}$ over $\mathbb{R}$. I’m gonna take a second to discuss some of the preliminaries of that topic and to maybe outline the proof of a basic assertion that’s generally considered general knowledge among those in the know. Worth noting is that the background of Clifford Algebras (and hence of the analysis of functions thereon) extends far more deeply than discussed here and so this should in no way be taken as an actual, worthwhile exposition on the ease/difficulty of the topic.

Moving on

As per the earlier entry, I’ve finally decided to give algebraic topology a rest. I’ll probably try to work a problem or two every day once I start back, but for the next few days, I’m going to focus on other things.

I went through the books on my bookshelf and came up with some that are going to work well for studying this summer. Here’s what I’ve uncovered:

• Differential Geometry
• Kobayashi and Nomizu Vols 1 & 2
• Auslander
• Differential Topology
• Lee’s Introduction to Smooth Manifolds
• Guilleman and Pollack
• Algebra
• Hungerford
• MacLane and Birkhoff

At some point, I may decide to break out some textbooks on Real Analysis and Complex Analysis, too, but those seem less pressing to my progress moving forward.

In addition to the above, I have some additional research lined up in hypercomplex analysis and in module theory (the study of $D$-modules, in particular), but I don’t own any textbooks for those.

You know that feeling when you really really enjoy something that also happens to coincide with what you do for a living, and when you’re able to remove the “work” aspect from it? That feeling that happens when you have an entire summer to just enjoy the thing you enjoy without being forced to do it outside an environment that works for you?

What has two thumbs and is completely basking in that feeling right now? This guy, and I’m nerd-cheesin’ so hard right now.

Catching Up

So I managed to go nine months without ever looking at this thing. That’s unfortunate. It is, however, an indication of how the past two semesters have been for me, time-wise.

Without moving backwards, I’ll leave this for an update:

• My GPA is pretty not-terrible.
• I managed to exempt all of my qualifying exams by scoring A- or higher in the classes needed.
• I’ve gotten to know some professors who seem not to hate me.
• I’ve become a better mathematician.

All of those are important, of course, but the last one is the one that matters most.

When I started this thing, I’d meant it to be a place for sharing lots of general math stuff. Now that I (hopefully) have more time, I’m expecting more sharing to happen. Just to get things off on the right foot, here’s part of a little something I worked on last semester.

Here, we’re considering functions $f:\mathbb{H}\to\mathbb{H}$ which are quaternion-valued functions of a quaternion variable. The function $f$ is said to be (Feuter) Regular at $q\in\mathbb{H}$ provided $\overline{\partial}_\ell f(q)=0$, where

$\overline{\partial}_\ell=\dfrac{\partial}{\partial t}+i\dfrac{\partial}{\partial x}+j\dfrac{\partial}{\partial y}+k\dfrac{\partial}{\partial z}$

The Cauchy-Fueter Integral Formula
If $f$ is regular at every point of the positively-oriented parallelepiped $K$ and $q_0$ is a point in the interior $K^\circ$ of $K$, then

$f(q_0)=\dfrac{1}{2\pi^2}\displaystyle\int_{\partial K} \dfrac{(q-q_0)^{-1}} {|q-q_0|^2} \,Dq\,f(q).$

Proof. The proof given largely follows the outline given by Sudbury. For an equivalent statement translated from the language of differential forms into the framework of vector fields and their integrals, see the proof on page 12 of Deavours.

First, let $K$ be a 4-parallelepiped in $\mathbb{H}$ and let $q_0$ be an element of its interior $K^\circ$. Define the function $g$ so that

$g(q)=-\partial_r\left(\dfrac{1}{|q-q_0|^2}\right)=\dfrac{(q-q_0)^{-1}}{|q-q_0|^2},$

where

$\partial_r=\dfrac{\partial}{\partial t}-\dfrac{\partial}{\partial x}i-\dfrac{\partial}{\partial y}j-\dfrac{\partial}{\partial z}k.$

Clearly, then, $g$ is real differentiable at every point $q\neq q_0$ in $K$. Moreover, a simple computation verifies that $\overline{\partial}_r g=0$ except at $q=q_0$. From other results (see Sudbury), it follows that $d(g\,Dq\,f)=0$ for all $q_0\neq q\in K^\circ$ by regularity of $f$.

Next, we use the dissection method from Goursat’s theorem. In particular, consider replacing $K$ by a smaller parallelepiped $\hat{K}$ for which $q_0\in\hat{K}^\circ\subset K$. Because $f$ has a removable singularity at $q=q_0$, it follows that $f$ is continuous at $q_0$, whereby it follows that choosing $\hat{K}$ small enough allows approximation of $f(q)$ by $f(q_0)$. Finally, let $S$ be a 3-sphere with center $q_0$ and Euclidean volume element $dS$ on which

$Dq=\dfrac{(q-q_0)}{|q-q_0|}dS.$

Such substitution is possible by way of a change of variables in the definition of $Dq$. This change of variables results in the following:

$\begin{array}{rcl} \displaystyle\int_{\partial K}\dfrac{(q-q_0)^{-1}}{|q-q_0|^2}Dq\,f(q) & = & \displaystyle\int_{\partial K}g\,Dq\,f(q) \\[1.5em] & = & \displaystyle\int_S g\,\dfrac{(q-q_0)}{|q-q_0|}dSf(q_0) \\[1.5em] & = &f(q_0)\displaystyle\int_S\dfrac{dS}{|q-q_0|^3}\\[1.5em] & = & f(q_0)\left(2\pi^2\right)\end{array}$

Dividing both sides by $2\pi^2$ yields the exact statement of the result. $\,\square$

The above is part of a (rather long) paper I finished last semester that’s in the infancy of a future as a potential pre-print/publication. I’m not going to link to it until I’ve done a bit more with it. Worth noting, too, is that this is the first time I’ve ever used WordPress’s incarnation of LaTeX, meaning there are many kinks to work out and much tinkering to do. I’m hoping that I can fit this blog into my plans for the rest of the semester, though, so maybe that will force me to do more with it.