analysis

Implementation for Interesting Proofs (Framework)

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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!

Function Algebras, and a resounding NO! to the Peak Point Conjecture

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A former professor of mine (who I’ve discussed here previously) still to this day does some of the best work of anyone I’ve ever known. When he and I were colleagues (we existed at very-near geographical locations), I didn’t know the capacity of his amazing work, rather I just knew his work was amazing and that he for all intents and purposes should’ve been at a bigger, more prominent school.

Now, I know a bit more.

One of the things for which he was pretty famous (subjectively, natürlich) was proving a 50-ish year old unsolved problem in manifold theory; another of his fortes, though, is in the study of function algebras. That’s where this little journey takes us.

A function algebra is a family \Lambda of continuous functions defined on a compact set X which (i) is closed with respect to pointwise multiplication and addition, (ii) contains the constant functions and separates points of X, and (iii) is closed as a subspace of C(X) where, here, C(X) denotes the space of continuous functions defined on X equipped with the sup norm: \|f\|=\sup_{x\in X}|f(x)|. Associated to such an A is the collection M=\mathcal{M}_A of all nonzero homomorphisms \varphi:A\to\mathbb{C}; one easily verifies that every maximal ideal of A is the kernel of some element of M and vice versa, whereby the space \mathcal{M}_A is called the maximal ideal space associated to A. Also:

Definition: A point p in X is said to be a peak point of A provided there exists a function f\in A so that f(p)=1 and |f|<1 on X\setminus\{p\}.

One problem of importance in the realm of function algebras is to characterize C(X) with respect to such algebras A of X. To quote Anderson and Izzo:

A central problem in the subject of uniform algebras is to characterize C(X) among the uniform algebras on X.

One attempt at satisfying this necessity was the so-called peak point conjecture, which was strongly believed to be true until it was shown to be definitively untrue. The purpose of this entry is to focus a little on topics related thereto including the conjecture itself, the counterexample and its construction, and the related results (including work done by Izzo and various collaborators).

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Hodge Dual part deux

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You may recall that I deemed yesterday Differential Geometry Sunday and posted a small expository thing on the Hodge Star/Dual operator. Apparently in my cloudy haze of mathematical mediocrity, I concluded my post without having touched on the derivations I actually intended to touch on.

Sometimes I feel like I need a vacation from my vacation.

In any case, I’m going to take a stab at saying some of the things I’d meant to say yesterday, but in order to ensure we’re all on the same page, I’m going to recall what exactly the Hodge Star/Dual operator is. Then, after the break, I’m going to show some of the cool derivations that come about because of it.

Let M be a manifold of dimension \dim(M)=n and let \omega=\sum_I f_I dx_I be a k-form on M for k\leq n. Here, I denotes a multi-index which – without loss of generality – can be assumed to be increasing. Then the hodge star operator is the operator which takes \omega to the (n-k)-form *\omega =\sum_I f_I(*dx_I) where *dx_I=\varepsilon_I dx_{I^C}, where I^C is the multi-index consisting of all numbers 1,\ldots,n not in I, and where \varepsilon_I=\pm 1 denotes the sign of dx_I dx_{I^C}.

So what we said is that for the most commonly-recognized example \mathbb{R}^3 with orthogonal 1-forms dx,dy,dz and the usual metric ds^2=dx^2+dy^2+dz^2, the Hodge Star operator sends 1 to dx\wedge dy\wedge dz and vice versa, it sends dx,dy,dz to dy\wedge dz, dz\wedge dx,dx\wedge dy respectively, and it sends (dx\wedge dy)\mapsto dz, (dy\wedge dz)\mapsto dx, and (dx\wedge dz)\mapsto -dy. But the question then remains: Why does anybody care?

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Catching Up

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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.

The End of Summer, or “Orientation”

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My intention was to make my next post (i.e., this post) be some cool tidbits I’ve been pondering concerning the Gamma function (see here: http://en.wikipedia.org/wiki/Gamma_function), but alas, I’m simply too tired to squeeze that one in for today. I’ll try to do some of that later today – you know, when I’m not galavanting around Tallahassee, attempting to establish residency status. But that’s neither here nor there.

Instead, I’ll update about the most recent departmental things that have gone on, most notable of which was the first day of orientation.

The first day of orientation happened yesterday (as it’s 2:41 am now) from 10:45am until…. I wasn’t really sure what to expect, and as is usual in such situations, I was a little uneasy about being around a whole slew of new people, jumping through hoops, etc. etc. I can honestly say, though, that I was pleasantly surprised.

I woke up and saw that it was raining. Surprisingly, rain has been the ongoing theme since arriving here in Tallahassee, but we were fortunate to wake up early, leave early, and arrive with a few minutes to spare. At 10:45, we met Torrece – the Graduate Admissions Coordinator – to receive info packets. Not long after (about 11-ish), we went to “the big classroom” (LOV 101, which seats 120 students) for some general information, etc. During this, we got to “meet” (in actuality, we got to see from afar) some of the faculty who were “important for us to know,” and after much ado about nothing (or so it seemed), we went upstairs for lunch.

Lunch began at noon. It was a quaint setup with a light meal, and by way of how the setup was designed (or so I suspect), we were put right in the middle of forced socialization. As is often the case in such situations, the information I heard was largely forgotten afterwards, but what I did take away from the session is that most of the new admits were pretty nice folks. That’s always a plus.

After lunch, we dealt with a small scheduling mishap, after which we split up by area of study (pure, applied, bio and financial math are the options here) to register for our first semester of classes. During the pure math registration, I got to meet the advisor (Dr. van Hoeij) for the very first time; I also got the first WTF? You’re a Ph.D. student in what now, trying to take which classes for what? Say huh? moment of the semester when I asked Dr. van Hoeij about a scheduling quirk stemming from my having registered for two early qualifiers.

Those are always pretty funny.

Apparently, because I’m signed up for two qualifiers, I have to sign up for two additional “backup” classes in addition to the mandatory three everyone has to take. That means that for the first week or two of the semester, I’ll be devoting myself to five graduate math courses (for which I may or may not be qualified), in addition to teaching and lesson planning and whatever else may be required. I guess there’s no easing back into academia for me, eh?

Anyway, this is my current schedule as it stands now. Changes will be made later this semster:

Measure and Integration I, MWF, 11:15am – 12:05pm

Topology I, MWF, 12:20pm – 1:10pm

Abstract Algebra I, MWF, 1:25pm – 2:15pm

Complex Variables I, MWF, 2:30pm – 3:20pm

Groups, Rings, and Vector Spaces I, TR, 11:00pm – 12:15pm

I know you’re jealous of my Mondays/Wednesdays/Fridays, right? 😛

Anyway…after registering for classes, we were ordered to go to some building or other (I still don’t remember the name of it) to do “other stuff.” The other stuff was us getting assigned twelve different identifying pieces of information, having our pictures taken for the website, setting up email forwarding, etc; of course, all this stuff (it took about an hour, an hour fifteen total) happened after wandering around – still in the rain – trying to find some building I’d never been to before. The good news was that I finished everything (at 3:15pm) before I was even scheduled to start “other stuff” training (3:30pm), so I guess it wasn’t all bad.

Honestly, it was a little more underwhelming than I’d expected it to be. As I said previously, later today is going to involve me trying to remedy some domicile information, and then the rest of the week is eaten up with more registration stuff. The bad news about that is that my Saturday Sunday qualifier in Complex Analysis isn’t going to get nearly the last minute cramming it ought to; the good news? I’m not really sure yet. 😛

Stay tuned for some Gamma function nonsense, and maybe some other interesting tidbits as the journey towards academic greatness continues.

Second Friday

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So it’s Friday….

I’m starting to think Fridays are the new Monday….

In particular, this is the second Friday we’ve been in Tallahassee (which I’m sure explains the title of the post), and so far, Fridays haven’t really been kind to us. Last week was little ado about nothing, but today’s been pretty awful: Lots of difficulties with people in administrative positions in the Tallahassee government plus some car trouble that’s undoubtedly the direct result of people not knowing what they’re doing equals a whole bunch of terribleness.

It might even be bad enough for me to go full Charles Barkley on this one and call it turrrbuhlness. But that’s neither here nor there.

As is generally the case, though, mathematics (or, more specifically, the pursuit thereof) provides the only ray of decency in such circumstances.

So far this week, I’ve made some advancements in that pursuit:

  • On Monday, we spent our first real day on the FSU campus. During that trip, we visited the bookstore (erroneously, as circumstances would prove) and the student ID office / Suntrust bank location. The result? I got my ID made, my campus bank account established, and the linking of the two completed.
  • On Thursday, I received an email from the current department chair, Dr. Case, indicating that I’ve received my office early due to my signing up for early qualifiers. I don’t know what this means yet since I haven’t been to the math building directly (I plan to remedy that absence early next week), but my office is apparently in 404/6 C. I’m not sure which building that’s in or anything, but eh: I suppose I’ll figure that out soon enough.

In addition to the above, I’ve also gotten back into a somewhat-professional routine of preparing for qualifiers, meaning that I’ve spent two legit days this week preparing for the Complex Analysis exam.

I’ve been using FSU’s old qualifiers as guides, along with Conway’s Functions of One Complex Variable I to help refresh some of the pertinent ideas there. Of course, Conway is a standard text for that particular curriculum, though I’m being constantly reminded of how idiotic it was to sell back my copies of Marsden & Hoffman’s Basic Complex Analysis and Gamelin’s Complex Analysis: Both of those texts are more elementary than Conway’s but both are superior in different ways. The good news, I guess, is that I have a couple supplementary texts I can use on my own – one of which is Ahlfors’ Complex Analysis, which is said to be very good – so there’s no lack of resources for me to consider. I’m not sure it matters much at this point, though.

I may try to post the occasional problem here so you guys can see what I’m workin’ with, but that’ll have to start later. Definitely not today.

Definitely not.

New To Town

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Well, we’ve been in town for about a week now and we’re finally starting to get settled in. In the midst of settling down, I have the overwhelming burden that is studying for qualifying exams.

Each qualifying exam is a four hour test meant to evaluate whether a graduate student has the required knowledge to officially move into Ph.D. candidacy or not. In my case, I’m signed up to take two such qualifiers – one in Complex Analysis and one in Groups, Rings and Vector Spaces.

I can say, honestly, that I have been studying for these exams off and on throughout the summer; if I’m still honest, however, I’d have to say that my studying has been more off because of all the other random chores and responsibilities that have been on for what seems to have been nonstop. I’d managed to get by with that under the pretense of, Eh: I don’t even know when the exams are going to be given so I’m sure I have plenty of time. After all, ignorance is bliss.

As of a couple days ago, though, my ignorance (and, hence, my bliss) was erased.

I’m scheduled to take my Complex Analysis qualifier on Sunday, August 26 (from 1pm to 5pm) and the other on Saturday, September 1 (also from 1pm to 5pm). That means that I have no more than 18 days – indeed, two and a half weeks – until I’m put in a very stressful, very uncomfortable situation for which I should be prepared.

I should be prepared, but alas, I’m fearful that I won’t be.

My game plan is to update this thing with my journey through mathematics: It began about a week ago, when we pulled into town, and it’ll be never-ending – exhilarating, refreshing, and frightening – for the next few years. I look forward to having folks along for the ride.