algebra

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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Study Plan, tentatively, + Algebraic Geometry Exercises

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So I think it’s probably best to have a rotating study plan schedule that allows me to do certain topics on certain days. So far, I’m thinking of having a rotation that looks something like:

Differential Geometry -> Algebra -> Clifford Stuff -> Algebraic Topology (optional),

and since yesterday was (unofficially) differential geometry day, I’m going to spend today doing algebra.

First order of business: Eisenbud and Harris. And, since I’ve been meaning to write down some of the solutions to exercises I’ve passed, I guess I’ll do that here.

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Movin’ on up (and down) (and up) (and down)….

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I decided to spend as much time as possible today studying after a few days of being nonchalant with it. I went to bed early-ish last night, woke up early-ish this morning, and hit the books with very few breaks in between.

As it turns out, this recipe gave me ample opportunity to learn new things. Who woulda thunk?

I started with my professor’s paper on M-conformal Cliffordian mappings. I made it through a couple more pages of that guy, verifying theorems and assertions as I went along. Then, right as I was on the precipice of real math, I realized how mentally taxing my morning had been and shifted direction a bit.

My new direction: Dummit and Foote. I started section 15.2 on Radicals and Affine Varieties. About 2/3 of the way through that section, I realized I really really need to learn some stuff about Gröbner Bases, so I decided to forego that and keep the ball rolling. I spent a few minutes flipping through Osborne’s book on Homological Algebra and upon realizing I’m far too underwhelming to tackle that guy, I shifted focus again to Kobayashi and Nomizu.

Of course, K&N has kind of worn out its welcome around here, and upon reading a page or two, I decided to break out a different Differential Stuff book instead. My target? Warner’s book Foundations of Differentiable Manifolds and Lie Groups. This book is a nice amalgam of Geometry and Topology, as evidenced by its somewhat nonstandard definition of tangent vectors. Maybe I’ll share some of that later.

Finally, I decided to shift my focus back towards Algebraic Geometry, whereby I broke out Eisenbud and Harris’s book The Geometry of Schemes and tried to stay afloat. Much to my own surprise, I was able to make it through fifteen-or-so pages without floundering completely and/or ripping all my hair out, so I’m hoping that maybe the information I’ve picked up in other places has done me some good. We’ll see for sure moving on.

Overall, I think I cranked out about 45-50 pages of reading today – and all (well, most) on material that’s completely new. It ain’t a Fields Medal, but it ain’t a flop either.

Until next time….

What dreams are made of

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So for those of you who haven’t noticed, I’ve been doing algebraic topology a lot lately.

Pretty much any time I’m not doing life stuff or helping parent my son, I’m on my laptop with a black Pilot 1mm G2 (aka the greatest pen of all time) and a legal pad and I’m frantically trying to figure out things…or I’m trying to convince myself that I finally understand the differences between the definitions of retract and weak deformation retract and strong deformation retract and that the differences aren’t just subtle nuances but are actually deep and significant and deep.

I’ve been doing algebraic topology a lot lately.

Whenever I spend lots of time doing the same thing in succession, I find my brain existing in a very unilaterally-focused state where nothing really registers outside the thing I’m doing lots of. In such moments, I find that I’m constantly working on  said thing when I’m walking in the park or when I’m grocery shopping or when I’m sleeping…

…I’ve been doing algebraic topology while sleeping….

Pretty much every night this week, I’ve snapped awake only to realize that the thoughts pre-snap consisted of homotopies or of commutative diagrams or of compositions of compositions of compositions of….

…I’ve been doing algebraic topology a lot lately.

Presently, it’s 6am. I’ve been awake for 3 hours because – let’s face it – I almost never sleep anymore. The first thing I did when I woke up at 3am? Log in to WordPress and break out the legal pad.

The good news: I’m making progress. The bad news: I’m starting to fear I won’t be able to make progress on any of the other topics I’d set aside for summer studies.

I think I’m going to be more pro-active in remedying the bad news there, though. As soon as I get done transcribing the solution (I’m sure you guys have noticed my solutions page, yes?) I just figured out, I’m going to break out a different book (perhaps even on a different subject?) and try my hands at that.

Ideally, I’d like to expand my solutions page (singular) to solution pages (plural) as I work through other meaningful texts in my life. I’d really like to take a stab at doing a large portion of Munkres’ problems, though I’m about 101% sure I’ll never be able to do all of them; I’d also like to take some time to add some solutions to differential something because – truth be told – I’m a differential something-er at heart. This is inescapable.

On the other hand, I really don’t want to ignore my original passion – algebra – though posting solutions there would be more challenging. Why, you ask? Well, working through Dummit and Foote has always been a goal of mine, but Nathan Bloomfield already has an absolutely tremendous WordPress blog set up for precisely such a work-through. So, long story short: Even if/when I do work through those problems, posting them seems completely pointless.

Now, however, I fear I’m just rambling.

Good night, world.

Algebraic Topology Notes

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Algebraic Topology Notes

A colleague of mine just shared these with me. This seems to have a more general introduction (i.e., an introduction with less assumed knowledge) than does Dr. Hatcher’s book. 

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.

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.