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!

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

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

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)….

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

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 page(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

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

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.