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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So today wasn’t really my day, overall. Generally speaking, I woke up feeling congested and nasty, I spent the whole day with a migraine, and I was only not-lethargic for about four hours total overall.

Unsurprisingly, then, I realllly couldn’t force my brain to do any real math. For that reason, I completely avoided reading new things and instead typed up the expository analysis entry during the middle part of the afternoon. I ended the night doing some solutions for Hatcher – Chapter 0 of which I’m hoping to knock out soon to begin Chapter 1 – and drawing (really really) poor diagrams in MSPaint. I’ve emailed Dr. Sjamaar from Cornell to ask how he gets his diagrams drawn, but thus far have heard nothing back.

Pleeeeeeease don’t leave me hangin’, Dr. Sjamaar: My blog is evidence that I’m in desperate need of your resource knowledge!

Anyway, it’s almost 2am and I’m about to call it a night.

Auf Wiedersehen.

Continuous, Nowhere Differentiable Functions

A few days ago, I posted about a conversation I had with my friend L. We spent some time catching up and, in so doing, spent a little time talking about this particular plot of space on the grand ol’ internet. He mentioned a couple blog topics for me to consider and also asked if I was contemplating research in algebra/topology; looking back, the fact that L’s an analyst, the fact that I have very few analysis posts here, and the fact that the topics he suggested were analysis topics made me realize I really do need to do a better job representing my enjoyment for analysis. Consider this entry step one of that, perhaps.

Rather than spending a bunch of time researching stuff I’d never seen before, I decided to type up a little summary thing of an interesting article I found online when I was a master’s student. For a little perspective as to why this particular article is important, we’ll have to take a trip into so-called higher education and examine the topic that generally serves as most people’s introduction to grown up mathematics, i.e., calculus. A (really really over-simplified, primitive, simplistic) synopsis of calculus can be summed up in this way: Calculus is a class that abstracts the unknown variable quantities thrown at you in Algebra I/II into unknown variable quantities that themselves can vary by way of limiting arguments.

And that’s basically it: In America, calculus is really just taught as the algebra of limits. As such, some basic limit-intrinsic notions such as continuity, differentiability, and integrability are touched on / hinted at, and at the end of fourteen weeks of being fooled into thinking you’re finally understanding what math is, you’re sent on your way. For most, that’s the end of the story, but for a self-selecting few, the journey through mathematics continues, and new techniques / ideas get thrown at you in hopes that they’ll stick and that you’ll be able to use them for something special….

…and at the same time, for that self-selecting few, it’s not uncommon at all for certain somewhat obvious questions to go unasked through the years. For example: It’s invariably shown in Calculus I that f(x)=|x| fails to be differentiable at x=0 because of the sharp edge there. It stands to reason, then, that combinations and scalings of the absolute value function with two, three, four, etc. sharp edges would fail to be differentiable at two, three, four, etc. values of x. This idea isn’t a hard one to grasp for a calculus student. But then the next question: How many points of non-differentiability can a function have? Or how about, Construct a function that fails to be differentiable at infinitely many points. Most students would be quick to adapt previous examples and notice that a saw-blade function with sharp points at each value x=n, n\in\mathbb{Z}, proves the existence of functions with infinitely many points of discontinuity. Again, no big deal.

So what, then? Can we have functions that are non-differentiable at uncountably many points? How about functions that are differentiable nowhere? By and large, these are ideas that escape lots of students – even students nearing the end of a traditional math major curriculum at an average American institution. I know this because I was once one of those students, and have since taught several myself: I see how students fail to comprehend non-differentiability and even the everywhere-discontinuity of functions like g(x)=\chi_{\mathbb{Q}}(x). It’s simply something that fails to register for the average student.

Coincidentally, it doesn’t always stop there. L was actually telling me a story once about a statistics professor we both knew who claimed, absent-mindedly, that most continuous functions are differentiable. That, of course, is a big statement, and for the inquisitive audience-member, the natural response is: Prove it. Hence the aforementioned paper…. Continue reading

The Intersection of the Collection of Dreams and the Collection of Maths, or Dynkin π-λ Systems

I’ve posted before about how easily my sleep can be dominated by math stuff after a hard day or thirty of being cooped up in an office, grinding away at theorems and postulates and proofs with hardly a break in the mix.

Over the summer, the same thing happens after only a medium-hard day or three.

I woke up twice this morning, about two hours apart, and both times I was thinking about a random piece of mathematics not related to anything I’ve been actively studying recently. When I finally awoke a third time – this time, for good – I of course couldn’t remember it at all.

Then, finally, I sat in silence and forced my synapses to make connections they didn’t want to make and eventually, after a solid twenty minutes of mental strain, it all came flooding back in.

This is an exposition about so-called Dynkin (π-λ) Systems and the corresponding Dynkin π-λ Theorem. Feel free to stick around.

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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.