Fields Medalists and Topology and Thesis Research and…

Today, I spent the day at IAS, listening to Alex Eskin talk about Teichmuller dynamics.

I don’t know why, but I somehow struggle on some deeper level when it comes to that topic. These talks always start relatively similarly with billiards and the (non-)existence(?) of periodic orbits thereof before providing a dictionary between billiards and Riemann surface theory, an introduction to basic notions in ergodic theory (Ergodic, Uniquely Ergodic…), and then – apparently at some point when my brain shuts down – there’s really deep stuff including conjectures by Fields medalists, etc. etc. Somehow, I understand all the pieces before brain shut-down, but even so, the shut down always seems to happen and leave me scratching my head and wondering wtf happened during.

Maybe it’s a tumor.

I’ve been focusing  more on stuff about universal circles. In particular, I’ve found some other documents online that summarize the Calegari-Dunfield paper a bit, and I’ve been using Calegari’s wonderful book to help get new views on things. It’s slow, but it’s progressing way better than it ever has.

Last week, there were three Minerva lectures at Princeton University by Maryam Mirzakhani. The creative ways in which she applies and broadens the scope of hyperbolic geometry is staggering, and as much as I’d like to say I understood a lot of things, I understood very small fragments of a handful of things. It was an amazing experience that I’ll cherish for a long time, but man – I was so tangibly outclassed during that it was almost embarrassing. Wonderful, but (almost) embarrassing.

Besides that, I’ve been working: Mostly boring monotonous things for Wolfram with the exception of breaking Wolfram|Alpha today, and then finally some progress on fixing the very badly-done FSU Financial Math pages. It’s a lot happening, but it’s all mostly enjoyable and I like being kept busy, etc. Always good.

Unfortunately (or perhaps fortunately for my progress on things that matter), I haven’t typed up any more interesting proofs or anything. At some point, I hope I can blog regularly without feeling like I’m missing out on more important things but honestly? Now is not that time.

I hope this finds everyone well, and if I don’t see you again first: Happy holidays!

S^3 (the most basic prime manifold) is prime

So a while ago, I was reading Hatcher’s notes on 3-manifolds. In there, he defines what it means for a manifold to be prime and states, casually, that the 3-sphere is prime. He later says that it follows immediately from Alexander’s Theorem as, and I quote: Every 2-sphere in bounds a 3-ball. And that’s it. Done.

Wait, what?!

Elsewhere, Hatcher expands his above statement: …every 2-sphere in bounds a ball on each side…[and h]ence is prime. Again, though, it isn’t accompanied by anything, and while this is clearly a trivial result, I just couldn’t see it for the longest time…I knew that it followed from a number of things, e.g. the fact that is the identity of the connected sum operation, that is irreducible (and that every irreducible manifold is prime), that one gets the trivial sum by splitting along a 2-sphere in which bounds a 3-ball in , etc. Even so, I didn’t want to leverage some enormous machinery to deduce the smallest of results and what I really wanted was for someone to tell me what I was missing. So I never stopped thinking about this, even after moving forward, until finally – it just clicked!

I figure other people who are as visualization-impaired as I may benefit from seeing this explained in greater depth, so in lieu of typing a blog post containing something new and attention-worthy, I figure I’d share this instead. Details after the break.

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!

Interesting proofs series?

One of the things I’ve always enjoyed about moving farther in mathematics is being able to understand/process old information from a variety of different contexts. Of course, I’m still no Benson Farb (if you’ve never heard Benson talk about the insolvability of the quintic using algebro-geometric machinery, you’re really missing out), but I really do enjoy learning new ways to visualize things I’ve known for a while.

For example, there’s a novel trapezoidal proof of the Pythagorean Theorem that makes sense if you know simple things like the area of a right trapezoid:

http://eigenstuff.com/post/128236655616

. Seeing stuff like that makes me happy, and it makes me realize:

It could be fun/worthwhile to begin a blog post series of Interesting Proofs which outlines various “interesting” proofs of fundamental(-ish) mathematical facts. I’m imagining a recurring series of blog posts connected only by the fact that they’re novel proofs of things that most people have known since adolescence.

You guys should chime in to tell me whether you’d be interested in anything like that. Pretends people actually read this nonsense

Week 3, Day 1 or Properties of Lie Brackets

Today is the first day of the third week of the semester. I know that counting down like this is going to make it seem longer than it already seems, but it seems so long that I can’t seem to help remaining conscious of the precise time frame I’m dealing with.

Such is life, I suppose.

I’ve noticed an amusing trend in my page views involving solutions from Dr. Hatcher’s book, namely that I’ve been receiving an abnormally-high level of page views lately, almost all of which seem to center on those solutions. I guess that means that the semester has started elsewhere too and that people find topology as difficult and frustrating as I do.

For those of you who fit this bill and who are reading this right now: My plan is to start doing more problems ASAP, so that page might get its first update in quite a while.

Today, though, I want (read: need) to talk about differential geometry. In particular, we spent some time in class last week discussing the Lie bracket and its properties, and because we have a derivation of one particular property, I wanted to take the time to put that here for my own benefit.

One week in

So, I’ve officially survived the first week of my (second) second year as a Ph.D. student. It’s hard to imagine sometimes that I’ve been a grad student for over three years.

I’m such an old man.

It’s been even less easy than I’d anticipated, too, which is nicht spaß; sadly, too, this is the first week where teaching responsibilities are actually a thing (we get week 1 off from such things), so now it’s difficult and time-consuming. Jajajajaja.

Rather than making this entry all about Yours Truly, I figured I’d pit stop in and use some of my (*gasp*) unoccupied time write up a little spiel I saw for the first time in my Riemannian geometry class last week.

In topology, there’s an idea called invariance of dimension which can be stated in many different contexts, situations, etc. This can be modified in the case of manifolds with differential structures, and because the idea of the proof seemed a bit cool, I decided to throw it up here for you guys. Throughout, the notation denotes a manifold of dimension with an associated differential structure.

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 of continuous functions defined on a compact set which (i) is closed with respect to pointwise multiplication and addition, (ii) contains the constant functions and separates points of , and (iii) is closed as a subspace of where, here, denotes the space of continuous functions defined on equipped with the sup norm: . Associated to such an is the collection of all nonzero homomorphisms ; one easily verifies that every maximal ideal of is the kernel of some element of and vice versa, whereby the space is called the maximal ideal space associated to . Also:

Definition: A point in is said to be a peak point of provided there exists a function so that and on .

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

A central problem in the subject of uniform algebras is to characterize among the uniform algebras on .

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).

Revisiting, and something light

Well, today was the fourth day of my official employment with Wolfram. This job is absolutely amazing; I couldn’t be more stoked. It’s saddening, of course, that I’m not spending my days engulfed in the books I’d been looking at earlier in the summer; it’s also a bit saddening that I have less time to spend with you beautiful people. Regardless, things are pretty amazing and overall, I couldn’t be happier.

I wanted to take some time to swing by here and say something, though, and fortunately for me, my TA duties this semester have yielded me something of precisely the right balance of depth (or lack thereof) and length (or brevity) to be fitting for tonight’s (this morning’s) pit stop.

On Wednesday (July 3), I was sitting in a precalculus class, doing Wolfram stuff and vaguely listening to what the instructor was talking about at the time. The topic? Logarithms. As someone who’s solo-taught precalculus before, I know precisely how little students understand – or like – or care – about logarithms. I also know how much we try to convince them to believe without their understanding which – among others – has to be a primary cause for their confusion and disdain.

One thing we try to get them to believe? The change of base formula. The change of base formula says that given a base , , the quantity is equal to the quantity

where , .

This information is shared with students at that level largely so they can feel comfortable evaluating an expression like in their calculators given only the capacity to utilize and functionality. Surely, they never really need to know it.

And then I realized…

In all my years in mathematics, I’ve never actually seen this rule proven before. That, of course, sparked my interest, and so I went back to my office and jotted the (surprisingly simple) proof on my whiteboard just to appease my curiosity. Here’s the way that goes:

Proof of The Change of Base Formula.
Let so that . In particular, then, it follows that for , ,

.

I think I may force my next round of precalculus students to know that. It keeps ’em fresh, on their toes, where they gotta baayayaeeee….

Did anyone just catch my reference to ‘Heat’? Or, rather, my reference to Aries Spears’ reference to ‘Heat’?

I hope everyone’s 4th was safe and that there were only minimal injuries due to inebriation, explosives, and general tomfoolery.

Until next time….

Half-June

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 fails to be differentiable at 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 . 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 , , 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 . 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….