Version 1.2

Added a short overview of the overall proof early on. It’s simply too long to not know how it all fits together going in. More overhauls might help other areas, but this was the most important.
I also want to talk about the topological proof, which is also closely related to Euler’s Formula. Future versions may include this.

During my vacation time, I found a couple days for an old project I’ve been wanting to complete for a while now:

Euler’s Formula and the Fundamental Theorem of Algebra

EDIT: V. 1.2 is up now.

For a few years now, I have wished that the Fundamental Theorem of Algebra had been written up at a much lower level, and that someone had given me such a write-up at, say, age 16. I am not sure whether this first draft is a success or not, but I have attempted to write what sort of thing I had in mind, complete with remarks that are included in undergraduate lectures, but not always present in undergraduate textbooks. The audience for this is supposed to be relatively general, though familiarity with Taylor Series would be an advantage.

The interplay between Euler’s Formula and the FToA is strong. Although students are often introduced to both in high school, the FToA is never proved (but Gauss sure was smart! He proved it five different ways! Wish you were him, don’t you?), and Euler’s Formula is reduced to the status of a bit of numerological trivia I like to call Bieber’s Theorem.

It’s not longish because it’s that hard. It’s long because I’m attempting to make it that easy. Though I suppose there’s always a little bit more to it than I thought that there was. Depending on the audience, a paper 2-3x as lengthy can often take a quarter as long to read, depending on how useful the expanded explanations are.

The theorem has a few other proofs, one of which in particular I want to sketch in a later draft. The proof presented in the first draft is much closer to a fully rigorous complete proof than the others. Some content has been left out, but it is content that is easier to fill in details for in a self-contained sort of way, when I get around to it.

Myself, Izabella Laba, and Alexander Volberg have kicked off the semester with a new paper, Buffon’s needle estimates for rational product sets. As we try to generalize old results without simultaneously weakening them, we run into a new obstacle – the parasitic lamprey.

Not that lamprey.

What we mean is the Linear Multi-Polygon Relation (LMPRe). That is, a vanishing sum of roots of unity. A very old subject that is not entirely understood, one could call it a “living fossil” of sorts.

In particular… Well, you’ll have to read the paper, but in any case, they are legitimate obstacles to Buffon’s needle problem. But as we look for ways to “sidestep parasitic lampreys”, we lose more and more freedom to ignore terms in our equations, as the parasitic lampreys “suck away” large chunks of our “good sets”. Eventually, the existing method will either break down, or it will succeed because of our new insights into cyclotomic divisors of {0,1} polynomials. I’m hoping for the latter, naturally. But we will see.

The debate over the appropriateness of the terminology is underway here.

I just got around to adding one more draft to that page which I completed some time ago, but hadn’t posted yet. Mostly minor changes, but one place about Poisson kernels wasn’t written properly before. This is more or less what’s actually been printed, except for the location of L’Animal Anonyme.

I think I’m done talking about NSA for now. But here’s all of it in one place, and a couple links for further reference.

1. “Free Ultrafilters” cigarette ad.

2. Slide show, intended to also be self-explanatory survey (for people who’ve been through “Baby Rudin”) For a while, NSA looked kind of redundant and pointless, but the proof of Tychonoff’s theorem was nice and short. It seems like things that depend heavily on the Axiom of Choice might be good places to apply NSA. Whether it’s a good idea to have a whole functional analysis course using NSA in places might be questionable. The slide show itself came out to only a half hour (going fast), so if I were to lengthen it, I might have added a proof of Robinson’s Theorem. The Transfer Principle might have been too much, but if I understood it better, I would at least include some big-picture ideas of why it’s true. A lot of *-transferred sentences are pretty obviously true, though, like the nonstandard Archimedean property.

3. Two posts on Terry Tao’s blog: [1] [2].

4. Remark – NSA isn’t that fashionable, and large portions of it seem sort of almost circular – you take the sequence arguments you already know and love, encapsulate it in some special notation with equivalence classes, and write up the same proof with different notations. The most important inequality will probably have to be transferred back and forth at some point, but it will certainly feel very familiar but scrambled a bit. It could be that either you need to get deeper into it to get any real payoff, or it could be that Axiom of Choice management in some settings is simplified by embedding the AC in the definitions of your basic objects.

Non-standard Analysis

April 4, 2011

These slides are based on A.E. Hurd, P.A. Loeb – An Introduction to Nonstandard Analysis.

Check them out. They’re supposed to be moderately self-contained, but standard analysis basic fluency is somewhat assumed.

EDIT: New draft. Some more examples, figures.

EDIT2: Presented this at the student analysis seminar. Found some typos, naturally. Fixed for posterity and for great justice.

Free Ultrafilters

April 2, 2011

Admit it… this is the first thing you thought, too.

(That’s Abraham Robinson, founder of Non-standard Analysis. I will have more to say about this soon.)

This is the one I just submitted to the graduate college today. Hopefully they’ll accept it.

Updated list of final drafts.

Thesis slides

March 21, 2011

I can’t imagine what you’d need them for, but here are some slides.

The first chapter of the thesis itself is a bit more self-explanatory, maybe.

This might be the one I submit. I just need to add a disclaimer about the use of color images (the print copy will be black and white), but while I’m posting it here, I may as well spare you that irrelevant disclaimer.