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.)

So my thesis is coming along. I hope it gets easier, but at the moment, it’s slow going.

Here’s the first partial draft for you to peruse.

Chapter 1 is brand new, but it’s entirely expository. All of my published research so far is about the same problem, so I went ahead and unified the notation and spent a great deal of time laying out background. Chapter 1 of my thesis is the most thorough and expository paper I have written so far about the broad strokes and main ideas of my research. Mathematicians (pretty much anyone who knows Holder’s inequality, say) wanting to know what’s going on with my research in medium-broad strokes without slogging through an actual research paper will hopefully find this part to be a light read.

Chapters 2-4 will be in ascending order of difficulty. Chapter 2 is a bit long, but it’s mostly due to choices in favor of more exposition. I might slowly wean the reader off of this as I go along. Chapter 3 will repeat 2 to a large extent, and chapter 4 will have so much technical bulk to it that I’ll have to choose points of exposition sparingly.

Please let me know if anything looks wrong or badly formatted. As far as formatting goes, there are some TeX tricks I haven’t looked up yet but need to to fix some things, so if you see anything that needs to be made prettier, assume that I don’t know the code.

Our new, improved paper is online now. Not only is the estimate much sharper thanks to one nice little lemma using Hardy space theory and another using something we call “analytic tiling”*, but you may even be able to read this one.

A 3^{-n}-neighborhood of a 1-dimensional Sierpinski gasket decays in Buffon needle probability at least as fast as C/{n^p}.

* – “Analytic tiling” is essentially our fancy name for the observation that if three unit complex numbers sum to 0, they form an equilateral triangle, and therefore, their third powers sum to a number of size 3. The name is motivated by the role tiling played in Laba and Zhai’s paper on product Cantor sets.

Cantor

cantor

Story: Though perhaps not a villain in the strict sense, there is no denying the wickedness that is Cantor’s jams. You might recognize him as the final boss of Karaoke Apocalypse, the monk with a slammin’ death metal sound that astounded punk rocker Mittag-Leffler, who declared that “it’s about one hundred years too soon for such righteous awesomeness to descend on the Earth.” Lead singer David Hilbert replied that Cantor death metal was here to stay, and that “No one shall expel us from the Paradise that Cantor has created!” At which point Cantor was crowned The Supreme Cardinal, which could never be exceeded. But the next move shocked everyone.

“Friends, rockers, fans, musicians,” Cantor addressed the screaming, enraptured masses. “Let it be known that there will always be a greater Cardinal! We must prove it… again and again, without end! I hereby declare a perpetual Karaoke Apocalypse!”

Who will be the next Greater Cardinal? Will it be you?

Powers: See for yourself:

Signature song: Continuum Hypothesis, probably the most awesome death metal song ever, but friggin’ impossible to sing because of a line that goes like this:

unsingable

…and then back down again in reverse. It’s questionable whether it’s even possible for a human voice to rise that rapidly, but then again, no one knows whether Cantor is a mere mortal.

Weaknesses: None, really. The final stage of Karaoke Apocalypse is pretty much friggin’ impossible. However, there is a choice to make. You have to win at Continuum Hypothesis if you want the good ending, but a lesser ending is available if you play the easier Power Set instead. In this ending, you become the next Greater Cardinal, but the next year, a new challenger defeats you by playing the Power Set twice, playing better than ever before the second time, and reaching a new, higher level of Cardinality. Your fame is short lived.

Rumor: While it is unknown whether any gamer has ever gotten the good ending without using a cheat code, Kurt Gödel claims that he has done it, but Paul Cohen argued that since it’s pretty much friggin’ impossible, no amount of evidence could ever conclusively prove that it had been done. But at the same time, if anyone could do it, it’s probably Kurt Gödel, and it’s equally impossible to prove that he didn’t do it, either.

Pro tip: Practice makes perfect! You might have to do the Power Set several times before you become the Greater Cardinal, so don’t give up!

Update

August 15, 2009

Convolutions and the Weierstrass approximation theorem has been updated and expanded. Now it also covers the “other” Weierstrass approximation theorem, taking a one-page tour of selected topics from the first chapter of a harmonic analysis text along the way for motivation. Also, a solution to the heat equation in one dimension has been briefly provided without proof in the remarks section.

So now you all should know what a convolution is and have some idea that they’re important.

In other news, a summary of the noodle paper is under review at Comptes Rendus Mathematique. If they choose to publish it, I’d like to link it here and say a few words about it having to do with some recent thoughts of mine concerning the lacunary circular operator, as well as possible connections to Furstenburg sets and incidence geometry. In particular, a comparison of the solution to the joints problem with the estimate taken from the multilinear Kakeya conjecture indicates to me that there is some reason to think that maybe incidence geometry won’t provide the answer to the sharpness of the LloglogL bound on the lacunary circular operator. I’d like to say something at length, but I want to share the paper with you first.

Here they they are, the write-ups for the MRC Summer School in Harmonic analysis, Carleson theorems, and
multilinear analysis
. Somewhere in there is my summary of Bennett, Carbery, and Tao’s On the multilinear restriction and Kakeya conjectures. (And here’s an obligatory link to Terry Tao’s blog.)

The complex analysis involved is relatively elementary, but intricate. But at any rate, Dr. Alexander Volberg and I now have an analog of the Nazarov, Peres, Volberg result, adapted to the Sierpinski gasket with a moderate loss.

Here it is. I have it labelled as a Level 3, but this one will probably keep you up late regardless of who you are or how much you know.

I have a new short paper which proves the Weierstrass approximation theorem in a manner quite similar to how it is done in Rudin’s Principles of Mathematical Analysis, but with much more motivation for the technique. In particular, the underlying idea, the convolution (by an approximate identity) is given center stage rather than relegated to the status of a magic trick. If you have read and vaguely understood the details of Rudin’s proof but are mystified as to how anyone could have thought of that or what the main idea is, then I hope that this paper can help you.

This paper is probably a bit advanced for a level 2 paper, but in principle, the prerequisites aren’t too intense, and I have some hope that an undergrad could get something from this without having taken Advanced Calc first – maybe a little help from an advanced friend or a professor might be necessary in this case.

Click here for the paper.

Edit: Version 2-0, with some introductory harmonic analysis and Weierstrass for trig polynomials. (8/15/09)

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