Vaughn Climenhaga

Being a mere grad student who's only recently passed his comprehensive exams, I don't actually have any research papers published yet.  Hopefully that will change soon.  In the meantime, here are a few links to various things I've written at one point or another, which are mostly of an expository nature, and are certainly of varying quality and value.

A book.  Definitely the biggest project I've been involved in to date.

• Lectures on Surfaces

• Continued fractions
• Topological pressure

• Water waves

A brief digression.

If you examine the background to these web pages closely enough, and if you happen to have taken the right course in abstract mathematics, then you may find yourself exclaiming, "Why of course! It's several iterations of a Peano space-filling curve!" If you have no idea what I'm talking about, then please, read on:

The piece of paper, before you've messed with it.

Suppose I handed you a pen and a piece of paper and asked you to cover the piece of paper with ink from the pen. As a first attempt, you might break the pen in half and pour the ink all over the page - if you weren't feeling quite so destructive, you might simply unscrew the top of the pen, remove the ink from the pen, and then pour it all over the page. In any case, I would chastise you for finding a loophole in my instructions, and would then modify them to specify that you must in fact transfer the ink from the pen to the paper via the traditional method of drawing and/or writing.

The piece of paper, after you've messed with it.

At this point you would probably start grumbling to yourself, and after about thirty seconds of angry scribbling would hand me the piece of paper, perhaps slightly crumpled and torn from the application of too much force, and looking something like this:

A rather unhappy piece of paper.

Sensing your frustration, I would refrain from stating the obvious fact that you did, in fact, leave a substantial portion of the page blank. Once you had calmed down a bit, I would point out that your method of scribbling wildly relies on the fact that the pen I've handed you draws rather thick lines, and that if I gave you a pen with a finer point, you'd have to do more work. For instance, if each line drawn by the pen was one-eighth of an inch thick, then you could cover an inch of the page's height by drawing eight horizontal lines.

Letting the centre of the one-eigth inch thick pen follow the lines shown, the entire page will be covered.

If, on the other hand, the pen drew lines that were one-sixteenth of an inch thick, then you'd have to go back and forth sixteen times. And so on and so forth - if I decided to be particularly spiteful and hand you a pen that drew lines only one-millionth of an inch thick, you'd have to be there for a while to get much of the page covered, since the total length of the line you have to draw increases as the width of the line decreases. (Remember that total length of the lines times width of each line equals area of the page, assuming that you don't let the lines overlap).

Letting the centre of the one-sixteenth inch thick pen follow the lines shown, the entire page will be covered. Note that the total length of the lines drawn doubles.

Now, suppose I wanted to be not just particularly spiteful, but infinitely spiteful (if you've taken calculus, think "limit as spite goes to infinity"), and give you a pen with no thickness whatsoever, so the lines you draw have zero width. Then to cover the page, you'll need to draw a line of infinite length, which will take you infinitely long (thus the spite). But what will that line look like? Filling the page with horizontal lines worked just fine when each line had some thickness - now that they have no thickness, if you try to start each line just below of the previous one, you'll never get anywhere!

So obviously, we have to try a different tack.  Go back to the case of having a pen whose width is, say, half the length of the main diagonal (which has a length of about 1.414 inches).  Then if you follow the path shown below with the centre of the pen, you'll cover the entire page.

Letting the centre of the 0.707-inch thick pen follow the path shown, the entire page will be covered.

Now cut the width of the pen in half, so it's only one-quarter the length of the main diagonal.  By adding a couple detours to the path, you can still find a way to cover the page.  Notice that the distance traveled by the pen doubles.  Also, the new path is really just four copies of the old one, rotated and moved around appropriately.  See it?

Letting the centre of the 0.354-inch thick pen follow the path shown, the entire page will be covered, but you have to draw a path twice as long as before.

Lather, rinse, repeat - do the same thing, adding exactly the same kind of detours to the path, and you can cover the page using a pen only half the width of the one you had before.  Do you see how the pattern works?  Can you draw the next step in the progression?

Letting the centre of the 0.177-inch thick pen follow the path shown, the entire page will be covered.  Again, the path length doubles, and the new path is four copies of the old one rotated, moved around, and placed end-to-end.

If you keep doing this, adding more and more detours to your path, you wind up with a path that you can follow, using a pen of no width whatsoever, and still cover the page!  Now mind you, the path that such a pen needs to follow is infinite in length - but that's no problem, we'll allow you to move infinitely fast.  The key point is that the path is continuous - you never have to take the pen off the paper and move it somewhere else, which wasn't the case with the horizontal lines we tried at first.

This path that you eventually obtain is called the Peano curve, or the space-filling curve, and is one of the more profoundly counterintuitive mathematical objects I've run across.  Thus I decided to use it as the background for this page - the three diagrams above (plus a couple more iterations) are all part of the background pattern, each with a slightly different colour.  See if you can make them out.  I dare ya.