Let’s say you have a picture, a piece of music, or a movie that you want to store on your computer. You can do this without taking up your entire hard drive, but why? Because there’s a way to look at each of these things so that they appear very simple: Imagine someone is making a movie of you reading this. You’re just sitting there. Maybe a fly is flying around the room, but not much is changing. That means each moment of the movie looks a lot like the one right before, and this makes it very easy to store the entire movie on a computer.
The fact that pictures and such are so simple allows you to do other cool stuff. Let’s say you find your favorite movie in the back of a second-hand store, but when you watch it at home, different marks pop up every now and then. Since movies are so simple, you can use a computer to fill in what you can’t see, and make it good as new.
Here’s a more exciting way to use the fact that pictures, movies, and other things are so simple: Suppose a doctor wants to take pictures of your insides to find bad things. This will take a while because the bad things are small, but your body is large, so a lot of pictures need to be taken. However, if they take fewer pictures in the right way, they can still figure out how the rest of the insides look by noting how simple they must be. This means you don’t have to lie still for so long while they take pictures.
These ideas are pretty new, and they have changed the way we approach some problems. Suppose you want to know what a very small thing looks like, but it’s too small to see, even with a glass. This happens a lot when you study things that keep people from getting sick — if we could only have a picture of these things, we could probably help people get better. But these things are so small that the best you can do is hit them with light and see what happens. Say the light passes through and makes a funny shadow on the wall. This shadow says a lot about the picture of the thing you’re trying to see, and using the ideas from before, you can actually figure out this picture.
In this case, it’s not that the picture is simple, but rather we can “lift” the picture to make it look simple. To see this, sit back and think about the picture you’re trying to see. Now imagine a large table, each cell of which showing that picture in different colors. Maybe the picture in the top left cell has more blues, and one in the middle has more reds. For every possible picture, you can form a table like this (call it a “lifted” picture). But you can also imagine other tables — tables with very different pictures in each cell, not just the same picture with different colors. With these in mind, lifted pictures are rather simple as tables (the different cells almost look the same). In fact, since they’re so simple, it turns out you can use the ideas from before to figure out the lifted picture (of the picture you want) from funny shadows on the wall. We hope this will help with taking pictures of small things so we can keep people from getting sick.
I’ve thought about each of these problems, but the way I work on them is different from most. My work usually deals with tables of numbers which are short and wide, and we want to put numbers in the different cells of a table in a good way (in some sense, these numbers tell the doctor how to take fewer pictures of your insides). It turns out that one way to do this well is to roll dice to pick the numbers for the cells, but I try to pick numbers without using dice (this seems to be hard). When I work on these problems, I sometimes think about large groups of points with lines between some of them. In some cases, you can use whether two points have a line between them to help build good tables of numbers. I also like to think about whether some questions are too hard for computers to quickly find the right answer, and when they can answer quickly, I like to help computers answer as fast as possible. Perhaps the prettiest thing I think about is groups of straight lines in space that meet at a single point but do a good job of filling the space. There are many ways to try to fill the space, but for some ways, the best line packing is rather beautiful. Such a line packing can also be used to build a good short and wide table of numbers.