There is a famous xkcd describing a space ship using only the one thousand most used words. Since then some people have tried to describe their work using only those thousand words, and even a quite useful text editor was built.
So, starting in this very line, I will only use the ten hundred words most used, and I will try to explain my work with them.
I study things that move around. I don’t study what makes those things move, or why they move, I just look at them moving and try to guess what will happen if they keep moving for a long time in the same way.
I also don’t study things that keep moving in the same direction and never come back, I only study things that move around inside some fixed space. You could say that I study spaces that move around inside.
One space of things that move around inside is the sky with all the stars as they move during one night or during one year.
There are many different ways that things can move around inside a space, some move around in weird or funny ways, some don’t really move that much, some move a lot. Part of my work is to tell how a given space is moving (I mean how the things inside that space are moving, the space is not really moving).
It is possible to use these spaces that move inside around to answer deep questions. The best known case is this: if you have a set with enough numbers (say the set has one over ten hundreds of all the numbers) then you can walk for as long as you want within that set taking always the same steps. Actually, people knew how to do this without spaces that move inside around, but using such spaces we know much more about which steps are allowed: we now know that the steps can be taken to be a power of a number with a three on top (yes, like a block with all sides the same, how is that word not more used?), or the step can be one of those numbers which can not be broken into smaller pieces without leaving rests, or even harder, a number which can not be broken into smaller pieces raised to a power of three.
Another important thing that was done using spaces that move inside around was that inside the set of all numbers which can not be broken into smaller pieces without leaving rests it is possible to walk for as long as you want, always with the same step (although you have to know for how long you want to walk before you start). An easier case had been found by Green and later he and a very well known person got the full problem done.
So far I only talked about things other people have done in my field of study. One problem I have tried hard to get (and did get something but not all of it) is this: Suppose all the numbers were painted with ten hundred colors. I want to find two numbers such that when you add them and you “times” them, you get numbers of the same color.
Seems so simple, and yet it is (so far) an open problem!