Terence Tao

Some other work that I spend a lot of time on is to prove what are called structure theorems or inverse theorems that give tests for when something is very structured.

So some functions are what's called additive.

Like if you have a function that has natural numbers, they're natural numbers.

So maybe, you know, two maps to four, three maps to six, and so forth.

Some functions are what's called additive, which means that if you add

If you add two inputs together, the output gets added as well.

For example, a multiplying by a constant.

If you multiply a number by 10, if you multiply a plus b by 10, that's the same as multiplying a by 10 and b by 10 and then adding them together.

So some functions are additive.

Some functions are kind of additive, but not completely additive.

So for example, if I take a number n, I multiply by the square root of 2, and I take the integer part of that.

So 10 by square root of 2 is like 14 point something, so 10 up to 14.

20 went up to 28.

So in that case, additivity is true then, so 10 plus 10 is 20, and 14 plus 14 is 28.

But because of this rounding, sometimes there's round-off errors, and sometimes when you add A plus B, this function doesn't quite give you the sum of the two individual outputs, but the sum plus or minus one.

So it's almost additive, but not quite additive.

So there's a lot of useful results in mathematics, and I've worked a lot on developing things like this, to the effect that if a function exhibits some structure like this, then there's a reason for why it's true, and the reason is because there's some other nearby function which is actually completely structured, which is explaining this sort of partial pattern that you have.

If you have these inverse theorems, it creates this dichotomy that either

the objects that you study either have no structure at all, or they are somehow related to something that is structured.

And in either case, you can make progress.