A bunch of circles and squares

Question

The grey region is simply the region you see below.

The answer will give you 10 letters.

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US's:

TSDFCK

RTOYLG

AWQMRZ

PUWSTU

SRNXEL

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Hint 1:

Added "mathematics-tag". Nothing fancy, but still some mathematical knowledge is required.

Hint 2

For the bottom part; You will have to do some small calculations (or maybe it's not even necessary. For some, just a look at them can be enough.) But first of all you have to make sense of the top image. What can it possibly represent?

Answer 1

Original Partial Answer (Complete Answer below this one):

Weeks ago I figured out that the gray section "IN" means

irrational numbers (I hope).

thinking that the values in order are

??, Real Numbers, Irrational Numbers, Rational Numbers, Integers, Natural Numbers

At first I thought the slopes of the lines might be important but I couldn't make anything out of those, but when I saw in the comments that

U.S. means unit square, I looked at the line length values

The values for each image are:

Image 1: $\sqrt{2}/2$, $\sqrt{5/4}$, $1/2$, $\sqrt{2}$, $1$, $\sqrt{2}/2$
Image 2: $\sqrt{5/4}$, $\sqrt{5/4}$, $1$, $1/2$, $\sqrt{2}/2$, $\sqrt{2}/2$
Image 3: $\sqrt{5/4}$, $\sqrt{5/4}$, $\sqrt{2}/2$, $1/2$, $\sqrt{2}/2$, $\sqrt{2}/2$
Image 4: $\sqrt{2}$, $1$, $1/2$, $1/2$, $1$, $1$
Image 5: $\sqrt{2}$, $1$, $\sqrt{2}/2$, $\sqrt{5/4}$, $1$, $\sqrt{5/4}$

I thought that each image's line marking (I through IIIIII) could correspond to

the letter in the string to the right of each image, in order.

I first thought that maybe the lengths that are

irrational numbers correspond to the correct letters, but that gives me more than 10 letters, since 18 of the 30 lengths are irrational.

So I then tried to make the letters corresponding to the

12 rational numbers fit somehow, but with no luck.

Then, grasping at straws, I had the idea of choosing only the lengths that had

no integer component i.e. $\sqrt{2}/2$, although irrational, would be ignored because it was divided by an integer. This method did yield 10 letters: S,F; R,T; A,W; P; S,X,L. However, I have not been able to make anything of it, so it is extremely likely that this is not the solution.

EDIT: OK, I've finally got it!

Using some hints in the comments, we must apply

Natural Numbers (NN) to the first square, Integers (I) to the second square, and so on.

From the first square, the

fifth value, C, is a natural number (1)

From the second square, the

third value, O, is an integer (1)

From the third square, the

fourth value, M, is rational (1/2)

From the fourth square, the

first value, P, is irrational ($\sqrt{2}$)

From the fifth square,

all of the values are real numbers, so all of them are chosen (S,R,N,X,E,L)

That leaves us with these ten letters:

C,O,M,P,S,R,N,X,E,L

When we reverse the order of the last 6 letters (Thanks @user39583), we get

COMPLEXNRS (Complex numbers, which is the answer to ??)