Could someone explain to me what a Teichmuller Space is? In the simplest terms possible, for someone who understands the basics of Manifolds, Topology, but barely any of the more complicated topics.
I've been using the following:
http://homeowmorphism.com/articles/18/Link-Hyperbolic-Teichmuller-MCG
https://schapos.people.uic.edu/MATH549_Fall2015_files/Survey%20Nathan.pdf
https://www.msri.org/workshops/739/schedules/19292/documents/2419/assets/23053
But I'm still confused.
I'm trying to do a reading on these but I seem completely dumbfounded at every attempt.
I am first trying to understand how the Teichmuller space of the Torus $T^2$ is $\mathbb{H}$. Could someone explain why this is and how it follows from the definition?
I think this would allow me to understand this best. 
Thanks.
 A: Here's a way to think about the torus. 
You know, I'm sure, that if you take a square and glue opposite sides appropriately then you get a torus. 
And you probably also see that if you do the analogous gluing with a rectangle then you'll also get a torus. 
And in fact even if you do the analogous gluing with a paralellogram then you'll still get a torus. 
So, let's take a general parallelogram, with one pair of opposite sides labelled "$a$" and the other pair labelled "$b$", and with appropriate arrows on the sides to specify identifications. But now we need to normalize the parallelogram. First, we'll replace our parallelogram by a "similar" parallelogram, scaling its size so that the $a$ side has length $1$. Next, we'll rotate and translate the parallelogram in the coordinate plane so that its lower $a$ side is the line segment $[0,1] \times \{0\}$, and so that the parallelogram sticks up into the upper half plane. With these choices, the origin $O$ is the lower left vertex of the parallelogram, and the upper right vertex $P$ is a point in the upper half plane. In fact, the original choice of parallelogram determines $P$, and $P$ determines the parallelogram. 
In summary, the choice of a point $P$ in the upper half plane determines a unique parallelogram (with one side along $[0,1] \times \{0\}$ and $P$ being the vertex opposite $O$), and that parallelogram determines a torus by gluing sides.
Furthermore, the conformal structure on that parallelogram descends to a conformal structure on the glued up torus (because the four vertices of the parallelogram become a single point on the torus, and the four angles at those vertices add up to $2\pi$). Also, the $a$ and $b$ sides determine a basis for the fundamental group of the torus.
Now there's something to prove, namely that the choice of $P$ gives a one-to-one parameterization of the conformal structures on a torus with a basis for the fundamental group. That's why we say that the upper half plane is the Teichmuller space of the torus.
