What's a quadratic map and how do I invert it? I'd like to get just the mechanics of multivariate-quadratic cryptography, so I'm trying to understand the definitions involved.  From Wolfram's website, I get that a quadratic map is for example a recurrence of the form $x_{n+1} = 2 + (x_n)^2$.  I will need an initial value too, right?  I choose $x_0 = 3$.  What's the inverse of this map and how do I get it?

I'm guessing an inverse of a quadratic map would be the transformation that takes $x_n$ back to $x_{n-1}$ so the using the example given in the first paragraph, $x_0 = 3, x_1 = 11, x_2 = 123$.  So given $x_2 = 123$, I can return to 11 by subtracting 2 and taking square root: that is, $x_{n-1} = \sqrt{x_n - 2}$ is the inverse map of my example.

It's just a guess.  I've never seen these things before.  Can't find any reference that explains this in baby-steps.

Also, I'm wondering what would be an example of a quadratic map whose input is say a vector.  My example is a 1x1 vector.  What would it look if my input would be a 2x1 vector?
In cryptographic applications a quadratic map is generally conceived as a system of n quadratic equations in n variables.  Each equation is generally represented as an n-long vector of n variables x and a matrix Mi (for equation i of the system.  The equation is then:

(x^T)Mi(x) = ai

Your are then given the n matrices Mi, and a resultant vector of the ai.  The inversion you are looking for is to find a vector x such that:

(x^T)Mi(x) = ai for all i

If the Mi are chosen at random then the problem of finding a solution to this problem is believed to be intractably hard (NP-hard in fact).  If the Mi are specially chosen (not random at all) one can make inversion easy.  Mulivariate Quadratic cryptography tries to take an easy to solve quadratic map and transform it into a seemingly difficult to solve map.

If you look at the Unbalanced Oil and Vinegar schemes:

https://en.wikipedia.org/wiki/Unbalanced_Oil_and_Vinegar

you get an idea of one such system.