Geometric invariant theory for geometers I am trying to learn "Geometric invariant theory" like it was introduced by Mumford. But I do not have a strong background in algebraic geometry since I work in geometric topology and geometry. 
So my question is if there is any nice reference where people explain geometric invariant theory from a geometric viewpoint. In particular, I am looking for a good reference where the analogies between algebraic geometry and differential geometry are pointed out.
 A: If you just want to get a feeling for invariant theory, here are some books that aren't necessarily comprehensive but nevertheless are enlightening at a more leisurely pace as compared to GIT, which would be useful for someone who isn't as familiar with algebraic groups and algebraic geometry:


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*Santos and Rittatore - Actions and Invariants of Algebraic Groups: Minimal prerequisites. A very gentle introduction to some aspects of invariant theory, including some motivation via Hilbert's 14th problem. This book also contains most of the required theory of linear algebraic groups.

*Dolgachev - Lectures on Invariant Theory: This takes a more geometric viewpoint and might be something you are interested in. This only requires some basic knowledge of algebraic geometry.

*Schmitt - Geometric Invariant Theory and Decorated Principal Bundles: this might also be interesting if you are interested in the geometric applications and the related geometry, though I haven't looked into this book very much, but Part 1 does contain a fairly leisurely-looking introduction to GIT
There is also Popov's and Vinberg's treatise "Invariant Theory" in the Ecyclopedia of Mathematical Sciences Volume 55 (Springer) which contains a good summary of the classical results in characteristic zero.
A: I would recommend a look at chapter 8 of the third edition of Geometric invariant theory by Mumford, Forgarty and Kirwan.  It describes a connection between GIT and Hamiltonian group actions in symplectic geometry.  
(edit) You may also like Moment maps and geometric invariant theory by Chris Woodward.
A: For a "more classical" point of view:
"An introduction to Invariants and Moduli " by S. Mukai.
Fogarty J. "Invariant theory" (Benjamin, 1969)
For a introduction to Mumford's:
P. E. Newstead, "Introduction to Moduli Problems and Orbit Spaces" 
Anyway you have to learn (before or after) a Gothendieck-categorical background:
Fondements de la géométrie algébrique (Grothendick).
The Hilbert schema chapter is very important (need the Hartshorne "Algebraic Geometry" as base)
Or in more gentle way: 
Fundamental Algebraic Geometry. Grothendieck's FGA Explained  - Fantechi B., Göttsche, L., Illusie L. 
A: Read the survey in my article and go over the references therein. It is written with exactly similar intentions you have asked for.
Kalafat, Mustafa, Geometric invariant theory and Einstein-Weyl geometry, Expo. Math. 29, No. 2, 220-230 (2011); addendum ibid. 37, No. 1, 92-95 (2019). ZBL1250.14034, MR2787619.
