Physical (GR) Differential Geometry? I am looking for problem lists or books which contain open problems in the area of mathematics motivated by physics. Ideally, I am looking for questions asking about which reduce to some calculation from General Relativity, which might shed light on a purely differential geometric question/conjecture.
If it helps, I have a bachelors degree in mathematics, but am currently doing a masters in Theoretical Physics. My primary motivation is to write my dissertation discussing such a topic. Within my masters studies, I have completed a course on Riemannian Geometry based on do Carmo's book, and a course on GR based on Schutz's book.
 A: See 
Alan Coley, "Mathematical General Relativity" (arXiv:1807.08628)

Abstract. We present a number of open problems within general relativity. After a brief introduction to some technical mathematical issues and the famous singularity theorems, we discuss the cosmic censorship hypothesis and the Penrose inequality, the uniqueness of black hole solutions and the stability of Kerr spacetime and the final state conjecture, critical phenomena and the Einstein-Yang--Mills equations, and a number of other problems in classical general relativity. We then broaden the scope and discuss some mathematical problems motivated by quantum gravity, including AdS/CFT correspondence and problems in higher dimensions and, in particular, the instability of anti-de Sitter spacetime, and in cosmology, including the cosmological constant problem and dark energy, the stability of de Sitter spacetime and cosmological singularities and spikes. Finally, we briefly discuss some problems in numerical relativity and relativistic astrophysics. 

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Particularly interesting (and pressing) is the problem of inhomogeneous cosmology that Coley's text considers in section 3.5. On p. 28 he writes:

An important open question in cosmology is whether averaging of
  inhomogeneities can lead to significant backreaction effects on very
  large scales.

This innocent sounding statement acknowledges that the "Backreaction Debate" in mathematical general relativity has remained inconclusive:
The standard model of cosmology assumes that it is accurate to neglect matter inhomogeneity on large cosmic scales. Under this assumption, the model famously finds a positive "cosmological constant". But various informal arguments as well as computer simulations suggests (not fully conclusively so far) that cosmic inhomogeneity may have a non-negligible effect on cosmic evolution which may account for some or even all of the effective cosmological constant.
This shows that there is immense phenomenological impact behind the problem of inhomogeneous cosmology in mathematical general relativity.
Of course there are other open problems, too. See Coley's survey!
