Is differential geometry used in solid state? I'm an undergraduate in physics interested in a career in solid state. While I know that any additional math is helpful--I am on time constraints, and can only take a few supplemental classes.
That said, is differential geometry used much in solid state physics? I'm aware of things like Fermi Surfaces, but wonder if much diff. geo. techniques are actually used.
 A: On the very theoretical side of solid state physics is the holographic AdS/CFT correspondence which links strongly coupled condensed matter systems to gravitational theories. Recent work has been done on describing things like phase transitions in this theory. For example models of superconductivity in the gravity dual are promising in describing difficult condensed matter systems in terms of easier differential geometry. 
See for example the final chapters of http://arxiv.org/abs/1409.3575 
A: Yes, especially in research-level topics. There are several research groups that work with finding ways to apply differential geometry concepts to solid state systems (although condensed matter seems to be the preferred term nowadays). See for example the book by Altland and Simons, Condensed Matter Field Theory, Chapter 9 "Topology". This book is suitable for a masters degree level class with a solid state class and a quantum field theory as prerequisites. The specified chapter has a (very!) brief primer on differential geometry, but I would probably have found it unintelligible if I had not had a proper differential geometry class first. The chapter will give you the very basics of what kind of research people are doing.
Some keywords to start with are topological (insulators|superconductors|order|field theories), quantum hall effect, composite fermions. Maybe these two talks (not by me) can be a good introduction: What is topological matter & why do we care? I, II.
A: There is theory of dispersion in crystals. One can say that the differential geometry is used there. I think it is Group theory + differential geometry.
A: Differential Geometry has been useful in Mechanics of Crystalline Solids with Dislocations. See for example:
http://arxiv.org/abs/1212.5125
