Just now I'm studyng for an exam in Toric Geometry, and the professor suggested "Toric Varieties" by Cox, Little and Schenck, "Introduction to toric varietis" by Fulton and "Symplectic toric manifolds", which are notes by Cannas De Silva about a Symplectic point of view in Toric Geometry. I think that Fulton's book is beautiful, but the most complete is surely Cox, Little and Schenck's.
From my own experience, I will advise you against every book of mathematical methods written specifically for physicist. From my point of view, is better to learn about mathematics from mathematically written books (it sounds so obvious but is not). For example, many people like Schultz, Geometrical methods of mathematical physics, but I prefer to learn about the common topics in Singer, Thorpe, Lecture notes on elementary topology and geometry. (I don't say it is not a good textbook, I only say I find difficult learning things on books written in a pretty informal way.)
The most complete work about methods of mathematical physics is probably
- Reed, Simon, Methods of modern mathematical physics,
that covers functional analysis, Fourier analysis, scattering theory, operator theory.
Since you are interested in cosmology, the best review on Loop Quantum Gravity is that by Thomas Thiemann,
- Thomas Thiemann, Modern and canonical quantum general relativity,
a 900 pages review, equipped with about 300 pages of mathematical methods (mathematical appendices are not a textbook however, but a collection of necessary results, eventually explored in some depth). References therein are very useful also.
Many people like
- Deligne et al., Quantum fields and strings: a course for mathematicians,
that joins a good part of your requests. (I haven't read it, however, I know it since is "famous".)
A celebrated book on methods of classical mechanics, concerning manifolds too, is
- Abraham, Mardsen, Foundations of mechanics.
Another is
- Choquet, Bruhat, Analysis, manifolds and physics.
All Arnold's books are always a great choice. (he wrote about ergodic theory and geometrical methods for differential equations, among the other things.)
There are a lots of more specific books, e.g. dealing with mathematical structure of quantum mechanics, but many of those are more and more specialized and is better to have very clear the general theory before try to get more involved into dangerous subjects such as, to say, quantum field theory. Once one has a strong background, the best opera on the subject of field theory probably is
- Zeidler, Quantum field theory,
an enormous amount of things (Zeidler style!) that covers all of the subject. Another excellent text on field theory is that of Haag,
- Haag, Local quantum physics.
EDIT. I'd like to add some book I've discovered more recently and I think fit very well:
Streater, Wightman, "PCT, Spin and all that",
Teschl, "Mathematical methods in Quantum Mechanics",
Bogolioubov, Logunov, Todorov, "Axiomatic Quantum Field Theory",
Lansdman, "Mathematical concepts between classical and quantum mechanics".
Best Answer
First of all, Mirror Symmetry is huge. As you said, there are many fields involved. To know how much you need to know depends on where you're working. Roughly, one can divide the whole mathematical aspects of mirror symmetry into two categories. 1) Analytic and symplectic, (mainly (complex) differential geometry/symplectic geometry) 2) Algebraic (containing Algebraic geometry, homological algebra, etc.) I've been around with people who're doing Donaldson-Thomas theory (One Algebraic geometry side of Mirror symmetry) and personally willing to know more about homological mirror symmetry these days. Unfortunately, I don't know much about the analytic aspect which is related to Gromov-Witten theory.
The connections between these two categories are related to conjectures, one is called MNOP conjecture and the other interesting one is the homological mirror symmetry program.
As for the books and references, if you want to know just very little about what's going on, you may find Mirror symmetry written by leading mathematicians as well as mathematical physicist useful.However, Mirror symmetry and Algebraic geometry by Cox and Katz satisfies me more than the previous book (because obviously it's more mathematics.)