While studying physics as a graduate student, I took a course at the University of Waterloo by Achim Kempf titled something like Advanced Mathematics for Quantum Physics. It was an extraordinary introduction to pure mathematics for physicists. For example, in that course we showed that by taking the Poisson bracket (used in Hamiltonian mechanics) and enforcing a specific type of non-commutativity on the elements, one will get Quantum Mechanics. This was Paul Dirac's discovery. After taking his course I left physics and went into graduate school in pure mathematics.
(I don't believe he published a book or lecture notes, unfortunately, though I just emailed him.)
In transitioning from physics to mathematics, I learned that the approach to mathematics is different in a pure setting than in a physics setting. Mathematicians define and prove everything. Nothing is left unsaid or stated. There is an incredible amount of clarity. Even in theoretical physics, I found there to be a lot of hand-waving and ill-defined statements and lack of rigor (which hilariously caused me a lot of anxiety). Overall, though, Mathematicians are focused on understanding and proving relationships between abstractions, whereas physicists are more interested in using these abstractions as tools. Therefore, the approach is very different: mathematicians don't care what the application is, they only want to understand the object under consideration.
Nevertheless, for a theoretical physicist looking to get a firm background in mathematics, you want to have the following core mathematical concepts, which will provide a foundation to explore any avenue:
- Linear Algebra
- Functional Analysis
- Topology
But the real list is something like:
- Set Theory
- Group and Ring Theory
- Linear Algebra
- Real Analysis
- Topology
- Functional Analysis
- Measure Theory
- Operator Algebra
Set, Group, and Ring theory are used extensively in physics, especially in Hamiltonian mechanics (see Poisson Bracket). Real Analysis and Linear Algebra are needed as a foundation for Functional Analysis. Functional Analysis could be described as an extension or marriage of Ring Theory, Group Theory, Linear Algebra, and Real Analysis. Therefore, many concepts in functional analysis are extended or used directly from Real Analysis and Linear Algebra. Measure Theory is important for the theory of integration, which is used extensively in applied physics and mathematics, probability theory (used in quantum mechanics), condensed matter physics, statistical physics, etc.
Topology and Operator Algebras are used extensively in advanced quantum mechanics and Relativity. Specifically, Algebraic Geometry is studied extensively in String Theory, whereas Topology is used extensively in General Relativity. Operator Algebras are an important area for understanding advanced Quantum Mechanics (ever heard someone talk about a Lie Group before?)
Some canonical text-books I would recommend:
- Linear Algebra: Advanced Linear Algebra by Steven Roman
- Real Analysis: Real Analysis by H. L. Royden
- Functional Analysis: A Course in Functional Analysis by John B. Conway
- Measure Theory: Measure Theory by Donald L. Cohn
Those are some decent text-books. I would say: give yourself two years to digest that material. Don't be hasty. Remember: mathematics is about definitions and proofs. Do not expect to see "applications" in any of those books. Just understand that the concepts are needed in advanced physics.
Unfortunately, though, I don't know of any text-book that forms a direct bridge between the two. If Achim Kempf had published his lecture notes, those may have worked, as essentially, he was doing just that.
Good luck!
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.)
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I am also a mathematics student who likes physics. My advice would be:
I think here at ETH there's a pretty good group doing mathematical physics, if you want to try to apply and to move to Switzerland.
Also general advice: If you are interested in a topic in physics, always look if Tong has written something on it. His lecture notes are usually very good.