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!
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
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Polygon Mesh Processing, especially Differential Geometry slides
(Discrete) Differential Geometry slides