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
Well, if you're serious about applied mathematics-and serious in that you don't just want "reciepe" books,rather applications that build on the meaty theory background you have-then you should avoid such texts and try and locate books that don't avoid theory,but merely downplay it. Those are the "real" applied mathematics textbooks.
You definitely need to strengthen your background in differential equations, you're pretty much dead in the water without that in your background. The best beginning textbook I know on differential equations is George Simmon's Differential Equations with Applications and Historical Notes. It is one of the most beautiful, richest textbooks you've ever find on any subject-it covers all the basics of ordinary and partial differential equations using only basic calculus and a context of not only physical applications,but incredibly detailed and scholarly notes on the historical founders of the subject. It's a must have for any mathematical library. The one flaw the book has is that it doesn't explicitly use linear algebra,but rather old fashioned linear equation notation. But by all means,don't let that chase you away from one of the best books there is on any subject. It really is the book to start with.
Another subject you'll need to be very comfortable with in terms of applications is linear algebra and there's no better book from an applied standpoint for this then Gilbert Strang's classic Linear Algebra With Applications. Be careful with this because there are several versions of this book and the one you want is this one-the other one is much gentler and less substantial. This is the one with the real red meat by one of the great applied mathematicians of our era-with a host of applications you won't find anywhere else,both standard and exotic. While you're at it, check out Strang's calculus text online at his website. It may be the best basic calculus text there is bar none-with more applications then just about any other text in it's wieght class. And best of all, it's free!
For financial mathematics,you really need a good background in probability theory. It sounds like you have a good working knowledge of the brute theory, but that'll only take you so far. You need to back up and learn some of the basics first, particularly the applications. One of the best books that currently exists on this subject for the beginner, which contains a host of applications, is Grinstead and Snell's Introduction to Probability. It provides not only a comprehensive undergraduate course in the subject,but it provides many applications, particularly to finance and the social sciences. Best of all,it's available online for free for download.
Lastly. a very good addition to your training will be advanced calculus with applications. Not "applied advanced calculus",which is mostly receipes in metric spaces. I'm talking about a relatively recent and very important trend in elementary analysis textbooks where real-life applications are included alongside a careful presentation of the theory of calculus. This is where the artificial separation of pure and applied mathematics-which is there for purely profiteering reasons in academia in my opinion-is taken down and both are discussed in an advanced calculus course in equal measure and import. There aren't many of these books yet, sadly-but the ones that do exist are excellent.The best one that currently exists is Jeffery Cooper's Working Analysis,in which a thorough course in advanced calculus of one and several variables is peppered with applications to physics, biology, numerical analysis, economics and so much more. This is a terrific text that most students of mathematics should have in thier library. A bit less advanced but also quite good is Donald Estep's Practical Analysis of One Variable. This is really an honors calculus book masquerading as an analysis book, but Estep writes beautifully and presents one variable analysis in complete unity with literally hundreds of applications, many fascinating insights and an interestingly original presentation. This one is well worth having as well,particularly for you since it's emphasis is on differential equations.Lastly and more sophisticated then either of the previous texts is Real Analysis and Applications: Theory in Practice by Kenneth R. Davidson and Allan P. Donsig.This isn't really a full blown analysis course but rather a supplementary text following up a standard intermediate analysis course based on baby Rudin or Charles Chapman Pugh's book. That being said,it has an amazing set of applications, including Fourier analysis and wavelet theory. It's definitely worth a look.
That should get you started. I'm sure the others will recommend other good texts as well and if I think of any others,I'll edit this post and add them. Good luck!