I'm not a big fan of full roadmaps and reading lists. Exploring mathematics is something that can be totally different depending on where and who you are. Any serious roadmap needs to be flexible and take account of the course: reading maths is a skill (one you seem well on your way to learning btw! but still...), initially you may find actual teaching easier to grasp- and your reading should work along side that. So here's my attempt at a flexible roadmap:
1) Buy some very carefully chosen books and read them cover to cover: There's a lot of baffling books out there- even some that look really UG friendly can have you weeping by page 5 in your first year- and you only want 4-5 to start with (any more will just be too expensive and you won't get round to reading all of them- top up via the library). My recommendations are: 'Naive set theory- Halmos', 'Finite dimensional vector spaces- Halmos', 'Principles of mathematical analysis- Rudin' and 'Proofs from the book- Aigner and Ziegler'.
[These, ostensibly, cover the exact same material as the book you have decided to buy- but to develop quickly I urge you to buy more mathematical texts like these: Halmos' and Rudin's writing styles are very clear but technical (in a way that the book you are interested in will not be), and will make you a better mathematician faster than any book that tries to 'bridge the gap' ever would. I also seconded Owen's call for proofs from the book: it is simultaneously inspiring and useful as a way of seeing 'advanced' topics in action- it's something you'll keep coming back to, right up to your third year!]
2) Do all of the excercises: Or as much as you can bear to- even if it looks like it's beneath you (if you're half decent- a lot of first year will!) you will be surprised as to how much it helps with your mathematical development (and the crucial high mark you'll need for a good PhD placement). This applies to classes and your 4-5 text books.
3) Ask your tutor about doing some modules from the year above: If you've read all of those textbooks and done all of the excercises, you will be ready. Get some advice from your tutor about what would be best and roll with it (most unis won't make you take the exam, so if you don't feel comfortable you're fine). Taking something like metric spaces or group theory in your first year will put you top of the pile.
4) Keep doing all of these things: Immerse yourself in maths- keep on MO, meet likeminded people and no matter how slow the course seems to be moving, no matter the allure of apathy: keep at it. Advice for later books would be pointless now, but there will be people who can give it to you there and then (use maths forums if you want). Oh, and never rule out an area- you never know where intrigue will come from...
Best of luck,
Tom
I think that, for the majority of students, your advisor's advice is correct. You need to focus on a particular problem, otherwise you won't solve it, and you can't expect to learn everything from text-books in advance, since trying to do so will lead you to being bogged down in books forever.
I think that Paul Siegel's suggestion is sensible. If you enjoy reading about different parts of math, then build in some time to your schedule for doing this. Especially if you feel that your work on your thesis problem is going nowhere, it can be good to take a break, and putting your problem aside to do some general reading is one way of doing that.
But one thing to bear in mind is that (despite the way it may appear) most problems are not solved by having mastery of a big machine that is then applied to the problem at hand. Rather, they typically reduce to concrete questions in linear algebra, calculus, or combinatorics. One part of the difficulty in solving a problem is finding this kind of reduction (this is where machines can sometimes be useful), so that the problem turns into something you can really solve. This usually takes time, not time reading texts, but time bashing your head against the question. One reason I mention this is that you probably
have more knowledge of the math you will need to solve your question than you think; the difficulty is figuring out how to apply that knowledge, which is something that comes with practice. (Ben Webster's advice along these lines is very good.)
One other thing: reading papers in the same field as your problem, as a clue to techniques for solving your problem, is often a good thing to do, and may be a compromise between working solely on your problem and reading for general knowledge.
Best Answer
I am not sure if this is an answer or a request for clarification.
There are a lot of different topics in math (and in physics) that go by the name 'topological quantum field theory'. Beyond the initial hand-waving about '(some) observables not depending on the spacetime manifold's geometric structure', they aren't always that closely related. You can't reasonably cover all of them in a thesis.
So your first task would be narrow your list of topics down. Where do you want to focus on? Are you interested in higher category theory? Operads? Knots? Quantum groups? Vertex algebras? Group cohomology? Geometric representation theory? The existence of smooth structures on manifolds? The topology of moduli spaces? The algebraic geometry of moduli spaces? This isn't anything like an exhaustive list. (And it's probably 10 years out of date!)
I don't think you need to know physics, but it's helpful to at least try try reading physics lectures notes. I've found Greg Moore's various reviews have a mathematician-friendly perspective, FWIW.