Roadmap for maths self-study for a physics undergrad wanting to pursue research in maths I am a second year physics undergrad. After spending about an year exploring mathematical physics (QFT,solitons, etc), I have realized that maybe theoretical physics isnt my passion after all, and I am thinking of changing my emphasis to maths, so that I would be able to pursue research after my undergrad in math. I would like to know, how should I go about studying maths so as to complete the maths undergrad syllabus which would enable me to apply for grad school after 3 more years, and also pursue a specialized research area simultaneously. Currently, I don't have anything on mind specifically. I have studied a bit of topology, and group theory both from mathematical physics textbooks. I am thinking of something on the lines of algebraic number theory, or algebraic geometry. What are the prerequisites for studying each of these? What should I study so that I would be able to study these by this summer? Also, what is the maths that I should know other than that requires for these areas. Please bear in mind, I come from a college, where shifting to a formal maths course is now impossible, and I have very little choice to chose my courses. Most of them are compulsory physics courses. 
I have heard algebraic number theory and algebraic geometry require lots of prerequisites, which I may not have the time to do. Could someone recommend similar fields where I am in a position to do research with less pre-study.
 A: As DonAntonio mentioned, there really isn't a choice of what to learn if you really want to pursue mathematics. Most of the undergraduate mathematics topics you will have to tackle eventually and most, if not all, of it you will use at some point. With that being said, I understand that you are asking which subjects to prioritize.
Assuming you are going into the two fields you mentioned, then Abstract Algebra in general would be a great idea to learn. You said that you've learned group theory from a physics book. Being a physics major myself, I can quite safely assume that you don't know quite enough group theory (especially at a rigorous level). I have always been told to study math from mathematicians and physics from physicists. It seems only logical that way. Brush up on your group theory, pick up a little ring and field theory and also linear algebra if you have not already learned it. Both of the fields you are interested in rely heavily on abstract algebra.
On the other side of the spectrum, analysis is quite central. Both real and complex analysis will be useful to you. If you are interested in algebraic geometry, then you will want to know more topology. If you are interested in algebraic number theory, then studying elementary number theory certainly wouldn't hurt.
There really isn't a priority order for the subjects I've listed above. Personally, I would just pick the ones which interest me the most and start from there. You may want to tackle several subjects concurrently and slowly pick away at them. Or you may want to focus on a single one. I will not debate here which is better, likely that will depend on you and how you like to learn. 
A: For self-study this might sound odd but I'd suggest you to study chaos theory. It connects physics and mathematics in an interesting way.
I find this book really intriguing. It will inspire you to find your field in mathematics. Along with the physics, you will find chaos theory related to the mathematics you already know, for instance topology.
This is a quite different approach on your main question, there are already great answers for the rest of the questions. Hope you find this different approach useful.
