I hear all these topics like Lie Algebra, Symplectic Geometry, Hopf Algebra, Clifford Algebra, Quantum Groups, Homology, Poisson Algebra, Semi-Riemannian Geometry and the list goes on.. As of the moment, I honestly have no idea what they are. But probably these topics are related or are even sub-fields of each other. Now as I've browsed through books/papers discussing these topics, I can say that such kinds of stuff (mathematically rigorous and related to physics in an abstract manner) really interest me. I already find it a pastime to browse through such books in my university library even though I don't really understand anything my eyes set on. Maybe people will find this weird or odd. But doing this gives me some kind of motivation and excitement to tell myself "wow, I want to say to myself that Iunderstand this stuff". ^^ I want to get to a level where I can understand and do research in these topics. My present mathematical background is of a beginning graduate student in physics. This includes analysis, vectors, ODE, some PDE, complex analysis, special functions, linear algebra, calculus of variations, and some tensors. Most of these topics are quite "calculational" or algebraic in approach. The paths towards these topics are quite clear and one can just follow the standard prerequisites in sequence. I want to make a transition from this kind of mathematics to a more geometric and abstract one like the topics I mentioned above. But for the "higher" topics above, I simply don't know where to start and what path/s of prerequisites to follow. Can someone help me with this? Thank you very much!