Basically you should know that people devote their whole life to one sliver of these subject areas unless they are absolute geniuses. You would be surprised how

Basically you should know that people devote their whole life to one sliver of these subject areas unless they are absolute geniuses. You would be surprised how little working knowledge a let's say expert in statistics has of category theory (you can pick basically any pair of subjects that you listed and could say that). Your list also seems highly idiosyncratic, so let me propose a different list based on what a typical undergraduate degree in mathematics could look like in Germany. First semester (half a year): - Analysis I (rigorous study of real analysis, based on parts of Rubin (a graduate book in the US but never mind) and German equivalent books) - Linear Algebra I (rigorous study of linear transformations, fields, vector spaces, simple group theory, eigenvalues, eigenvectors) - Numeric Analysis 0 (introduction to matrix algorithms (QR, LL, Eigenvalue decomposition, Householder), simple optimisation) 2nd Semester - Analysis II (multivariate real analysis, derivatives, Lebesgue Integration, maybe differential forms) - Linear Algebra II (Simple Ring theory, R-Modulus, some more group theory, depending on lecturer simple homological algebra, chain complexes and spectral sequences) - Numeric I (ODE solver methods (Runge-Kutta etc.)), alternatively Statistics (you need a good foundation of Lebesgue theory for that) alternatively Optimisation I 3rd Semester - Analysis III (simple analysis and integration on manifolds, differential forms, stokes theorem, some simple functional analysis, many different things possible here, depends on the lecturer) - Algebra I (only if you totally want to go this route), more Ring theory (commutative Algebra), Field Extensions, Galois Theory (one of my favourite things in mathematics, super beautiful, but virtually unknown outside mathematics) Maybe - Numeric Analysis II (Simple PDE solving, finite differences, finite elements, needs Analysis II / Analysis III) alternatively - Complex Analysis (Holomorphic functions, residuals, etc., depending on the lecturer this can mesh really well with what you already learned in Analysis I and II) Beyond this point there is lots to explore, many people diverge in their interest pretty rapidly from this point. Basically what I'm trying to say is that you absolutely need to no Linear Algebra and some Algebra rigorously for practically any field of mathematics (or physics) and the same goes for Real Analysis. Everything is build on top of that. I would ignore more foundational stuff like Mathematical Logic / Set Theory etc. because anything that you need from there will be introduced as you go along. The same goes for anything applied, unless you have a foundation in rigorous real analysis and linear algebra, you won't be able to appreciate anything applied beyond a superficial level (for example all the decompositions in numerical analysis have nice geometric interpretations). In terms of pure mathematics the first two semesters are absolutely necessary, if you don't invest massive amounts of time in these foundations, there is no chance to understand anything more advance, even though you might fool yourself that you do. In order to test whether you only think that you understood something or actually understand something, you absolutely have to do lots of exercises. Ideally there should be someone more experienced that checks some of them and gives you feedback. Once you've mastered these fundamentals I suspect you will have been exposed to more advanced concepts along the way and maybe have a better judgement of what you find interesting (beyond what is known in popsci circles). As books I recommend Linear Algebra Done Right, Sheldon Axler http://linear.axler.net http://linear.axler.net and Baby Rubin Principles of Mathematical Analysis, Walter Rubin they should not be your only references, but are pretty solid jumping points I think. Whenever you don't understand a step in a proof try to fill details in yourself or consult other books.