> Group Theory i still have a copy of Fraleigh -- a first course in abstract algebra. not sure if it is the best, but it did the job. I really enjoyed the two

> Group Theory i still have a copy of Fraleigh -- a first course in abstract algebra. not sure if it is the best, but it did the job. I really enjoyed the two real analysis & functional analysis courses when i attended university, but alas, the reading material for these courses were notes produced by each lecturer, they're not available as published books. > 5).Topology(cup = donut) i recall covering the material for "cup = donut" style results in an algebraic topology course, i think in 3rd or 4th year, after first being drilled with 2-3 years of pure math including real analysis, including lots of basic stuff about topological spaces, continuous and smooth functions, measure theory, some abstract algebra, etc. 3). Probability 4). Statistics MacKay's "Information Theory, Inference, and Learning Algorithms" is a great read: http://www.inference.org.uk/itila/book.html http://www.inference.org.uk/itila/book.html 5). Optimization Sign up for this course: https://www.coursera.org/learn/discrete-optimization https://www.coursera.org/learn/discrete-optimization edit: you didn't mention PDE, but i still have a copy of Evans -- partial differential equations serving as a monitor stand. you probably want to have 3 years of pure math including linear algebra, lots of real analysis & some differential equations under your belt first.