Not a mathematician, but have spent some time self-learning math. I have not gone through the following books entirely, however I've used them when needed to learn or clarify a concept:
> Group Theory
- I would look for a good abstract algebra book that also covers groups, such as A First Course in Abstract Algebra, Fraleigh or Basic Algebra, Knapp. Another popular recommendation is Abstract Algebra, Dummit and Foote.
> Topology
I think you should first look at a real analysis course and understand metric space topology for Euclidean spaces. If you have taken calculus, you can try and go through Baby Rudin. For topology specifically you can look into Munkres or Introduction to Topological Manifolds, J Lee.
> Measure Theory
T. Tao has a good book on this topic.
> Differential Geometry
Introduction to Smooth Manifolds by J Lee
> Probability
Probability Theory, Achim Klenke
> Game Theory
Game Theory by Maschler, Solan & Zamir
School admissions should be reviewed like (some) conference papers, i.e. single-blind review. The name of the student is never known when considering him/her.
It's a weird system, in Australia we all get a standard score on leaving high school and if your score is higher than the next person choosing your degree preference you win the spot. Different for medicine only AFAIK.
I'm fond of the system in Texas. If you graduate in the top 10% of your class, you get automatically accepted to any public university in the state (well, any one you apply to).
I had zero interest in extracurriculars or anything of the like, so I was never going to get into a competitive university. But I'm smart, and I have a near-photographic memory, so I aced most of my classes without having to study too much, so I was able to graduate in the top 10% and then went to UT Dallas, which is possibly the best tech school in the southwest. I'm really, really glad I went to UTD; I made lifelong friends there, the people I hung out with really helped shape my interests, I'm still running into UTD alumni everywhere I go, etc.
This might be besides the point of the post, but I'll ask anyway.
@Greg, perhaps you could talk a bit about what you think makes you a productive engineer and problem-solver. What's your workflow like, how do you approach a problem, or learning a new concept, what tools do you use, etc.
> Group Theory - I would look for a good abstract algebra book that also covers groups, such as A First Course in Abstract Algebra, Fraleigh or Basic Algebra, Knapp. Another popular recommendation is Abstract Algebra, Dummit and Foote.
> Topology I think you should first look at a real analysis course and understand metric space topology for Euclidean spaces. If you have taken calculus, you can try and go through Baby Rudin. For topology specifically you can look into Munkres or Introduction to Topological Manifolds, J Lee.
> Measure Theory T. Tao has a good book on this topic.
> Differential Geometry Introduction to Smooth Manifolds by J Lee
> Probability Probability Theory, Achim Klenke
> Game Theory Game Theory by Maschler, Solan & Zamir