Reddit mentions of Elementary Applied Topology

You should look up Professor Robert Ghrist , he is extremely visual and such an eccentric person he makes his own font, graphs for his math books, you will not be disappointed.

Try this book: https://www.amazon.com/Elementary-Applied-Topology-Robert-Ghrist/dp/1502880857

He also has a series in Multivariate Calculus which you can check out. The topic isn't advanced but the books are beautiful!!

(Sorry I dont know how to insert a link on reddit as I'm on mobile)

To go a different direction, Elementary Applied Topology by Robert Ghrist might be a book that is different than most that your boyfriend has. If he's already big into Algebraic Topology he probably already has a few texts on the topic. This one would likely be different since it focuses on the applications of algebraic topology.

I've already asked on stackexchange but I didn't get a satisfying answer (link). Ouch! It seems it was even put on hold :( I guess I don't know how to write questions for stackexchange.

Anyway, I think that if I want to do research I'll need to learn math properly. The problem with applied math is that a superficial knowledge is often enough to understand classic methods (which are written in textbooks) but not to approach cutting-edge methods (which you can only find in very theoretical papers).

Here's my extensive list of books to read (in this order):

Diff. Geometry

1. Apostol: Mathematical Analysis (as a review)
   
2. Herstein: Abstract Algebra
   
3. Lee: Introduction to Topological Manifolds
   
4. Hatcher: Algebraic Topology
   
5. Arnol'd: Ordinary Differential Equations
   
6. Lee: Introduction to Smooth Manifolds
   
7. Lee: Riemannian Manifolds: An Introduction to Curvature
   
   

   Topology (there are prerequisites in common with Diff. Geom.):
   

   
   

8. Ghrist: Elementary Applied Topology (link)
   
9. Edelsbrunner: Computational Topology: An Introduction (link)
   
10. Edelsbrunner: A Short Course in Computational Geometry and Topology (link)
    
    See here for applications of topology to data analysis.
    
    Let me know if you have any suggestions or thoughts and thanks for the help.