Computational Topology and Lie Group Theory

Question

I study Machine Learning and my limited background in math is enough to understand all the popular algorithms and methods.

However, recently, Topology has been successfully applied to Data Analysis and Lie groups have been used to explain why Deep Learning works so well.

My limited math background is:

1. Analysis 1 & 2
2. Linear Algebra
3. Discrete Math / Combinatorics
4. Probability
5. Statistics
6. Convex Optimization

I'd like to study both Computational Topology and Lie Group Theory.

I didn't asked two separate questions because I suspect there might be some overlap in the prerequisites.

What books/papers/tutorials should I read and in which order?

I found some interesting recommendations but they're usually tailored to math students who have a different background than mine. I'm willing to fill any gaps in my knowledge but I'm not sure where I should start.

Answer 1

I haven't studied Computational Topology or Lie Groups, but I can maybe recommend some stuff to get you started in that direction.

First off, you will need a good understanding of Algebra and Point-Set Topology. Here are some topics you will probably need to know:

Abstract Algebra

• Groups: actions, homomorphisms, isomorphisms, subgroups, quotient groups, Fundamental Theorem of Finitely Generated Abeliean Groups
• Rings: ring homomorphism, quotient rings, ideals
• Modules
• Fields
• a little bit about Vector Spaces maybe

Point-Set Topology

• Definition of Topology
• Continuous Functions
• Homeomorphisms
• Product Topology
• Subspace Topology
• Quotient Space
• Compactness and Completeness

For books I used Dummit and Foote for Algebra and Munkres for Topology. The topics in algebra correspond to parts I, II, and III in D&F. The topics in Point-Set Topology correspond to chapters 1-3 of Munkres.

From there you can study Algebraic Topology (you should be able to start studying this after finishing the initial Topology subjects and knowing groups and rings). I hear Hatcher is pretty good (and it's free online). I'm guessing from that you will most need to study homology (I believe that's an entire chapter).

You will also need to study Differential Geometry for Lie Groups, but I don't know enough about that to suggest a book. But the books above should keep you busy for a while anyway.

But like I said, I don't know too much about the subjects of Computational Topology and Lie Groups, so take this advice with a grain of salt.