What math do I need to understand gauge theory?

[Muu9]

Note that I'm not interested in using it for physics, but instead for deep learning.

 

[jedishrfu]

First off, I have to say this ML approach looks very interesting and I need to read more about it.

Here's a blog on the recent arxiv paper:

https://towardsdatascience.com/an-e...uivariant-convolutional-networks-9366fb600b70

and the paper itself:

https://arxiv.org/pdf/1902.04615.pdf

Working backward:

They mention Algebraic Topology which leads to:

https://www.math.umb.edu/~oleg/algebraic_topology

where they mention point set topology and linear algebra as prereqs

Point set topology would require group theory, abstract algebra and set theory

So the learning order would likely be:
1) Set theory
2) Group theory
3) Linear Algebra
4) Abstract Algebra
5) Point set topology
6) Abstract topology

@Euge, @fresh_42 or @Mark44 would have a better take on this though.

 

[Muu9]

Can you see if this proto-book has similar prerequisites? https://arxiv.org/pdf/2104.13478.pdf
It seems to use quite a bit of formal signal theory

To what extent are topics like fibre, tangent, and vector bundles the focus of topology vs differential geometry? I feel like the latter would be more focused

Would this book be sufficient for the first 3 steps? Also, is this the appropriate thread for my question?

 

[jedishrfu]

having read a little bit of the book, it seems you should also take these standard math courses in parallel:

- Calc 1,2,3
-Differential Equations
- Geometric Algebra

while they may not have direct bearing on the ML topic, Calc 3 in particular works with vectors, surfaces and volumes and Differential Equations can lead to Partial Differential Equations which also does a lot with surfaces and volumes. It doesn't hurt to know more math. Geometric Algebra brings together many of the vector and tensor concepts that again may help in understanding some of the transformations that are done.

Set theory is important as a means to learn mathematical proof using the set definitions as a basis. Proof becomes more important as you move up the chain to Group Theory, Abstract Algebra, Pointset Topology and then Abstract.

When I was a Physics student I jumped into Abstract topology as a junior and was overwhelmed by the definitions and the proofs. At the time, I thought my understanding of proofs from HS Geometry would get me through. Nope. I would get lost is the nested nature of the definitions and found I couldn't prove a thing.

I guess I thought Abstract Topology would have more in common with the popular rubber sheet topology but its all very theoretical. My prof was kind and gave me some leeway, knowing that I was not a math major but that was more than forty years ago.

There's a book called All the Math You Missed but Need to Know for Graduate School by Prof Thomas Garrity

https://www.amazon.com/dp/1009009192/?tag=pfamazon01-20

that might help here too. It summarizes some of the courses I mentioned earlier as chapters in the book ie Point Set topology, linear algebra, calculus 1,2,3, abstract algebra, and parts of geometric algebra in the form of differential forms.

 

[Muu9]

little bit of the book

Are you referring to the geometric deep learning proto-book I linked to, or the abstract and linear algebra book I linked to?

Can you explain how differential forms, differential geometry, and geometric algebra are related (as courses, not as mathematical concepts)? Which order should I study them, and are any a special case/subtopic within another?

 

[jedishrfu]

Geometric algebra neatly ties together vector analysis, tensor analysis, differential forms and differential geometry.

As an example, the vector cross product only makes sense in 3D as the area of a parallel and as a normal to a surface element. However, in geometric algebra it makes sense as a bivector in 2D as well as higher dimensions.

https://en.wikipedia.org/wiki/Geometric_algebra

Here's a 40+ minute tutorial on geometric algebra: