I get emails from undergrads about wanting to intern with me. One discussion that keeps recurring is about what maths should undergrads learn to break into deep-learning theory. I have no definitive answer to this. A message for the undergrads at UManchester would be that they should try to take MATH20401 and MATH20411 - knowing these two gives a very good grounding for seriously doing deep-learning.
I have had students who knew just basic engineering mathematics like in the book by Erwin Kreyzig - and they could read the top-notch deep-learning theory papers from Day-1. Lastly, let me list here some of the key references in mathematics that I read most carefully, during my 3 years of undergrad. So here goes that list,
If prefixed with a (Course) means that I credited a course where this was the textbook followed.
(Course) First 9 Chapters of Algebra by Artin
(Course) Most of Calculus by Apostol (Volume 1 & 2)
and (Course) Differential Calculus in Normed Linear Spaces by Kalyan Mukherjee
> This above book was quite significantly influential on my ways of thinking!
More advanced calculus from,
Calculus on Manifolds by Spivak
I picked up most of what I knew about ODEs then from,
(Course) Mathematics for Physicists by Philippe Dennery and Andre Krzywicki
(Course) Mathematical Methods in the Physical Sciences by Mary L. Boas
> Naber's book was definitely my most favorite book to read during early undergrad.
I had particularly gotten interested in Riemannian Geometry and I read many of the chapters of the following two books,
(Course) Morse Theory by Milnor
PS:
In retrospect its kind of curious to note that I had studied nothing (absolutely nothing!) of probability or statistics as an undergrad!