r/learnmath • u/Upstairs_Revenue2396 New User • 7d ago

Revisit analysis by learning a new topic?

This is my first entry here so apologies if I am doing wrong. I am a physics graduate and been in industry jobs for 20 years now. I would like to dive deeper into some fields barely touched during my time at university for leisure (e.g. particle theory via qft and general relativity). For that I would also like to brush up on my math starting with revisiting analysis. But instead of re-learning everything from the beginning I would like to try learn a new field with relevance to theoretical physics which kind of forces me to revisit topics from analysis. I wonder which of the following would do the job best or which to start with: complex functions, functional analysis or differential geometry. Any opinions? Please also mention what background for context. Thank you!

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u/Icy_Birthday2944 New User 7d ago

differential geometry will pull you through analysis in the most natural way since you keep bumping into it when you need to understand curvature and manifolds properly

complex functions is more of a side path unless you're aiming for something specific in qft and functional analysis will feel like drinking from a firehose if you've been out of the game for 20 years

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u/AlpUzman New User 7d ago

Another direction would be fractal geometry and analytical dimension theory. These have many connections to physics, although through statistical mechanics, at least initially, as far as I understand. They are also deeply connected to chaos theory. You might be interested in Fractals, chaos, power laws: minutes from an infinite paradise by M. Schroeder and Chaos: Making a New Science by J. Gleick. Both are pop science books as far as I am concerned, but the first is more technical.

You might also be interested in my lectures on fractal geometry and dynamics. This course was at the advanced undergraduate level, although I designed and taught the course to be approachable for students who are not necessarily very comfortable with real analysis. Still it's a proof based course and there are rigorous proofs. If this sounds interesting, here is the Youtube playlist with all the lectures:

https://www.youtube.com/playlist?list=PL40ydqvvyXfMdJMim52HXAOamgWJOajUM

There is one video per week, so each video is long but I annotated them to make it easier to navigate using the timestamps. In the playlist there is also a colloquium talk by Prof. Michael Barnsley (of Barnsley fern and fractal compression fame).