Specialty shift and book skipping problem....

[Geofleur]

My area of specialty has been geophysics for the last fifteen years, but I have been increasingly drawn toward mathematics, not the pure sort but the sort inspired by physics. Things like investigating the mathematical structure of physical theories and looking for connections between them.

Toward that end, I decided to do some retraining/retooling, but I keep getting ambivalent about which books to read, which is a problem because books of this nature take a long time to work through! Specifically, I've been skipping back and forth between Hassani's Mathematical Physics, Frankel's Geometry of Physics, and Choquet-Bruhat's Analysis, Manifolds, and Physics.

The problem is that each of these books has pros and cons, and that I cannot, I think, master what is in all of them. I am looking for a solid basis for pursuing more mathematically inclined physics, but I need a geodesic path to it. I cannot read a million books. What would constitute this "mathematical core"? And how can I make myself stick to a course of study instead of zigzagging all over the place? To be extra clear, it's not that I cannot concentrate, it's that I keep weighing the benefits of this or that book!

 

[DEvens]

Are you sure that "optimising" is the right way to go here?

Figure out what you want to achieve. So, you want to learn about mathematical physics? Are you sure that "with the least possible amount of work" (or something like that) is an appropriate or desirable qualifier?

Usually books of this sort will have a part that is specific, and a part that is pretty similar to several other books. If you read one or two, then start on a third, hopefully you will be able to go lightly over the duplicated parts. "Oh, it's that theorem again. I know that."

As to focus. My usual method is to create a ritual. I set aside a time, place, and a few other conditions. I set up these things and start work. After a few times doing that, it becomes routine. In effect, I put on my study hat and study shirt and study slippers. And I sit in my study chair in my study. And my brain just clicks over into "time to do this stuff" mode.

 

[Geofleur]

I think of it more as an optimization wrt time rather than effort. I spend a lot of time studying. But I go very slowly indeed through Choquet-Bruhat, for example. I want to reach competency, whatever that means, in minimal time. You are right though, I do get the "I've seen that theorem" experience fairly often. Perhaps it's not as bad as I thought it was to skip around. I still get the feeling that I am being inefficient.

 

[DEvens]

Heh. If you didn't go slowly through Choquet-Bruhat, that would be remarkable. Especially the first time. That's some stiff stuff. There are not many people who could learn that stuff quickly. When a book has a quote like "Happiness is a Banach space" you know the ride is likely to be bumpy. Fun. But bumpy.

Maybe what you need is to audit a couple classes at your local university? Assuming there are useful classes at your local uni, which is not automatic. Lots of universities won't have anybody who has worked on any of that stuff.

 

[Geofleur]

I had always thought that happiness was a Hilbert space. Then I read that quote and thought, "We don't even get an inner product, man!"

 

[micromass]

My area of specialty has been geophysics for the last fifteen years, but I have been increasingly drawn toward mathematics, not the pure sort but the sort inspired by physics. Things like investigating the mathematical structure of physical theories and looking for connections between them.

Toward that end, I decided to do some retraining/retooling, but I keep getting ambivalent about which books to read, which is a problem because books of this nature take a long time to work through! Specifically, I've been skipping back and forth between Hassani's Mathematical Physics, Frankel's Geometry of Physics, and Choquet-Bruhat's Analysis, Manifolds, and Physics.

The problem is that each of these books has pros and cons, and that I cannot, I think, master what is in all of them. I am looking for a solid basis for pursuing more mathematically inclined physics, but I need a geodesic path to it. I cannot read a million books. What would constitute this "mathematical core"? And how can I make myself stick to a course of study instead of zigzagging all over the place? To be extra clear, it's not that I cannot concentrate, it's that I keep weighing the benefits of this or that book!

A lot depends on what your goals are. There's a lot of math out there. What is the physics you would like to know? In what detail do you like to know the math? Etc.

 

[Geofleur]

A lot depends on what your goals are. There's a lot of math out there. What is the physics you would like to know? In what detail do you like to know the math? Etc.

Well, that is part of my problem. In the past I have studied PDEs - Morse and Feshbach, Courant and Hilbert type stuff. I don't know if this exists or not, but if there are general methods of extracting information about the solutions to such equations without actually solving them, I would like to know those.

Apart from that, I am interested in the underlying mathematical structures of physical theory. In particular, geometrical or algebraic methods of gaining insight into them or relating them to one another. It's a little hard to decide where you want to go when you only see it vaguely, but there it is.

 

[Geofleur]

I mentioned this in a separate post, but I think it's relevant here. I want to be able to see physics from a unified, high level perspective, like how category theory views the rest of mathematics. But a physics version of this. From that thread, I get the idea that some kind of algebraic structures would be important...

 

[kith]

The most efficient and enjoyable learning experiences for me were always when I discussed stuff with peers. So ideally, I would look for people who are doing similar things as you. Practically, there may be not many of them.