Cool and useful math I should learn? [3rd year +]

[mcabbage]

Some background: I'm a mathematical physics (it's not focused on the field titled mathematical physics specifically, but rather it's a general math and physics degree) student in Canada.

(Update if you saw my last post about struggling in linear algebra 2 - I have been doing very well in groups and rings (the course that comes after it). I think most of my struggle was due to some burnout and time management issues.)

I cover the standard undergraduate curriculum in math for physics - ODEs 1/2, PDEs, linear algebra 1/2, applied complex analysis, probability, and statistics. Also some applied math courses on quantum theory

I am also completing the majority of an analysis focused pure math curriculum (real analysis 1/2 which include topology, functional analysis, groups and rings) this has been going well this term since proofs and derivations come more naturally to me than rote computations (sign errors, misremembering procedure, etc)

I have completed the core classes, and I'm only left with 3rd and 4th year courses.

Rather than taking more courses (which is too much work and i'd learn a lot of useless-to-me stuff), I teach myself a lot of material (I mostly teach myself everything in all of my classes anyways - I learn better from books and discussions than from lectures). There are a few courses I don't know enough about - how valuable would it be to learn some:

Fields and Galois theory (prereq to some of the others listed)
Commutative Algebra
Representation Theory of Finite Groups
Measure and Integration
Algebraic Geometry
Geometry of Manifolds
Lie Groups and Algebras
or Algebraic Topology?

As these are offered at my school they would be the easiest to find study groups for and people to exchange knowledge with. Other suggestions are welcome!

If anyone is curious, see the AMATH and PMATH calendars for courses available to me.

 

[fresh_42]

Fields and Galois theory (prereq to some of the others listed)

1. Commutative Algebra
2. Representation Theory of Finite Groups
3. Measure and Integration
4. Algebraic Geometry
5. Geometry of Manifolds
6. Lie Groups and Algebras
7. Algebraic Topology

Assuming your major interest is still physics, then among those you listed, I would disregard the "finite cases" 1-2, which many of it can also be learned by book, if you are really interested in. I would add to the list of minor importance also algebraic geometry and algebraic topology. Not that they wouldn't have their justification to be on the list, however, what remains is more important:

3. Measure and Integration
5. Geometry of Manifolds
6. Lie Groups and Algebras

Of course it will depend on how the lectures are planned, i.e. where the emphases lay. You can teach Lie theory from an analytic point of view as well as from an algebraic one. Nevertheless, I think they are very important and I hope they cover Noether's theorem. Measures, integration, manifolds and Lie theory are basic skills for advanced physicists in my opinion.

As of the disregarded algebraic subjects (alg. geometry and alg. topology), I'd say it depends on what is in it. They are large fields and lectures can vary a lot. E.g. if your interested in cosmology, algebraic topology might be of some interest. However, the 3 courses I extracted from your list are already heavy stuff and essential for physicists. The more as Manifolds and Lie groups are closely related and all of them serve as examples for the more abstract algebraic subjects.

 

[FactChecker]

I see nothing that really emphasizes Fourier transformations. You might want to take a look at the Stanford Lectures for class EE261 (). I highly recommend it.

 

[mcabbage]

I see nothing that really emphasizes Fourier transformations. You might want to take a look at the Stanford Lectures for class EE261 (). I highly recommend it.

We do Fourier transforms in Partial Differential Equations as well as Calculus 4. Fourier analysis is also emphasized in Real Analysis 2 as well as in my Quantum Theory courses. I will check out this for good measure though! Thanks!

Assuming your major interest is still physics, then among those you listed, I would disregard the "finite cases" 1-2, which many of it can also be learned by book, if you are really interested in. I would add to the list of minor importance also algebraic geometry and algebraic topology. Not that they wouldn't have their justification to be on the list, however, what remains is more important:

3. Measure and Integration
5. Geometry of Manifolds
6. Lie Groups and Algebras

Of course it will depend on how the lectures are planned, i.e. where the emphases lay. You can teach Lie theory from an analytic point of view as well as from an algebraic one. Nevertheless, I think they are very important and I hope they cover Noether's theorem. Measures, integration, manifolds and Lie theory are basic skills for advanced physicists in my opinion.

As of the disregarded algebraic subjects (alg. geometry and alg. topology), I'd say it depends on what is in it. They are large fields and lectures can vary a lot. E.g. if your interested in cosmology, algebraic topology might be of some interest. However, the 3 courses I extracted from your list are already heavy stuff and essential for physicists. The more as Manifolds and Lie groups are closely related and all of them serve as examples for the more abstract algebraic subjects.

Alright, I am definitely interested in cosmology, but I suppose I can learn the relevant alg top when it comes up in my studies. I will make sure I study a healthy amount of measure theory and integration. Luckily, our undergrad GR course spends about 2/3rds of the course doing differential geometry and tensors, so I will be well off for geometry and manifolds if I combine the course with self-study. As far as Lie theory goes, I can try to schedule the course but I don't think it will fit into my degree. Might as well learn that myself too (alongside friends who are interested).

Do you have any book suggestions on these topics?

 

[fresh_42]

Do you have any book suggestions on these topics?

I do.

• V.S, Varadarajan: Lie Groups, Lie Algebras, and Their Representations - the author is a specialist for Lie groups, the book is more on the analytic page of the matter
• Peter L. Olver: Applications of Lie Groups to Differential Equations - a reference with Noether's theorem
• James E. Humphreys: Introduction to Lie algebras and Representation Theory - the algebraic page of the matter with the classification of semisimple Lie algebras

However, they might be a bit thick to study by book, except for Humphreys which is good to read.