Advice on what math I should self study and take next semester.

[xdrgnh]

Alright so far I took calc1-3 which included vector calc and I'm currently taking LA. The LA class is only a quarter long and afterward I'm taking DE and Complex variables. I'm wondering what math should I take next year. I'm thinking either taking Analysis or Abstract Algebra. What do you think will be more time consuming? The syllabus for abstract is What It Means to "Solve" a Polynomial Equation
2 Fields
3 Solutions of Cubics
4 The Euclidean Algorithm for Polynomials
5 Simple Field Extensions
6 Galois's Method
7 The Galois Group of a Cubic
8 Permutation Groups
9 Normal Subgroups
10 Quotient Groups
11 Solutions of Equations of Degree Four
12 The Group of the Dodecahedron
13 Unsolvability of the Quintic
14 Review

. The syllabus for Analysis is This course covers the study of basic topics in analysis with emphasis on methods. Sequences, series, functions, uniform convergence, continuity, partial differentiation, extreme value problems with constraints, Riemann integrals, line integrals, improper integrals, integrals with parameters, transformations, Riemann-Stieltjes integral, uniform and absolute convergence of integrals. Beta and Gamma functions.

 

[chiro]

Hey xdrgnh.

What kind of mathematics are you inclined to get into?

Usually in mathematics we have pure, applied, and statistics. There is of course overlap, but in a subject like mathematics you have to focus on a specific area and then build up specific experience.

It would help if you get some specific experience of some kind in one (or more) of these areas and take it from there.

From what you have described you have good foundational pure (and somewhat applied) mathematics courses which will help you start to specialize a bit more.

If you are interested in pure mathematics I recommend you do at the minimum topology along with your analysis course. From the sounds of what you are doing you seem to be more interested in pure over the other areas.

Along with topology I recommend a good algebra course if you want to go the pure math route.

 

[xdrgnh]

I'm a physics major who wants to go into theoretical physics.

 

[Jorriss]

Algebra and Analysis are equally good. They will both make you more mature. I think most people, myself included, have found Analysis more difficult and in turn, makes one even smarter!

In terms of usefulness though, there is little to distinguish - Algebra probably edges out a bit though.

 

[bpatrick]

Algebra (if you have to choose between algebra and analysis). I'd argue that an advanced linear algebra course would probably be more valuable than either algebra or analysis. You can never be too good at linear algebra. but there may be no advanced LA course at your institution or it may require algebra prerequisites.

If you feel completely compelled to take an analysis class, I'd choose more complex analysis (since you said you'll be taking complex variables) instead of real analysis. for a physicist, complex analysis is without a doubt more valuable than a single intro course in real analysis. complex analysis will get you more familiar with the complex plane, calculus, some PDEs, and more vector ideas, which are all great topics to gain more experience in as a physicist.

Maybe it's required for you during your physics degree, maybe not (sometimes they are just there as electives), but: math methods in physics. find a course or two on that and fit them into your schedule. those would likely be more valuable than continuing to take pure math electives. the topics are quite specialized and will give you extra practice in vector calculus, complex math, stats, counting, Fourier analysis, gamma/erf, PDEs, etc...

Discrete math + computer programming: useful for if you end up doing some computer stuff later or if you end up going experimental instead of theoretical ... usually discrete is a good intro to counting and algorithms too. Overall pretty useful. I'd imagine some of the exposure might help if you end up taking a statistical mechanics class later, there's lots of counting and large number math involved in that, not that it's very hard, but it might make it more intuitive if you have exposure to combinatorial math.

 

[micromass]

I'm a physics major who wants to go into theoretical physics.

I'd take the analysis class. This analysis class won't be particularly useful to you, but it's a prereq for other useful analysis classes.
For example, if you want to understand quantum physics, then you absolutely need to know functional analysis (to an extent). But your analysis is a prereq for functional analysis.

The algebra class won't be too useful in general (it has applications however!). If you ever need it, then you can self-study it.

 

[Fredrik]

I'd choose analysis for the reasons micromass mentioned. It will make it easier for you if you ever want or need to learn the mathematics of QM. You would probably need something like five math courses to get there, but a course on analysis is the logical first step (for a person who has already studied linear algebra).

Abstract algebra isn't as useful. I'd say that it's useful to understand the definitions of terms like "group", "ring", "field", "homomorphism" and "isomorphism", and a few theorems about those concepts, but these are things you can study on your own.

I agree with bpatrick that a second course on linear algebra would probably be more useful than either of the two options you suggested. This is especially true if the first one focused on systems of linear equations and that kind of stuff, rather than on linear operators and spectral theory (eigenvalues and stuff). The latter is much more useful in QM.

 

[SophusLies]

I highly agree with what micromass and Fredrik has said. I'm mostly a physicist with a mix from pure math and in my experience you can always get more linear algebra especially for quantum. Analysis was the first math class that made me feel like I could organize my technical thoughts into a clear concise manner. It was also very useful for PDE's.