Which class should I take next semester, Complex Analysis or Topology?

[drkatzin]

Hi,
I'm a junior undergrad majoring in math and physics, and am deciding between complex analysis and topology for next semester. (I'm planning on doing theoretical physics for grad, something on the more mathematical side, so topology would likely be used).

Complex Analysis
Pros: ESSENTIAL to master for physics, offered only in spring
Cons: they switched the textbook to one I've already read a good chunk of (analytic functions, contour integration, taylor/laurent series). Don't know how much more I can get out of the class, other than mastering contour integrals and figuring out when, not just how, to use complex analysis. But I think that stuff would be taught in physics classes anyway.
Textbook: Brown and Churchill (used to be Ahlfors)

Topology
Pros: beautiful subject, prereq for algebraic topology (used in higher level theoretical physics).
Cons: The book is so good that I can probably teach myself everything in the course. The course also moves pretty slowly, starting with a review of general math topics I already know (e.g. cardinality, functions, relations)
Textbook: Munkres

Which one do you think I should take?
Thanks!

 

[MathematicalPhysicist]

Both of them are rather essential in either math or theoretical physics.

It really depends on you which one you think you can handle by yourself.

I myself took both type of courses with two terrible lecturers, I used for Topology Munkres as well (someone should tell the publisher to lower down the price of this textbook).
Basically you don't need a lecturer if you have Munkres for a book. :-)

Best of luck either way.

 

[Newtime]

Both of them are rather essential in either math or theoretical physics.

It really depends on you which one you think you can handle by yourself.

I myself took both type of courses with two terrible lecturers, I used for Topology Munkres as well (someone should tell the publisher to lower down the price of this textbook).
Basically you don't need a lecturer if you have Munkres for a book. :-)

Best of luck either way.

I agree with this but I would also like to say that a rigorous course in complex analysis will certainly (hopefully) be more than contour integrals and Laurent series. Do you have a brief description of the course from the catalog or something?

 

[axiomatic7777]

Does "read a good chunk" = understanding all the concepts AND did all/most of the end of the chapter exercises? If so, take topology. It's an interesting subject although what you get out of it depends highly on your instructor. Have you taken any kind of real analysis before?

Just reading won't do you a lots of good if, for example, you only have a concept of how to do contour integrals, instead of really knowing how to do them from doing lots of exercises. Contour integrals are reallllllly important - make sure that you actually master it. If you are going into theory, you also need a good grasp of analytic functions and all the implications. No, just knowing how to define what analytic functions are is not enough.

 

[drkatzin]

Sadly I didn't do enough exercises, complex analysis is probably the way to go

Complex analysis:
Studies the basic properties of analytic functions of one complex variable. Conformal mappings and the Poincare model of non-Euclidean geometry. Cauchy-Goursat theorem and Cauchy integral formula. Taylor and Laurent decompositions. Singularities, residues and computation of integrals. Harmonic functions and Dirichlet's problem for the Laplace equation. The partial fractions decomposition. Infinite series and infinite product expansions. The Gamma function. The Riemann mapping theorem. Elliptic functions.

 

[drkatzin]

Btw my pure math background is real analysis, measure theory, analysis on manifolds, algebra (linear, groups, rings, galois)

 

[CharmedQuark]

Brown and Churchill is a great book, however the exercises can be incredibly difficult, so having a lecture can help with those.

Munkres is definitely doable without a lecture.

 

[drkatzin]

Thanks! I'll go with the consensus -- read Munkres on my own and take Complex Analysis

 

[╔(σ_σ)╝]

I used Brown and Churchill for complex analysis. It was a good book for an Engineering student but I don't know how much you would get out of it for physics.

I didn't get much insight with the book. The excersices where either long and tedious or short and "number" cruchy. A few were challeging but the hints were good.

I guess it depends on the way the course is taught.

I can't help but mention that the course I took was for engineers and some things like conformal mappings, riemann surfaces ...etc... were not covered. Also I was able to follow along without a real analysis backgorund.

It was really a class on integration and finding laurent series much of which I forgot anyway.

 

[planethunter]

I would def do topology. I took complex analysis using Visual Complex Analysis, which required you to totally think outside the box- and didn't like it with such a difficult subject. I think it also depends what area of theoretical physics you want to do. If you want to work in areas such as string theory, topology would be better

 

[╔(σ_σ)╝]

I would def do topology. I took complex analysis using Visual Complex Analysis, which required you to totally think outside the box- and didn't like it with such a difficult subject. I think it also depends what area of theoretical physics you want to do. If you want to work in areas such as string theory, topology would be better

Visual Complex analysis is a really good book after you get past the notation and all the geometry. Unfortunately, I didn't have the time to read much of the book and I my geometry was too weak for the book.

 

[muppet]

If you're a theoretical physicist, it probably depends on what kind of physics you're interested in. For quantum field theory it's absolutely ESSENTIAL that you get the hang of complex analysis, contour integrals in particular, wheras most field theorists probably never study topology, certainly not at the level of munkres. For general relativity on the other hand, it's perfectly possibly never to look at a complex number, whilst it might be handy to be able to read mathematical texts about paracompact hausdorff spaces locally homeomorphic to R^n. If in doubt I'd take complex analysis, but if you really want to get mathsy you'll probably need both.

 

[MathematicalPhysicist]

If you're a theoretical physicist, it probably depends on what kind of physics you're interested in. For quantum field theory it's absolutely ESSENTIAL that you get the hang of complex analysis, contour integrals in particular, wheras most field theorists probably never study topology, certainly not at the level of munkres. For general relativity on the other hand, it's perfectly possibly never to look at a complex number, whilst it might be handy to be able to read mathematical texts about paracompact hausdorff spaces locally homeomorphic to R^n. If in doubt I'd take complex analysis, but if you really want to get mathsy you'll probably need both.

Well Munkres isn't that much difficult I might say, the real hard stuff is when you go to algebraic topology (though in Munkres' Topology there's a taste of this).

Well in QFT I guess you really need to get to grips with Lie groups and Functional analysis, and they use topological terminology.

And ofcourse there's the case of topological defects, and TQFT and many more...
It's like the saying:" Q: How much math should a theoretical physicist know?
A: More!". :-)