Math Course Advice -- Harvard freshman planning to double major in physics and mathematics

[Quantum55151]

@mathwonk Sorry for the delay in my response. I was preparing for finals.

It is not entirely clear to me what preparation is expected, or usual, for success in 55.

Auroux writes in the course syllabus that no formal knowledge of linear algebra, group theory or analysis is required, but a familiarity with proof-writing and abstract reasoning as well as a commitment to a fast-paced course is. What this actually means in practice is a separate question.

As to learning differential forms, there was a thread here on PF devoted to that topic some years back, and they went through a very nice book by David Bachman, available from him free online:
https://faculty.washington.edu/seattle/physics544/2011-lectures/bachman.pdf

I'll check out, thanks!

As for Rudin's Principles of analysis book, it is famous for being very precise but very unmotivated, so I never recommend it for learning. But analysis professors love to recommend it, so you should take a look, maybe it will work for you. I prefer books by Spivak, Apostol, Berberian, Fleming, Lang, Simmons, and although quite difficult I admire Dieudonne'.

Do you know anything by Elements of Classical Analysis by Marsden and Hoffman? This is the book used in 25b.

On the topic of the student body in math 55: yes one will naturally feel intimidated at times, maybe most of the time. I definitely felt that way when there.

Even though you were the oldest student?

 

[Quantum55151]

But isn't that okay with you, since you intend to be a physicist and not a mathematician?

True. At the same time, I still want to major in math and do physics research that is more on the mathy side. So I do want to get a solid grasp on the material.

 

[mathwonk]

I don't know the book by Marsden/Hoffman. Reviewers on amazon say its helps that they teach n dimensions starting from 2,3 dimensions., which sounds user-friendly.


oldest ≠ best. when I was a freshman in math 11, there was another freshman who lived across the hall from me who was a year or 2 younger, and taking math 55. he occasionally helped me in math 11. I guess I wasn't so much intimidated by these very gifted students as impressed. The intimidation factor was greater later as a postdoc interacting with the professors.

 

[mathwonk]

please let us know what you choose and how it goes. good luck!

 

[VWFeature]

Hi everyone!

I would like to get advice concerning my math course for the spring semester.

For context, I am a Harvard freshman planning to double major in physics and mathematics with the long-term goal of doing research in high-energy theory and/or mathematical physics. I am currently finishing math 25A, a proof-based linear algebra course. For the spring semester, I have the option of either taking 25B, a proof-based real analysis course, or moving up to Harvard's infamous math 55. In the spring, math 55B covers topology (both point-set and algebraic) and complex analysis with a brief two weeks of real analysis in between.

Now, 25B is an objectively easier class, because it attempts to cover much less material in one semester than does 55B and also assumes less mathematical maturity on the part of students. For me at least, this might translate into a better understanding of real analysis than what I would gain from 55B where the professor tries to speed run through Rudin over the course of...2 weeks lol. At the same time, based on what I've read (and please correct me if I am wrong), I don't think real analysis is particularly useful for theoretical physics (at least not the kind that is taught in 25B; integration and measure theory might be a different story, but that is taught in a different class altogether). Topology and complex analysis, on the other hand, seem to be much more relevant to the kind of physics that I want to do. The other nice thing about 55B is that it would allow me to save time by basically knocking out three undergrad math courses in one semester which in turn would allow me to take more advanced undergrad classes and/or grad classes sooner. On the flip side, the trade-off will consist in how well I will actually learn the material in 55B as well as the level of difficulty of the class which, although a far cry from the stuff you'll read on 55's Wikipedia page, is nevertheless non-negligible.

What do you think? At any rate, I can always try out 55B for the first few weeks and then drop down to 25B if necessary. But I would still appreciate any advice so that I have a better idea of my spring plans and can plan out my winter studies accordingly.

P.S. I am attaching the syllabi for the two courses in case anyone wants to take a look.

Know yourself. You have way more math ability than I did. I took Physics 12A, bombed it, was grateful for a C, took Physics 1 w Paul Bamberg & aced it. (This was 1973)
Bamberg advised me about going on to Physics 112 and was insightful ( I didn't.)
I recommend you speak w him. He's a great teacher and straddles your departments.