Suggestions on independent study courses/book choices? Real analysis, fourier, ?

[dh363]

Hey, I was wondering if you guys could offer any course guidance on independent studies I could try to take my senior year. I have some ideas, but I was wondering whether you guys could give me any recommendations/book recommendations.

My background:
I initially wanted to go into a more algebraic field, something like algebraic geometry or algebraic number theory, but it's going to be harder for me to go into any depth with algebraic independent studies, for the professors I am closest with here are both specialists in Harmonic analysis. Sure, they would probably still know the subjects at the level I'm learning them but I was thinking of specializing in something more analysis related, for the professors I'm working with would be able to provide better insight and modern perspective into these subjects (I'm not particularly better at algebra or analysis). Here are the classes I've taken so far.

Calc I, II, Multivariable Calculus
Linear Algebra
Analysis I, II
Ordinary Differential Equations
Algebra I
Algebra II
Probability Theory
Differential Geometry (Barrett O'Neil)
Complex Analysis (Ruel & Churchill, though prof's notes gave a more rigorous treatment, though still very much at an undergraduate level.)

I have currently gotten through these textbooks through self study:
Hardy & Wright's Intro to Theory of Numbers
GF Simmon's intro to topology and modern analysis

and I am currently reading through Munkres' Topology on my own (and working through the problems).

Senior year, I am currently planning on doing a higher level Real Analysis independent study, that goes into more depth on Measure Theory, Lebesgue integration, Banach Spaces, Hilbert Spaces, etc. However, I also want to get an introduction to Fourier Analysis, as I think I may want to look into doing harmonic analysis. Are there any rigorous, good theoretical treatments of Fourier analysis that do not assume too much knowledge of Lesbeque integration and measure theory from the start? (introducing it in the middle of the book or something is not a problem since I will be taking it concurrently with Real Analysis).

Right now, first semester senior year I've been looking into simultaneously doing Elias Stein and Shakarchi's Real Analysis and Fourier analysis books from the same series (stein was my professor's advisor and I think he likes his books!). Is anyone familiar with these textbooks? The Fourier analysis intro is volume one of the series, while real analysis is volume three. Does anyone know whether this makes one the prerequisite of the other/would i have problems doing them simultaneously? I was thinking first semester senior year doing this and second semester going further into real analysis, and doing an introduction to harmonic analysis.

Or, if I end up really liking differential geometry, I may try to do an independent study with O'neill's semi-riemannian geometry, or do carmo's riemannian geometry. Thoughts?

Any suggestions are welcome! I just want to best position myself for applying to a specific group when I apply to grad school. Since I'm not terribly picky about the subjects, it seems either differential geometric or an analytic path would be most realistic given my options.

Thanks

 

[micromass]

First of all, deciding to do analysis just because most professors in your undergrad institution specialize in it is a huge mistake. If you like abstract algebra, algebraic geometry or algebraic number theory more, then you absolutely need to go into that. You got to do what you think is more interesting, not what your professor think is more interesting. You do not want to end up in a PhD with a subject you don't care about and regretting that you didn't pick what you originally wanted.

That said, there are many good book on harmonic analysis. The Stein and Shakarchi book is certainly a good one. I know his real analysis volume is the third book, but I don't really think he assumes much knowledge of his previous two volumes anyway. There might be some things missing, but they're not going to be essential. Another book I like is "Fourier analysis and its applications" by Folland. You might want to look at that too.