I'm new here, and I hope this is within the scope of the website. I'll try to ask few advisory type questions in the future... I'm a college junior, and 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, especially textbook recommendations.
My background: The professors I am closest with here (and whom I will probably ask to write my recommendations for graduate school) are both specialists in Harmonic analysis, so I'm thinking of going deeper into more advanced analysis coursework. I didn't come into college wanting to go into mathematics, so keep in mind that I only started taking mathematics courses last year. Nonetheless, by the end of my junior year I'll have taken:
Calc I, II (AP BC calc)
Analysis I, II (Wade...)
Ordinary Differential Equations
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've gotten A's without too much difficulty in all of my classes, and I have currently worked through these books through self study: Hardy & Wright's Intro to Theory of Numbers (no exercises, tried to work out proofs of theorems myself before reading them in the book). GF Simmon's intro to topology and modern analysis (did all of the problems. Did not get up to the last few chapters, though.)
and I am currently reading through Munkres' Topology on my own (and working through the problems).
Therefore, I have experience in analysis to the degree of finishing Wade, and I have developed quite a bit of topological knowledge through Simmons, Munkres.
EDIT: So I cut out quite a lot of what I was thinking because apparently I should really work through baby Rudin and learn Lebesgue Integration earlier. I have winter break (in which i usually work extremely hard on maths), next semester, and all summer (minus possible internships/research time) to go through baby rudin, and learn as much measure theory/lebesgue integration as possible.
Given this new addition to my background, what would be the best suggestions for real analysis/fourier analysis texts?
Any suggestions are welcome! I just want to best position myself for applying to a specific group when I apply to grad school.
Also, feel free to recommend other classes I really should take, but not in place of answering my questions. I have a lot of free space my senior year, so I can take these independent studies while still filling in any other holes in my learning.
Thanks so much!