I currently have finished my 2nd year as an undergraduate and I will have a very easy time graduating from my school in a year. I started out somewhat rocky these first 2 years and I was wondering whether it would be better for me to stay in school an extra year or apply this coming fall semester. I'd also like to ask at what level of graduate school I would be seriously considered.
I'm a Caucasian male.
At a Top 50 school:
At an Undergrad level:
Ordinary Differential Equations
with all A/A-'s except for Diff Geo and Complex and Fundamental Proofs. Those were a B-/C+/B- respectively.
Advanced Undergrad Level:
Real Analysis(self-studied and passed qualifying exam)
Current cumulative GPA: 3.30
Current Major GPA: 3.53
My GRE Math score is a 910.
I also have experience as a grader, a TA, and separate on-campus job during my second year.
I have a well-known advisor in his field writing a letter of recommendation and I'm not sure who else to ask.
I have attended an REU at a top 10 school this summer.
I have co-authored a paper in analysis and did additional research in a separate field in computer science.
My personal statement will most likely talk about a love of harmonic analysis or something. I'll ask my advisor about that. I plan to specialize in harmonic analysis so I would appreciate if you could advise some programs with good programs in that direction. I'm only 19 so it won't hold me back all that much to apply next year; however, it would also be pretty expensive for me to do that. I could just use the next year to blow through my school's grad classes if I need to. Also relevant may be the fact that I double-majored physics as well.
I've also just started to buckle up and work since I received a wake-up call. The only reason any of my undergraduate grades are underneath flat A's is missed HW assignments which I understand is incredibly stupid. If I were to stay for another year, I'd probably average a 3.8ish in some Grad courses. Although I'd definitely only take like 3 each semester and an additional undergrad course in a foreign language, since I know that's a requirement for most schools.
I'd also like to ask if it's a good idea to self-study a course over the summer, like I did last year for Graduate Real Analysis. As in, do Grad schools look at it fairly positively. Thank you preemptively for any responses. I know I'm a slightly ridiculous case, but I greatly appreciate the advice.
One last point, when talking about possible acceptances to schools, what are my chances for UCLA and UChicago, I know they're probably dismal, but two of my ideal advisors are at those schools. Again, thanks for any advice you may provide.