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I am applying to 11 graduate schools in mathematics. I have enjoyed studying mathematics, and the main reason I am applying to graduate schools are simply because I want to learn math. I would be lucky if I can work in academia in a future, but I would probably be happy if I end up working in an industry or a government, or something completely unrelated to mathematics.

Recently, however, I am starting to think that maybe my mathematical ability is not good enough. For example, here are my grades in the "big three" upper division math courses:

Analysis I-II-III: B+/A-/B+

Algebra I-II-III: B+/B/B-

Topology I-II: B/? (will be taking the second one next quarter).

(Note: My school uses quarter system)

Not so impressive. My excuse for the B- in Algebra III (and B+ in analysis III) is that I was simply doing too much during that term (too many classes and work hours), but that's about it. For other terms, I felt like I always did reasonable in one exam and then bomb the another exam in the class, which usually brought my average to B/B+ at the end.

On the brighter side, I got A's on most of the other math courses, such as complex analysis, probability, dynamical systems, combinatorics, number theory, and all the other lower-division math courses (calculus, diff eq, linear algebra, elementary analysis). I also did a reading with one professor on Hilbert space, and I did quite well on it too (and enjoyed the subject too!).

My MGRE score is 710/65%. I guess this is okay... I've heard both good and bad things about this score (bad ones are mostly coming from online, and good ones are mostly coming from my professors that I have talked to). BTW, I am applying to these schools: U Wisconsin, U Washington, UNC Chapel Hill, UCSB, UCSD, Penn State U, Indiana U, U Arizona, U Colorado Boulder, U Nebraska-Lincoln, and U Mass-Amherst.

My current interest is in analysis (both pure and applied), but I am also interested in learning more about probability (I just took a course in it last quarter, and I really enjoyed it!). And of course, I am still open to other fields as well.

I feel like this isn't something I should be concerned at this point, since I will be applying to graduate schools anyway. My current solution to this problem is this: Let the grad school decide whether I am good enough for grad school. If I did not get into any of these schools, stop, maybe I should do something else. If I cannot pass a qual/prelim exam (of course, I am willing to do my best to avoid this), stop, and do something else instead. The only problem with this is that what "else" should I be doing instead of math... It would be great if someone has a good idea about this. At this point, operations research sounds kind of fun, although I don't know much about it.

Okay, I think I will stop here because I feel like I am starting to ramble. Feel free to post something if you have something to say. Let me know if you have any question as well.

Thanks!