Hello everyone. I've applied to Fall 2018 PhD programs with the goal of studying graph theory. I'm quite worried that I won't get in anywhere. If anyone could generally evaluate how my application compares with others, I would be very appreciative. Thank you.
Applied: Berkeley, Princeton, Emory, South Carolina, MIT, Rutgers, Georgia Tech, Arizona State, Iowa State, Carnegie Mellon, Cambridge, Oxford, Waterloo, McGill, UBC, Simon Fraser, Birmingham, LSE
Undergraduate: State school, ranked 100ish in the USA
GPA: 3.82, 3.80 for math (dragged down by one C-)
GRE: 170 Verbal, 169 Quantitative, 6.0 Analytical Writing
GRE Math: 790 (77th percentile )
Demographics: White male, 23 (I have been out of school for two years)
Upper level courses: Analysis I and II, Stochastics, Graph Theory, Linear Algebra, Abstract Algebra, Combinatorics
I received an A in all of these classes except for Analysis II (A-) and Combinatorics (C-). I took Combinatorics in Korea as a sophomore without any previous mathematical experience, and part of the course was instructed in Korean. I mentioned in my SOP that getting my ass kicked in this course contributed to my motivation for starting to study mathematics seriously.
Research experience: Nearly ready to submit a paper in which I prove a new complexity theorem for tripartite graphs. My graph theory professor has been helping me to edit this paper for almost two years now.
Awards: National Merit Finalist, Phi Beta Kappa, graduated with distinction
Work experience: By next fall, I will have two years of experience teaching Cambridge A-Level Math and Physics to high school students.
Recommendations: I believe that my graph theory professor gave me a very strong recommendation. I submitted my complexity result to him as a final project for his class, and he seemed very impressed by my result, and he took the initiative to work with me to get my result published. He has seemed very enthusiastic about helping me edit my paper. My other two recommendations are probably average.
Other: I have some other random qualifications. I have a Java Bronze Certification, and I have language certifications in Mandarin, Spanish, Korean, and Japanese.
I'm worried because I didn't decide to finally major in math until my last year, so I don't have many upper-level math courses. I wish I could have taken complex analysis and algebraic combinatorics, but my scholarship ran out after four years, and I had to graduate. My graph theory professor said that applying for top schools would be aiming too high, and he gave me a list of medium-level schools with good graph theory faculty, but he said that even these schools are extremely competitive.
I'm really sorry to flood the forums with another one of these. If anyone could give any insight about how my application compares to others who apply for PhD's to study combinatorics or graph theory, I would be extremely grateful. Thank you.