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There are certain concerns that say this might be duplicate of this question: How does the admissions process work for Ph.D. programs in the US, particularly for weak or borderline students?. However, I believe that is not the case. That question talks about a low overall GPA. I do not have a low overall GPA but just an accidental F. I think that is different.
My situation is a little bit special so I am sorry if I did not write a perfect title to summarize the whole story. I will briefly tell the case and my situations and then ask the questions in detail.
I am an international student in pure math at a US college with 10-20 ranking and I plan to be a mathematician from the very beginning since I was young. However, I got into an accident in the first half of my 3rd year. I missed the deadline of the final of my PDE class and that final counted up for 40% of the grade, making me fail the course. This made me so depressed, since my school will show both grades even if you retake the course. I can retake the course but some of the grad schools that I hope to get into have an application deadline before my fall semester grade come out, which is really troublesome.
Also here is some of my overall performance in college
1.After fail, I still have an overall GPA about 3.8
2.I had top performance on other my courses and even I can do better than grad students in grad level course. For example, I just got a solid A+ in commutative algebra and functional analysis for the last semester.
3.I attended a wide range of math courses in graduate level like algebraic number theory, knot theory and geometric topologies with A grade.
4.I will retake PDE in fall 2016 and I am very likely to get a rec letter from the professors of that course. Also, the professor of PDE course in fall 2015 agrees to write a supportive letter for me confirming my situation.
5.I got independent study experience in pure math. (You cannot do original research in pure math as undergrad, usually)I did a very successful reading and presenting course in hyperbolic manifold (with 6 A+ credits) and I will do another courses like this in OPUC in fall 2016. However, that is all I can get. I was rejected by all REU I applied since unfornately all REU I was interested in did not accept non native students.
6.Because of 5, my other two rec letters will be from a very recognized professor in my department and from my department chair.
7.I have a high GRE general/subject test score.
Now is my question:
Based on my overall performance and the rescue actions I took, How badly this F will affect my PhD application?
If I want to apply for a school whose deadline of application is earlier than the time when my fall grades come out, how can I show that I have come back and fixed my mistakes?
I apologize in advance for grammar mistakes and awkward expressions if there are any. I really love mathematics and I really want to get into top tier school. Thanks so much.