I am a senior at a top 15 public university in the states and planning to apply for graduate school in mathematics. Having been a top 100 Putnam performer and won a couple of medals at IMO in my high school, I believe I have a strong enough mathematical background to be a competent student at a top 5-10 graduate program. However, my application will not be stellar.
In the beginning of my junior year, I got severely addicted to a computer game and my grades suffered from it. Now I have one C in an economics class and an F in summer English class as well as two B's in upper level maths classes. Luckily, I have completely recovered now and performing very well in all of my current classes.
On one hand, my addiction problem in the past would certainly make it look like I am not strong enough to get through long and hard five years of a maths PhD student. But I do know that a researcher in pure maths will be my career and is what I am passionate about the most.
On the other hand, my adviser at the current university believes that I belong to a top school. He said:
...a student of your talent really needs to go to one of the best maths programs...
Now I am taking a second course in complex analysis with him and he seems to like me.
Because of my financial problem, I can apply to at most a dozen schools and want to make the best out of it. I thought that top 20-40 is my reach and top 40+ is my safety range. I had completely ruled of the idea to apply to a top 20 program. But what my adviser told me made me confused.
So my question is will it make sense for me to apply to a top 20 school? I have a really tight budget and do not want to waste spot for a school that I do not have a shot at.