I will try to formulate my question so that the answer is not opinion-based.
My question: Considering a undegraduate math student planning to pursue a PhD program, what qualities and achievements can outweigh bad grades? For example, is there any exam or test somewhere in the world that is able to show that my bad grades do not reflect my mathematical knowledge? By the way, is there a Phd program in the world whose acceptance depends on an exam instead of having a good school record?
I will talk about my background since it will become clearer the purpose of my question.
I study Mathematics in one of the best universities in Brazil. Next year I will start my master's degree and, later, I intend to do a PhD with the purpose of being a researcher in the area of Mathematical Physics.
Unfortunately, due to health problems, I left the university in the middle of a semester which made me fail in all the subjects belonging to that semester. In the end, I stayed away from the university for a year. I returned to university in the second semester of 2019. However, although I did not fail in any subjects (except for that semester I mentioned), my grades were as low as possible. Looking at the criteria for being accepted into PhD programs, I believe it is practically impossible for me to get into a PhD program at an excellent university if I don't do something that outweigh my bad grades.
Knowing that on this site there are several researchers and university professors who have already had PhD students under their guidance, I ask: what qualities and achievements can outweigh bad grades?
Unfortunately there is no longer the possibility of participating in mathematical olympiads for university students since I am 26 years old and I am about to graduate.
My current project is to study the following books and prepare a good master thesis. I also intend to learn the Lean and C++ programming languages to have some differential in my curriculum.
The books I pretend to study:
- Mathematical Physics: Classical Mechanics
- Mathematical Gauge Theory
- Mathematical Methods in Physics
- Global and Stochastic Analysis with Applications to Mathematical Physics
Thank you for your attention!