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I will try to formulate my question so that the answer is not opinion-based.

My question: Considering a undergraduate 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 intend to study:

Thank you for your attention!

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    $\begingroup$ Evidence of original research activity? That would eclipse any grades. $\endgroup$
    – Carlo Beenakker
    Commented Feb 1, 2021 at 20:39
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    $\begingroup$ Given that you are asking this question, I suppose you have an answer to the following: what convinces YOU that your bad grades do not reflect your mathematical knowledge? That seems a good place to start, but I don't see any of this mentioned in your question. As an answer to your question, if you've been actively involved in side-projects, research projects, or similar, then these will look good $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Feb 1, 2021 at 20:42
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    $\begingroup$ @CarloBeenakker I thought about it, but I believe that this possibility is beyond my reach since the Mathematical Physics area is apparently very advanced and it is necessary to learn several mathematical tools to discover some unpublished result. $\endgroup$
    – user173409
    Commented Feb 1, 2021 at 20:52
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    $\begingroup$ @Carl-FredrikNybergBrodda's comment might (depending on how you read it) sound negative, but I think it is getting to something important. Why do you want to do a PhD in mathematics, and why do you think it is a better choice than the many many things you could do? Why do you think it is right for you, and, just as importantly, why do you think you are right for it? Of course I think a PhD in mathematics is a good choice, but I know that when I am considering candidate graduate students, I definitely want to know what their motivation is, and why they made this decision. $\endgroup$
    – Theo Johnson-Freyd
    Commented Feb 1, 2021 at 23:53
  • $\begingroup$ If you write a good Master's thesis (as you plan to do), then your grades prior to that will become significantly less important. $\endgroup$
    – Gordon Royle
    Commented Feb 2, 2021 at 8:37

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What you need are strong letters of recommendation that explain your bad grades and attest to your mathematical intelligence. These letters should carefully explain those reasons for your weak grades and point to evidence for your actual abilities (presumably, other grades that are not weak.)

There are programs that won't want to take a risk on you, but there are also probably programs that will be willing to take a risk on you, and the later is all that matters.

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