Bad semester - Math PhD application I wanted to bring up an issue I faced recently and figure out how much my graduate school application will be affected by this.
For some background, I am a sophomore math major at a good public school. In the first semester of my second year(i.e Fall 2020), I faced many problems that led to a case of terrible depression. I ended up getting an F in an art history class and two Bs in Probability Theory and another math grad class. There were many other internet access issues piled onto the existing problems as well. It would be an exaggeration to say that I even participated in the class. I only submitted about half the assignments and got an average score on the final exams.
I had a 3.91 GPA in my first year taking the major upper-div math courses(Abstract Algebra, Linear Algebra, Real Analysis) and a graduate class in commutative algebra. I got all As in my math classes with one or two A-s in some unrelated classes.
In my second semester of my second year, I am trying my best to fix this issue. I will likely get 5 As this semester including 3 key math graduate classes. I think I'll maintain this streak in my third year as well. At this point, my GPA is around 3.5 . By the time I apply to graduate school, I can hope to get it to around 3.7-3.82. My math GPA will be higher than that. I am currently aiming to get into a good PhD program(say a top 20 program, very roughly). I already have some pretty good connections with some faculty who are quite renowned in their fields and may end up getting good letters. I may also have some research opportunities lined up within the next year. I also took the math subject GRE and did well on it.
Will I be penalized for the drop in grades that semester when it comes to graduate school applications? Many of the other factors(GRE, recommendation letters) look decent enough on paper for now except for my GPA. Most of the people I know who got into a math PhD program have a near-perfect GPA. It seems almost like a minimum requirement at this point.
I'd be especially glad to receive a response from someone very familiar with the math PhD admission process or another student who faced similar circumstances.
It is unlikely that a single bad semester will be decisive, given everything else, especially good letters. You aren't alone in stalling for the final semester, of course. If it is burn-out, then you need to deal with that explicitly as the overhang can be pretty bad as you start a new program. You need to start with a fresh and eager perspective.
But &quot;top 20&quot; is an incredibly narrow range of schools to apply to. I'd suggest, instead that you cover yourself by taking a broader attack against the full range of research (R1) institutions. There is a lot of competition as you note yourself.
And, other than the small effect on your GPA a failing grade in art history is unlikely to be a block for a mathematician. You do have to fulfill all graduation requirements, of course.

From your description, I've assumed this is the US.