Apologies for my verbosity.
There have been more than a few questions from applicants with less than stellar grades. My situation is significantly different from those I have read, but I believe broadly applicable to others.
I am 30 years old and will soon be finishing undergrad with multiple degrees. My late teens and early twenties were tumultuous and despite far above average SAT/AP scores, I did poorly. I was majoring in fine arts and philosophy, struggling with my mental health, and earned grades ranging from A to F. I dropped out.
In my late twenties I returned to my local, unranked, state university majoring in Mathematics, Computer Science, and minoring in Philosophy of Science and Technology. I have earned good grades (18 A's, 2 B's, 3 C's) and have a few excused grades. I earned two C's while working far more hours than allotted by grading 8 sections (240 students) and tutoring: I did not complete all my coursework though the work I did do earned ~100%. The third C was due to being burned out. The C's are most recent, in my majors, followed a period of 4.0 semesters, and were concurrent with some A's.
I intend to earn ~ 8 more A's and possibly B's before finishing, and will take more electives than necessary.
My cummulative gpa is low (3.373) due mainly to the early failures. I have a huge number of credits, so each new A raises it an insignificant amount.
I wish to apply to PhD programs in mathematics.
Assume the early semesters and EX grades cannot be erased. Keep in mind an older student at an unranked, local, state university. All other things being equal:
Are EX grades truly neutral or are they negative?
To what extent does the multi year gap between the tumutltuous period and my current success mitigate those early failures?
To what extent does the fact that my poor grades were in completely unrelated areas and are not even applied to my degree mitigate my low gpa?
Will the above matter and be seen considering unofficial cutoffs (i.e. the "on the ground reality" of what happens in admissions committees)?
To what extent will the several C's diminish my "upward trend"?
I believe my TA duties were excessive though indicating this in my SOP would just be complaining. Truly though, is my performance during that period indicative of an inability to handle the workload of a PhD student?
How can a CV and SOP present these issues in a mitigating way while still leaving room to discuss the students purpose, passion, and strengths?
I have TA experience but not research (none was available at my department). I will have good to excellent LOR's from professors of diverse type. I will be (probably the only student at my insitution) taking the Putnam exam. All indications are that my GRE scores will be in the very high range, and my subject GRE score will be in the middle to high range.
- What tier of departments should a student with my background be aiming for?
I am especially concerned because I have found few programs with faculty in the (sub)specialties I am considering ranked outside the top 15. Most universities have at most one rarely offered course and no faculty in my area.
- Is a weaker student wishing to specialize in the uncommon strengths of a top department competitive with a stronger student wishing to specialize in more common areas? (i.e. I may apply to a top university not just "because it is a top university" but because it is a good fit).
I apologize for the multiple questions, they are intertwined so please consider them a focusing of "how do these various factors affect the prospects of an older student?"
Thank you for reading.