This question was motivated by

What to look for in applicants to graduate programs (in mathematics)?

Lost soul: loneliness in pursing math. Advice needed. [closed]

Tl;dr: If you want to skip to the main point of my question, please go to "III.) My Question".

I.) Justification of why I think my question is appropriate on MathOverflow. (Not directly related to my question.)

This question is on mathematics, career, and personal life.

There seems to be many question pertaining to math and life on MathOverflow that are voted closed but end up getting a lot of votes. (See for instance this question: Lost soul: loneliness in pursing math. Advice needed.)

I think this is an indication that:

1.) The community is still in debate as to whether questions of this sort are appropriate for this community. (See Jonathan Chiche's comments from the aforementioned post.) I am not active enough on MathOverflow Meta to know the discussions surrounding this issue.

2.) The number of upvotes suggests that many people find these questions helpful. (I think this is because there aren't any communities on the Internet dedicated to math-related personal/career matters, and MathOverflow is the closest that comes to such a community. Nevertheless, a lot of these psychological/personal/career problems are prevalent in many student's/mathematician's lives, and so there is a demand for answers to these questions.)

II.) Assumptions.

i.) Suppose that I am a Junior in college majoring in mathematics at a top research institution in the U.S. I have "mostly" good grades in mathematics (say, around 2/3 of my grades are A's or A-'s), but I have some B's and B+'s. I've taken a good number of graduate level courses with at least half of them A or A-'s.

ii.) Earlier in my college career, there was a noticeable dip in my grades (A's becoming B's and B+'s) due to personal issues.

iii.) My goals:

• (Short term:) enroll in a top pure math graduate program in the U.S. (ideally a top 6 institution) where there is a strong research community in my field as well as other fields in mathematics which may be helpful to my own research or broadening my perspective.

• (Long term:) become a professor in pure mathematics at a top 30 institution (or equivalent) studying geometric topology or some related field.

III.) My question:

i.) Here is a comment by Igor Belegradek regarding recommendation letters and personal statements in the aforementioned question:

"If you have had a bad semester with poor grades due to health/personal issues it may be worth explaining what happened."

However, I have heard some professors suggest the exact opposite of this, namely the only thing that really matters in applications is the student's capacity to do math research. Indeed, if I were to place myself in the admissions committee / graduate school's point of view, this attitude makes a lot of sense: it is more beneficial/risk free for the school to admit students who are happy, healthy, have a clean track record, and can be consistently productive than to admit students who were/are unstable (in any sense of the word) or were (or are still in) situations where they will experience hardship in their personal lives.

ii.) So, here are my questions:

a.) First of all, is it appropriate to talk about personal matters in one's personal statements or rec letters, if it has negatively impacted one's academic performance?

b.) What are grad school admission committee's views in regards to personal hardships? I know the short answer to this is "it depends," so I would appreciate getting the various viewpoints available on the MathOverflow community. I am sure it also depends on the school; school-specific viewpoints would be much appreciated.

If you think my question is ill-posed or not appropriate for the community as it is, please let me know in the comments so that I can adjust it.