I completed my masters in mathematics in my home country in June 2020. I didn't had success in 4th semester of my masters with PhD Applications. So, I thought I would try next year in both my home country and abroad (Europe). I didn't apply earlier to any of my home country's universities.
So, I started studying for the qualifying exam in my home country while side by side studying some more mathematics to apply for PhD programmes abroad. But I fell in depression and wasted 1 month and 1 week in stomach infection till now.
The depression problem: I took the exam for PhD admission in my home country in December and received the result in early January. The result was lower than I had hoped for, although I don't yet know if I've been accepted.
My father is using this to make cruel comments to me, hurting my mental health. This contributed to me being depressed and missing several application deadlines for European universities. I completed only one application on time. Since then I have received medical help for my mental health problems and for 1 week I could not study due to pretty bad stomachache.
Question 1: Should I email the professors which will be sending letter of recommendation, and tell them what I have been doing since I completed my masters in June 2020? It would be only list of topics I have studied till now.
Question 2: If so, should I also include episodes of depression along with stomach infection?
Question 3: If so, should I also describe the reason behind the depression?