I have recently finished my first year Math PhD coursework/TA duties. I would like to explain my background a bit and then explain why you should not be worried about your experience(s).

I was a nerdy kid and did many math competitions, etc. until high school. I nearly failed out of high school due to excessive partying and laziness (you get the gist) and proceeded to work for my family’s business for 3 years, entering college at the age of 21. I started my first year as an English major, and quickly rediscovered Math and my love for it. As an undergraduate, I found the courses doable and not so challenging until I encountered Real Analysis. Why was it so hard? Here is my answer:

Your first encounter with conceptual or proof-based mathematics will always be challenging. Most curriculums are “banking methods” - they deposit information, you return that information on an exam, you almost certainly forget the previous deposit and the cycle continues, and this happens for each course (my experience). Basically, you solve well-defined problems given an explicit procedure that will do the trick... then, Real Analysis. Multiple solution methods, having to identify the question being asked, understanding intuitively what from your toolbox will be needed to solve the problem - the average student, even in Mathematics, will have a hard time making this leap. The one simple fact is that time and discipline are a mathematician’s best friend. One can cram for a computational exam, but success in higher level math (in my experience) is directly correlated to the consistency with which you study the material, i.e. 30 minutes per day beats 3.5 hours once per week.

The truth is, a Math PhD is not reserved for the gifted (that helps). Rather, it is reserved for the individual who can consistently struggle through challenging material and self-doubt. Believe in yourself, maintain balance between your studies and hobbies (important), and you will develop into a promising mathematician. It is helpful (I think) to equate the PhD pursuit to professional sports... THINK ABOUT THE TIME IT TAKES TO REACH THAT LEVEL. Do not be upset that your progress does not fit into your timeline, progress in every endeavor is nonlinear. Do something good for your brain and body every day, dull the negative self-noise, and I think you will be proud of yourself a year from now. Feel free to reach out if you wish to speak further.

P.S. You’re a young person on r/math, that’s saying something.