Having completed the first year of my PhD program without the absence of great pain, I figured it might be beneficial for me to summrarize my first year as it’s been a long learning process. Making the leap from a small undergraduate program to a fairly good PhD program came with the realization that there were a lot of holes in my abilities and mathematical maturity, so the point of this post is to summarize those, reflect on what I could have done based on the lessons I learned, and look forward to what I want to do this Summer and onward.

Looking Back

I’ll start off by saying that CSU East Bay did not prepare me for a PhD in mathematics. I love the people I spent time with there, and my faculty were amazing and all came from strong schools, but our department is small and I was a minority among my peers, many of whom had no desire to continue studying after their undergrad. This was perfectly fine and I’m glad they cater to the demographic that needs them. However, I only partially had the urgency to really work beyond my undergrad experience in preparation for the PhD. I had young energetic faculty to learn from, even one who recently graduated from my PhD institution, but even with all the hours I spent talking and learning from them privately, there were many things I could have gotten more insight on. To name a few:

  1. What subjects or gaps should I be actively learning or filling in that will prepare me for coursework in the PhD program. This one is quite obvious in retrospect, but never having taken point set topology or not knowing what a group action was did hinder me at the start of my first quarter.
  2. Productivity and Workflow. I.e. how to balance my time in the PhD program between coursework and TA responsibilities. What are some note-taking or LaTeX habits or apps that would benefit the process.
  3. How does one learn mathematics (it never hurts to hear what others have to say on this matter)
  4. I should have spent more time solving problems on various subjects

With that being said, I don’t think the insights I gained were meaningless. Although not being a first generation college student, I was the first in my family and social circle to pursure a graduate degree. Much of my time with my faculty was spent on the application and refining it, so I am thankful that I was able to make it into the school that I did, especially with the background I had.

Looking back on my undergrad, it certainly didn’t help that I attended community college and then transferred. Doing so somewhat fixes where you start as a junior which leaves you with only 2 years to advance quickly through course work and do some kind of undergraduate reserach. On the topic of undergraduate research, I have some regrets about my research as I could have gotten much more out of it if only I had the humility to ask more dumb questions. It was my first time reading any kind of research paper and working on a problem with no answer, so I attempted to tackle these things the same way I work on problem sets and read introductory textbooks. Needless to say, it didn’t go so well. My professor and I did end up producing quite a pretty paper, but I don’t consider the result anything too impressive. Sometimes, I feel my time would have been better spent solving problems and self-studying more, especially since the results weren’t complete in time for graduate school applications anyway. Nevertheless, I’m happy with what I could do in a crazy 2 years of realizing I want to pursue mathematics and trying to figure out my way through that whole process.

The First Year

I made the unfortunate decision to take both a year-long sequence of Algebra and a year-long sequence of Real Analysis. They’re each already difficult enough on their own, so the combination of both was especially brutal. I more or less had to sacrifice one in order to succeed in the other. Naturally, coming from a undergrad department full of analysts, I sacrificed Algebra in order to succeed in Real Analysis. However, in reality, I just barely scraped by both, which is much more than I could’ve asked for.

There are some things I’m especially displeased about with how I chose to study and learn this past year:

  1. I was not honest enough in asking questions and being open about my lack of background. Being surrounded by peers that, at least on the surface, seem like they know what they’re doing put a lot of pressure on me to also sit back in silence and nod my head to the strange language being spoken during lecture. I squandered several opportunities with incredibly kind and generous professors to be open and properly learn from them. I now know that graduate courses are a lot more fluid in that professors will make certain adjustments based on what students know and don’t know, so for the future, I am trying to take advantage of this and most importantly, being honest. I should also post more on MSE about topics or things in texts that I don’t undestand.
  2. I did not do enough to prepare myself to learn. By this, I mean reading the text. posted notes, or rereading the previous lecture. Too often did I go into a lecture, having completely forgotten the content of the previous, when in fact, had I just skimmed the previous one, my understanding of what would be lectured on would have increased dramatically. I think, for myself, that I need to typeset my notes, or some kind of notes. My handwritten scratch does not suffice as something to reference, and the idea that “I have the textbook to reference later, so why should I take notes” is no longer feasible. Graduate textbooks lean too far towards brevity for me to use them as references when I haven’t even learned the material the first time over. An alternative to this would be to forgo notes entirely for a large amount of example problems and corrolaries of theorems worked out. This would naturally lead to rereading and internalizing the material. Of course, TeXing this would be a good idea.
  3. I started Winter and Spring quarter extremely strong, but due to unforseen circumstances, my routines and discipline were thrown off each time, and I always found myself scrambling by the end of the quarter to finish work while cramming for finals. I’m hoping this is just a symptom of being online.

However, I will also say that I think I was too hard on myself and I lost a lot of the joy of studying mathematics that I had in undergrad. COVID certainly didn’t help and being remote for one’s first year in a PhD program is a ridiculous idea. I avoided online courses my entire life, so to suddenly have to take extremely tough courses online while TAing online during a global pandemic just sounds like as much of a disaster that can happen. My mental health and physical health suffered intensely, and only now do I have the opportunity to begin to mend them.

On a brighter note, I did learn an incredible amount, both on the side of actual mathematics and also on how to be a good graduate student in mathematics and what it means and what it does not mean.

  1. I learned how little grades were taken seriously in graduate school, at least at UCR. It felt like every syllabus was a sliding bar in order to help those who were genuinely trying to pass. Weights on homeworks, midterms, and finals didn’t seem to ever be fixed. I’m allowed to say this because I know for a fact that I should have failed my second quarter of algebra.
  2. I was told by many upperclassmen that it is okay to feel like you don’t know what you’re doing your first year. I’m not quite sure when that feeling is supposed to end, but I do feel much more “on my feet” as opposed to this past year of crawling and falling.
  3. It is unhealthy to spend all of one’s time studying math. Mathematics benefits from time spent on it as well as time spent doing other things, like a physical activity. Inspiration to solve a problem, or renewing your eyes and mind can work wonders on what you’re studying. So many times have I been stuck on reading or solving a problem, taken some time off to go for a walk or to sleep, and the next time I come back, the understanding or what I’m reading or the idea I’m looking for comes much faster, if not instantaneously.
  4. I enjoy working on problems in analysis much more than algebra, but I did have fun in Galois theory.
  5. No one reads the textbook as much as you think they do (except that one person), but please read as much as you can.
  6. It is an important trait for mathematicians to always be open to criticism and doubt, both from our peers and ourselves and to never become complacent in our work or understanding. Try to never settle on something you feel is complete. If a paper is done, ask a friend if they can review it. If a problem is solved, think about how it can be generalized, or consider why each part of the hypothesis is necessary, or think about the converse. If something seems quite trivial or elementary, think about how you might teach it to those that don’t see it as trivial.

Looking Forward

From the second year, there is also a heightened importance in quickly finding a research direction and some potential advisors, or even better, finding an advisor. I came in without much in terms of a research direction. Algebra may be a bit out of my reach with how I performed in my sequence, but I think the most important thing for me is to find someone I am happy working with. Mathematics is infinite and it will always be there, but the human aspect of the PhD process is not. I’m very quickly recognizing that I do need someone who has the patience for all my silly questions, but also someone who can push me quite hard to learn high level material and to make progress. It’d also be nice if they shared a similar sentiment about how mathematics needs to be more inclusive and equitable. I really just want someone kind. I know there is great value (careerwise) in finding an older experienced advisor whose name carries a certain weight to it in their field, but from what I’ve seen (which isn’t much), the younger faculty generally fits my criteria a little more closely.

I think I have made it clear that I need to do a better job learning mathematics and I’ll take the steps to do so, but one piece of advice I really want to adopt is something I heard from an interview with Inna Zakharevich and adjusted slightly. Briefly, it is to absolutely do at least one little thing a day. There will be days where one does not feel inspired at all to look at mathematics, but for the sake of one’s “muscles”, just doing something small like solving a simple problem or making some progress on reading will gradually add up and build up your discipline to do a little more later one. The important thing is to take it slow and be honest.

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