To My Aspiring Mathematician Friends,
I would first like to thank Professors Thomas Koberda and Aaron Silberstein, the founders of the Fellowship for Pre-Graduate Studies in Mathematics program. The two of them, along with the program as a whole, have greatly guided and supported me in my journey through undergraduate and master's studies. On this occasion, I would like to share my personal experiences along this path. My reflections will primarily focus on studying mathematics and conducting mathematical research. These are just my personal experiences and are highly subjective, so they do not represent a universal path to success. Everyone must find the path that suits them best.
First, let me introduce three key stages in my life. In high school, I specialized in mathematics and participated in the International Mathematical Olympiad (IMO). For a university degree, I studied at the University of Science in Ho Chi Minh City, and I pursued my master's studies at the Hanoi Institute of Mathematics. These three stages have shaped me in a unique way.
I began studying advanced mathematics during high school, which gave me a certain mental preparation for university studies. As a result, I started working on research papers as early as my first year of university. Honestly, I think this was a bit premature for a student who had just begun formally studying advanced mathematics. If I could go back, I would spend more time building a solid foundation before diving into research. This is a major difference between me and most other students: I was exposed to mathematical research very early.
Later, I decided to pursue a master's degree at the Hanoi Institute of Mathematics. Looking back now, I see that this was a wise decision. However, it also represents a second difference in my path compared to those who participated in the IMO. Typically, IMO participants tend to study abroad early, either during their undergraduate years or immediately after graduation, because of their strong academic foundations, language proficiency, and robust networking. However, I chose to continue my master's studies in Vietnam. This decision allowed me to further develop my academic expertise, acquire essential skills, and foster valuable relationships within the research community—areas I had not focused on during my undergraduate years because I was too preoccupied with writing papers.
The third unique aspect of my journey lies in the order of my academic focus. Typically, mathematics students concentrate on mastering foundational knowledge in their undergraduate years and focus on research during their master's or PhD studies. Generally, the approach is to "study first and research later." However, I did the opposite: I focused on research during my undergraduate years and concentrated on learning in my master's program. This was an unusual process, and I do not recommend that anyone follow this approach. As a result, my academic experience was different from that of most mathematics students, different from those who took part in IMO, and different from other master's students in the field. I don’t know of anyone else whose trajectory was similar to mine.
Next, I will elaborate on my "subjective advice" for studying mathematics at the undergraduate level and share my personal experiences during the "master's degree phase." In my undergraduate years, as mentioned earlier, I focused more on mathematical research rather than thoroughly learning foundational skills. The advantage of this approach is that I had high-quality papers in English, especially my undergraduate thesis, which was an expanded version of my first research paper. This opened up many opportunities for me to present at conferences, and it also made my CV more impressive in terms of research, paving the way for my future path. However, this approach also had significant drawbacks. Because I concentrated on research, I did not take many of my other courses as seriously as I should have, resulting in poor grades and weak knowledge of the fundamentals. Moreover, I focused entirely on mathematics without developing other essential skills, such as LaTeX formatting and presentation skills. This was extremely detrimental at the outset of my research career. Over time, I felt my knowledge was impoverished, and as a consequence I experienced considerable stress. Therefore, my advice to mathematics students is as follows: focus on studying all the core mathematics courses thoroughly and develop essential skills like English, LaTeX, and teaching skills. Only when your thinking has matured should you start research, and you should not aim for research papers right away. Instead, try to write a professional document in English using LaTeX. A paper in English is enough for exchanging ideas with foreign professors or at least to apply for the Fellowship for Pre-Graduate Studies in Mathematics.
Regarding the "master's phase," I put this term in quotation marks because during this phase, I did not only study at the Hanoi Institute of Mathematics but also learned from Professor Thomas Koberda. Both modes of study helped me become better at presenting and, more importantly, provided me with a broad and deep enough knowledge base to carry out long-term research. This could be considered a "re-learning" phase for me, due to the gaps left in my undergraduate years. I revisited subjects like Measure Theory, Differential Geometry, and Differential Equations that I had not fully grasped during my undergraduate studies. In particular, studying under Professor Koberda allowed me to access many advanced fields not widely taught in Vietnam, such as fractal geometry, hyperbolic geometry, and algebraic graph theory. Through formal learning at the Institute of Mathematics and additional learning with Professor Koberda, I received a high-quality educational program comparable to those at top universities in the world. I consider myself very fortunate to have had the opportunity to "relearn" in this way. In terms of skills, I made significant progress compared to my undergraduate years. My English skills improved during my master's program with the support of the Fellowship program, and by the end of my master's studies, I achieved an IELTS score of 7.0. However, the most important skill I developed during this phase was presentation and teaching. Through weekly seminars with Professor Koberda and other professors at the Institute of Mathematics over the course of two years, I gave many talks on mathematics in both English and Vietnamese. This process gradually helped me improve my presentation skills, and I no longer felt exhausted when presenting as before. It is important to note that teaching skills are critical for pursuing a PhD abroad, particularly in the United States. Thus, the "master's phase" provided me with a significant advantage in applying for graduate programs, not only in terms of expertise but also in terms of skill development and procurement of recommendation letters.
In conclusion, I hope that mathematics students can find the path that suits them best. This article is just my experience, and as you can see, my journey has many unique aspects that I do not recommend for everyone. However, I hope my reflections will help guide you in choosing your learning methods and shaping your future directions. Finally, I would like to affirm one thing with certainty: the Fellowship for Pre-Graduate Studies in Mathematics is a valuable and reliable program for mathematics students, where professors provide the highest level of assistance and guidance, like a fairy tale come true!