In this blog post, I will write about my past life as a math student, and how my perspective changed about pure math throughout these years.

I hope that through my experience and some of the general lessons I learned, this article could serve as a source of inspiration or ideas for people that are walking along a similar career path as me.

1. The Start of a Mathematical Journey

I was fortunate enough to encounter many good teachers when growing up, enabling me to discover to joy of math. In my younger days - probably before high school, I enjoyed math, for the satisfication and self-confidence gained from problem solving. Starting from high school, through self-studying, I found the beauty of abstract mathematical theories.

It is hard to concisely describe the beauty of math in a few words, but roughly speaking, it is a mixture of:

1. The intellectual joy of exploring something that is both science and philosophy - understanding the inner workings of the world through the elegance of logic and the rigor of proofs.
2. The inherent beauty of its abstract nature - its self-contained world of concepts that can exist independently of physical reality yet still profoundly shape our understanding of the world.
3. The thrill of discovery - the unexpected surprises and patterns that emerge from frameworks built upon abstract symbols, blending both intuitive and novel ideas, as well as moments where we see seemingly unrelated theories intersect with each other.
4. Its profound usefulness - as the universal language of science and technology, it provides tools to model and solve real-world problems.
5. The learning experience - I have found that many texts are written so well and beautifully that left me in awe¹, and continued to experience the joy of problem-solving through working out exercises, that are both challenging and rewarding.

From this list, I hope that you have been convinced that I am fervent lover of math. I wanted to do math as a living before entering college, but I had concerns -

1. I always felt like in order to be a good mathematician, both effort and talent are the most crucial factors - I do not think I have what it takes to become one, as I am not a math olympiad person, nor someone that had obtained success in individual research in my highschool days.
2. I was ignorant about the job market for math graduates and was way too narrow-minded - I though it was either staying in the academia, or unhappily waiting my life to end in a place that doesn’t require any expertise in “true” abstract math (such as being a high school or cram school math teacher). I also had the impression that becoming a professor at National Taiwan University (my alma mater) is going to be extremely difficult, as a lot of the professors at that time did their PhDs in highly extinguished universities in the states.
3. I thought being an engineering student with abstract math skills can be a plus.
4. I recognized that I still know nearly nothing about what a math researcher’s life was.

I ended up making the best use of my college entrance exams scores, and enrolled in NTU as a student majoring in mechanical engineering. However, I discovered that I still have a thirst for math in my first semester:

1. I found myself spending a lot of my free time in the math department’s library, reading books and exploring topics in maths such as general / algebraic topology, category theory, linear/ abstract / commutative / homological algebra.
2. While I can appreciate the practical aspects of the fundamental classes taught in the mechanical engineering department, I feel like at that time, something was missing - the theoretical aspects of things. For example, in Calculus, while memorizing ways to solve integrals or derivatives were fun, I wanted more proofs and systematic theories.

After this realization, I started taking classes in the math department in the next semester, and successfully transfered at the start of my sophomore year. It was one of my happiest moments in my life, as the transfer process wasn’t easy (at least in NTU), so at that time, I made up my mind to completely isolate myself in the world of math, disengaging in any meaningless social interactions, and dedicate every time and energy to learning math, to get exceptional in at least one subfield.

2. A Loner’s Struggle