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My Second Ph.D. Year
21 Jul 2019
(中文版原文連結)
In the blink of an eye, my second year at Harvard had come to an end. Now, the summer is halfway through, and soon, I’m going to cross the middle line of my Ph.D. life. Compared to two years ago, I have perhaps made some progress and improvement, yet I still fall short of what I had expected of myself. During the CCC (Computational Complexity Conference) this summer, my friend Lijie gave me lots of his thoughts and advice. Although those might not be completely applicable to me, they indeed remind me to rethink my future plan and direction for the rest of the time.
What have I done in the past year?
In retrospect, this was a really frustrating year. The drama in life influenced my focus and time spent on research. Meanwhile, the choice in research direction was also not very ideal. The results in both research and learning seem to be below the bar.
This year, I have mainly been focusing on algebraic complexity, in particular, matrix rigidity and border Waring rank. The former is a research program that studies a combinatorial problem (i.e., circuit lower bound) in an algebraic setting. The ultimate goal is to (explicitly) construct a matrix that is far away from low-rank matrices into $\ell_0$ norm. This will imply super-linear lower bounds for logarithmic depth circuits, which will be a breakthrough in our field. I really like the idea of matrix rigidity because it turns a seemingly complicated and arbitrary structure (i.e., circuits) into an algebraic problem that is much more convenient to think about. Nevertheless, after almost 50 years since its conceptualization, there has been very limited progress in matrix rigidity. In the past year, my collaborator Sasha and I have tried multiple old and new approaches to attack the problem and even lowered the goal. However, it always feels like we are just paddling on the surface of a bottomless ocean. As for the other line of research about border Waring rank, initially, I just found the math to be interesting and then spent a semester learning representation theory to understand the paper. But once I finally understood what’s going on in the paper, I further realized that more math (e.g., Lie algebra, algebraic geometry, etc.) is needed for me to make some real contributions to the problem.
Looking back, I seemed to pick and dig into a research problem simply depending on my interest and neglect whether me or the people around me have the ability to tackle the problem. I know we were always told to try our best and not underestimate ourselves. But in terms of pursuing frontier research, now I believe it is very important (and sometimes necessary) to arm yourself with enough gears and surround yourself with the right resources and experts. Like what Lijie asked me: “Why should this problem be solved by you?”. Other than having more time to think about the problem, what’s the strength I have over those experts in the field?
Positioning myself
Perhaps this goes back to a basic but important question: what is my position in the field (or even the world)? Every researcher has their own strengths and weaknesses. Some excel in high-level intuition, while some are skillful in complicated calculations. Some know various math tools, while some are aware of the methodologies in different fields. And of course, some people, like me, may not be good at any specific thing… But anyway, I guess it is very normal for a Ph.D. student to feel that they are not good at anything. However, this should not be the case after graduation. One has to be the top person in the world in at least one skill to survive in academia. So then this boils down to the next important question: what skill do I want to cultivate for the rest of my Ph.D. and how?
In my case, I have always expected myself to be a broad-minded person (in TCS, math, or even in whole academic fields) and, at the same time, specialize in a few small directions. So far, it seems that I am not doing a very good job along either axis. On the broad side, I might get exposed to a wide variety of subjects without significant internalization. For many things, I am still at the level of “having heard the jargon/terminology before but not yet having the ability to have a meaningful conversation on them.” On the deep side, I really want to specialize in arithmetic circuits and boolean circuits. But I am currently still in the situation in which “I know many important results and key intuitions, but I am not yet able to fluently jump between different concepts and, let alone, ask interesting new questions.”
So, in the end, I have always vaguely positioned myself in certain directions. The problem is that I have not pushed hard on myself towards these directions. I am sometimes too lazy and easily get distracted by other stuff, hence deviating from the main line. As there is not much time left, I should carefully think about how to push myself back to the road I want.
Next step
Balancing time for self-development and a specific research problem is crucial. One must carefully consider before investing a significant amount of time. Do you truly enjoy this research problem? What is the meaning of this research to you? Are you ready to tackle it now? The two directions that I’ve been working on in the past year seem too specific, and I’ve lost sight of the big picture. Additionally, I’m not entirely prepared from a technical standpoint.
As someone who wants to specialize in arithmetic circuits and boolean circuits, the key problems like lower bounds and derandomization exist, and there are a few different circuit models with their own frontiers and barriers. Currently, I only have a vague understanding of the state-of-the-art results in most settings and lack a crystal-clear understanding of what the bottleneck is and what the potential breakout point could be. Therefore, the next step is quite apparent: read more, think more, internalize the knowledge, and build my library. I really hope to regain my good old habit of keeping notes rather than having blurry concepts in my mind.
In addition, mental stability is crucial, particularly in TCS, where smart people abound, and the field moves rapidly. One must find a way to be less influenced and keep moving forward, and most importantly, positioning oneself and not isolating from others. Sometimes, I feel that choosing this path is self-torture. Nevertheless, the sense of fulfillment and satisfaction it brings is extraordinary. We only get one shot in life, so let’s not leave any backdoor for ourselves and keep striving!