Research, ah, research. Research has always been the medium of progress and innovation in human civilization. From the most primitive forms of discovery, to the most complex and groundbreaking forms of advancement, research has always taken multiple forms, be it in a playful and casual sense: twiddling one’s thumbs and airily thinking, or in a deliberate and serious sense: reading books, websites, and computing without distraction. Regardless, all of these forms of thinking and “work” are research. Yet, it seems that research, in the current world, has taken a familiar yet unprecedented turn; whereas before the leap to providing education we had been closed-minded and unaccepting of new progress from people of diverse backgrounds, we now take a similar leap in the alternate direction: becoming closed-minded to the meaning of research, and the meaning of making progress in different fields in such a fast-moving and elitist field of study.

Today we will be talking largely about mathematics research, especially high school mathematics research. When I first began learning advanced mathematics beyond the conventional elementary work, I did not really consider the idea of research, and was not equipped with the tools necessary to even confront the concept of “making progress”. In the field of mathematics, “progress” is defined very differently from other fields. Rather than simply writing a paper or publishing a book on some lesser-studied topic, the complexity and crudeness of mathematics research converges on one thing: deriving some fundamentally unique and original idea, and from that idea, constructing a web of ideas to work back toward our current arsenal of mathematical ideas. Though this seems easy in word-form, it is far from that. With the enormous breadth of knowledge already dominating the current realm of mathematics, it’s almost impossible to even touch the surface of all current fields and understand everything in utmost detail. Conceptually, yes, but technically, not at all. The amount of time investment required to even begin conducting significant research in any field is enormous; to even begin is a culmination of over 10 years of decent schooling, from high school to undergraduate to graduate and then to pursuing a PhD. This truly illustrates how much human knowledge has truly progressed, and how amazing it is that several thousand of the brightest minds are able to pursue something so representative and so pure of the homo sapien sapien species.

Yet, as competition becomes increasingly fierce, with some of the most competitive schools such as the India Institute of Technology, Tsinghua University (清华), Oxford University, the Massachusetts Institute of Technology, or countless more becoming even more and more elite, the idea of conducting any form of research prior to university has become more a matter of “proving oneself to college admissions”, or to demonstrate some degree of interest in a field, and has placed itself on a pedestal representing the “ultimate objective” of a highschool student to appear dedicated to universities. This is nonsense. Before we continue, let us define two different types of highschool research. The first variety of research is transparent; this type of research consists of work that is either interdisciplinary or focuses largely on applications or experiments. This area of research tends to be somewhat easier, and unique and original problems are not difficult to come by. We will call this Type I research. The second variety of research is research that is incredibly gruelling, and relies more on theory and rigour than practice; this type of research consists of rigorous work on theory as opposed to applications, and problems in this area are extremely difficult to come by. We will call this Type II research. The majority of highschool students who have the opportunity to conduct research choose - or perhaps are required to - to conduct Type I research. Many papers on lab experiments, biological analyses, and some form of chemical description are published each year. Type II research, alternatively, only produces a mere few papers. Why may this be? When I first faced Type II research, many things were new to me. I was entranced by the idea of publishing something significant in a journal, and was looking forward to achieving something spectacular. I had been told repeatedly and had seen on social media how people had conducted research in many different disciplines, and it seemed to me that research, especially original research, was a requirement for being accepted to a good university, and was something to “check off” on the long list of objectives in applying to university and living an optimal pre-college life. This mindset is what produces poor quality and ugly research. While I observed many of these research projects, I noticed something in common: they were all Type I research. Rarely, if at all, did I see Type II research: perhaps something in rigorous mathematics, or perhaps something in physics that did not involve some crude application? Not at all. This surprised me a great deal. While I was certain that these projects were great research projects, something bothered me, and I couldn’t pinpoint it exactly.

When prestigious universities review applications from highschool students, they search for well-rounded, well-composed, and curious students. The last criteria, “curious”, is where research falls. This implies that regardless of whether a student conducts Type I or Type II research, universities will almost always place these students in the same categories. This does not consider the difference in time and effort spent between a student conducting Type I and Type II research. The culture surrounding Type II research differs drastically from that of Type I research. In a way, Type I research is far more accessible to a general audience in that it is both visualisable and relatable. One instance of how Type I research and Type II research differ in modern culture is through cinema: Type I research is depicted as a remarkable thing with many moving parts and science-fiction-esque concepts, whereas Type II research is a jumble of unrelated ideas - and even if the film is good, it often misses the real rigour. There are two reasons for this: the audience does not care about rigour, and the director or screenwriter does not know what to portray; these people know that what they are depicting is exceptional, yet they do not have the necessary tools to convey its significance. When we speak about universities, appeal falls more on relatable and understandable things than on rigorous work. There is not much I can say about this except that it is a failure of scientific - specifically mathematical - communication. The reason that many beautiful ideas in mathematics and similarly disregarded fields are not well-understood is because they are viewed as foreign, and especially in America, there is a long-standing culture of avoiding foreign things. This is very unfortunate, as with even just a little bit of inspiration, mathematics and its beauty are always waiting with welcoming faces.

What does making research progress mean? Type I research does more to expand current knowledge horizontally, as opposed to Type II research which extends itself vertically. New ideas in Type I research are frequent, so often there is a long list of papers and books that is released, with ultimately identical objectives depending on the trend of a certain time period. The nature of Type I research is to travel stagnant with occasional climaxes, where a revolutionary new idea is introduced every once in a while. What follows is a boom of new ideas and diverging papers, but not long after everything turns gray and bleak as people wait for the next climax. In Type I research fields such as biology and chemistry, the main restriction of progress is technology. Our abilities are as good as our own barriers. Type II research approaches things differently. In Mathematics, there are always diverging ideas, and while revolutionary ideas are truly monumental, the realm of possibility is even more endless than one can imagine, and people strive nonstop, nearly incessantly. It’s almost as though I discover a new idea or entirely new field of study every day. Mathematics has no bound on progress. The only thing preventing progress in mathematics is our mental capacity, not the technology that we rely on. I am no humanist, but the fact that I study mathematics is the most human thing that I do. While technology will eventually - and already does - offer the most advanced forms of computation and artificial intelligence, the field of mathematics is what will never become obsolete, because it is simply that impossible to replicate. Type II research is just that: investigating a problem that seems so unreal and so disconnected from reality, pursuing something so inherently human and anthropogenic.

And even with all of this said, Type II research remains a mystery to many, especially to college admissions. While scientific communication of Type I research is crucial to nurturing a developing society, the push for Type II research to be placed in the forefront of modern human progress is not only important, but necessary. Now that I look back, I realise what had disturbed me the most about Type I research projects: the melancholy of the fact that the only thing that can resurface in modern society is something that can be generally understood, and not something that can be appreciated to its fullest.

Maybe this is just a naïve interpretation of something that I don’t fully comprehend, but these are some of my thoughts.