The Art of Doing Research

I write this post especially to share my experience in doing research. Now that I have graduated from ITB already, I continue my past thesis project as undergraduate student to a whole new level. I worked in a much more general situation than previous research project and it is kind of abstract, especially to those who are not familiar with my field, harmonic analysis.

So, basically my past thesis project is about strong and weak classical Morrey spaces \mathcal{M}^p_q(\mathbb{R}^n), the set of all measurable function f where \|f\|_{\mathcal{M}^p_q} < \infty, where

\|f\|_{\mathcal{M}^p_q} = \sup\limits_{\substack{a \in \mathbb{R}^n \\ r > 0}} |B(a,r)|^{\frac{1}{p} - \frac{1}{q}} \left(\int_{B(a,r)} |f(x)|^q\ dx\right)^{\frac{1}{q}}.

My main objective was to prove that strong Morrey spaces are contained in its weak spaces w\mathcal{M}^p_q(\mathbb{R}^n), and the inclusion is proper. I studied the structure of the spaces and the behaviour of some nice integral operators defined on it.

To learn something new like that, I spent most of my time reading the papers my supervisor gave me and conducted a little experiment. Do not imagine this experiment conducted in a laboratory. There is no (physical) laboratory at all! My research kit is just a stack of paper and a nice pen. Sometimes I need good food and good music. That’s all. I can just do my research in a restaurant or in a cafe or in my own room or even in the toilet (although that’d be unwise). I spent most of my time imagining, since there is no way I can touch, feel, or see the Morrey spaces. Very cheap yet very abstract.

Turns out, with high effort and lots of luck, I could prove what I was supposed to do. Up until this point, I am still amazed by that miracle.

Now I am working with the generalized version, i.e. generalized Morrey spaces \mathcal{M}^p_{\phi}. You might not want to see the definition. It is rather disgusting lol. But as the name suggests, it is a generalized version of the classical Morrey spaces. With certain \phi, we can have \mathcal{M}^p_{\phi} = \mathcal{M}^p_q. What I want to show now is that two norms in two different Morrey spaces are not equivalent. To prove that, I have to find a certain function that satisfies my hypotheses. I am spending much of my time now just to find a single function.

Now that I have been stuck for at least a month, I learn that

  1. Research needs patience and perseverance. You have to know that you will fail most of the times. At first, it seems frustrating. But hey, successful people must learn how to fail. As long as the number of failure is not infinity, I should be okay.
  2. Not many people will be able to help you. Unlike undergraduate or graduate courses which may be taken by students at the same time, research is way more personal. You should be the one who know the most about your object of study, among with other fellow researchers working in the same field.
  3. Ideas can come at an unexpected time. Sometimes when I just happen to have some ideas or just think of some approach, I immediately jot it down in a piece of paper. I fear that the ideas will never come to me again lol. Some people even say that if you do not dream about it, you haven’t done your research well. Lucky I dreamed about it once.
  4. The feeling of successfully proving something you want to prove is just indescribable. I can smile all day and the day after if that happens. On the contrary, I can be very gloomy if for a month or so, I still couldn’t get the correct approach of solving my problem. Yes, I am referring to what I feel now.
  5. You just cannot stop doing research, once you begin. For someone who is at times can be very curious (to the point where it’s kind of annoying) like me, being able to know more only shows that we actually do not know anything. That is why once you do research, you know what you know and you do not know, and since you do not know that you do not know, you want to know more. That happens all the time.
  6. Good food and good music are good catalysts. Well, since research in pure mathematics only requires paper and pen, of course I have to maintain my mood. That is, by eating a lot of delicious food. And singing. And traveling. And writing like this. This is just my way to regain my mood of doing research. I do hope after writing this and finally going back sleeping, ideas can come to me.

The more I do research, the more I am sure that this is my world. I just love doing research.

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