I’ve been quite busy with PhD interviews recently, and I’ve found the experience to be very rewarding. I see interviews as a great opportunity to engage in meaningful conversations with experts. During the Q&A sessions, I often ask research-related questions that I’m truly passionate about—some specific, others more philosophical. One question I frequently ask is:
While the way this question is phrased may vary, the underlying issue remains the same. Below are some of the responses I’ve encountered, based on my personal recollections, so they may not fully reflect the professors’ original views.
Another question I mostly care about is strategies for conquering (theoretical) challenges. It is very rewarding to obtain professional guidance on this question!
What is good research?
First, a professor from the business school shared her perspective: when evaluating a paper, she typically looks at two key dimensions: first, whether the paper introduces innovations in methodology or mathematics, and second, whether it has a broader audience.
I also had a conversation with a renowned expert in optimization, who offered his view: the ideal research is simple and natural. However, he noted that such research is rare and, unfortunately, it’s hard to pinpoint exactly how to achieve that simplicity.
Lastly, in a discussion with another professor from the business school, he highlighted a particularly meaningful type of research: the kind that challenges widely accepted beliefs. For instance, when everyone believes conclusion A is true, your research demonstrates that this accepted view is actually biased.
Coincidentally, today I attended a lecture by Prof. Christopher Ryan, who discussed his book Paths to Research. One chapter of the book addresses Research Aesthetics, and it outlines six possible perspectives on what constitutes good research. These perspectives closely align with the views I’ve shared above, so I’d like to present them here:
- Novelty: Introducing new problems or new methodologies.
- Usefulness: Providing solutions to real-world problems.
- Difficulty: Demonstrating mathematical or intellectual brilliance.
- Generality: Addressing research problems that have broad relevance across various fields.
- Surprise: Something that challenges conventional wisdom (economists especially love this).
- Simple, yet profound: The beauty of research lies in its simplicity and depth.
Different people emphasize these dimensions differently. Personally, I prefer simplicity and generality, as they allow us to uncover the shared, fundamental nature of seemingly different things. Prof. Ryan mentioned in his talk that great research can encompass all six of these dimensions. The key is to tailor your presentation to the preferences of your audience. For example, for those who appreciate complexity (dimension 3), you might showcase your elegant proofs, while for others who value surprise (dimension 5), you might share compelling stories. The ideas presented in his book are insightful and inspiring, so I highly recommend giving it a read!
Strategies for tackling challenges
I asked two professors at Columbia University for their advice on tackling challenges. They shared some valuable insights!
One professor suggested that verbalizing your ideas or challenges can be an effective strategy. Simply talking through a problem with someone else can help clarify your thoughts. Another useful approach is keeping a record. When working on a problem, you may come up with multiple potential solutions—writing them down allows you to systematically test each one. If a method doesn’t work, document why it failed or note any counterexamples. This practice prevents you from revisiting discarded ideas unnecessarily and helps you refine your thinking. Lastly, the professor emphasized the importance of internalizing the problem so well that you can think about it even without notes—whether you’re walking or engaged in other activities.
Another professor highlighted the power of simplification when tackling difficult problems. Breaking a problem down into a more manageable form—sometimes even reducing it to a toy example—can provide valuable intuition. In fact, identifying the right subproblem to start with is often the hardest part. I have personally found this strategy incredibly helpful when working on theoretical problems!
