May 23, 2025
Welcome to the first entry in my Holiday in Sicily series. Over the next 13 days, while vacationing in Sicily, I’ll be posting here every other day. In today’s post, I reflect on the practice of writing research memos—why I do it, how it helps, and how surprisingly enjoyable the process can be.
At his retirement conference from Oxford, Nick Trefethen gave a talk on writing research memos. He shared seven compelling reasons for doing so:
Nick, who was my PhD advisor, passed that habit on to me, and I’m so grateful he did. I’ve written over 200 memos—60 during my PhD, 50 more at Columbia, and 90 since returning to France. Now my students are picking it up as well: Victor wrote just under 20 in his first year, and Esteban, who just started, has already completed his first one. Nice work, guys!
But what is a memo? Before I highlight the reasons I’ve found this habit so valuable—and add a couple of my own—let me explain.
A memo is a short research report, typed in LaTeX and generated as a PDF—I like using a two-column format. It synthesizes the progress (or lack thereof) made over a short period of time, typically a week or two. A memo blends formal mathematics when the work is theoretical, with updates on coding efforts, numerical results, and figures. It’s meant to be concise, rigorous, and practical—a snapshot of where things stand.
To keep track of them, I maintain a summary table in Notion (which I love and highly recommend). Each entry includes a brief description of the memo’s content and a tag corresponding to the project it belongs to—e.g., Nonlocal PDEs, LSM with random medium, or SciML—so that everything can be easily sorted and filtered.
Now, back to the reasons this habit has been so valuable.
One of the worst feelings I had during my BSc and MSc was knowing I’d done a computation before, but not remembering how, and being unable to find the draft. We can’t remember everything—and that’s okay. But we must know where to look.
Whenever I do a new computation or learn something that requires a bit of digestion, I type it up in a memo. Weeks, months, or years later, when I need it again, I just open the memo and—boom—everything comes flooding back. It’s a wonderful feeling.
As Nick mentioned in his talk, books take years, papers take months, but memos take just a day or two. That makes them a perfect tool for staying engaged with research on a daily basis.
This is especially valuable during a PhD—particularly in mathematics—when progress can be slow and setbacks frequent. Getting stuck is part of the process, but it can take a toll on your mental health. Impostor syndrome is real. Writing memos regularly, even just to document things that didn’t work, helps maintain momentum. And it ties back to Reason #1: when you revisit a paused project, a well-written memo reminds you exactly what failed and why—saving you from repeating the same mistakes.
And beyond the practical benefits, there’s something deeply rewarding about finishing a memo. Printing it out, proofreading it, and adding it to the growing stack gives you a tangible sense of progress—a small but satisfying moment of self-achievement that keeps you going.
I’ve had some fantastic blackboard sessions—brainstorming with Mikael Slevinsky on nonlocal operators on the sphere, and more recently with Luiz Faria and Pierre Marchand on convergence rates for curved boundary elements.
But every productive chalkboard exchange was always followed by something more: a quick photo of the board, then a memo or two. That’s where the real value crystallized. The blackboard is great for sparking ideas, but the memo is where those ideas are clarified, organized, and made durable.
When it comes to presenting results—especially to colleagues or supervisors—nothing beats a well-written memo. Writing forces clarity. It sharpens your arguments, ensures consistent notation, and gives structure to your thoughts. You’ve already done the mental work of deciding what matters and how to communicate it.
By contrast, unprepared explanations at the board often fall flat. The handwriting is difficult to decipher, the argument becomes unclear, and the notation is often inconsistent or poorly chosen. Too many times I’ve watched someone try to explain a result on the fly, only to get tangled in their own logic or lost in unclear symbols. I’ve had to interrupt just to restore some clarity—and that’s frustrating for everyone involved.
A memo avoids all of this. It respects your own time, and it respects the time and attention of your audience.
This one is crucial, especially for graduate students. Mathematical writing is hard. English writing (for non-natives) is hard. LaTeXing is hard. And you only get better by doing it often.
It’s then the role of the supervisor—as a mentor—to help you improve by pointing out what needs fixing: shaky reasoning, unclear explanations, inconsistent notation, awkward English, LaTeX quirks, formatting issues. That’s why reviewing memos is so valuable. Nick used to read his students’ memos before meetings, and I try to do the same. You get better by practicing; I support that practice by giving feedback. Each memo is an opportunity to sharpen your skills and grow as a researcher.
I usually write papers in a week. Why so fast? Because each paper is built from half a dozen polished memos. Writing the paper becomes a matter of stitching them together into a story. The mathematics, proofs, and most figures are already done. All that’s left is to make it look good—clear, neat, and polished. It’s fun.
I wrote my PhD thesis in seven weeks. It had five chapters, plus an introduction and a conclusion—each chapter roughly corresponding to a research paper. Because the memos and papers were already polished, the writing process was smooth—and enjoyable. The thesis was accepted with no corrections (about 1% of maths PhD theses at Oxford achieve that).
When I’m doing pen-and-paper research, I keep all relevant memos printed on my desk or displayed on my screens. This way, if I need a specific result, I don’t waste mental energy recalling it—I just glance at the memo.
Was it \(i\pi\) or \(-i\pi\)? Did the result require a compact operator or just a bounded one? I don’t need to remember—I just look it up. And that frees up my mind to focus on creating new ideas and making deeper connections.