Niche and deep vs broad and new

May 27, 2025

Welcome to the third entry in my Holiday in Sicily series. Today’s reflection comes from a quiet morning walk in Taormina. It got me thinking about how researchers choose between digging deeper into old, technical questions and venturing out into broad, unfamiliar territory. Both paths have their charm—and their challenges.

The classic path: niche and deep

Much of traditional academic research follows a well-worn pattern. You take a known problem and generalize it—bit by bit. In my corner of applied math, this often means scattering problems, and the progression looks something like this:

  • 1D → 2D → 3D;
  • Constant coefficient → smooth coefficient → non-smooth coefficient;
  • Time-harmonic → time-dependent;
  • Laplace equation → Helmholtz equation → Maxwell’s or elasticity equations.

This is the incremental path. It’s rigorous, methodical, and deeply satisfying—like a well-reduced ragù. You start with something simple and add layers until you have something rich, complex, and complete.

Pros:

  • You become an expert. You know the literature, the open problems, the tools.
  • You can guide students and propose research topics with confidence. You know what’s too easy, what’s too hard, and what might become a PhD or postdoc subject.
  • You can make sharp conjectures. You see structure and patterns others miss.
  • The theory is well developed. You have energy estimates, compactness arguments, variational formulations—tools that work.

Cons:

  • It gets harder to publish. Journals want novelty, not the fifth lemma on yet another variant of a known method.
  • The community gets smaller. Your work may be elegant, but fewer people notice. The impact fades.
  • It can feel slow. Progress is incremental, and big leaps are rare.

The adventurous path: broad and new

Then there’s the other route: jumping into something new. A new model, a new field, a new intersection. Maybe it’s inverse problems meets neural networks. Maybe it’s PDEs meets large language models. It’s less like a slow-simmered sauce, more like wandering through a Sicilian street market—you don’t know what you’ll find, but it’s exciting.

Pros:

  • It’s easy to publish. Everything is new. Even basic observations can be contributions.
  • The community is large and curious. People are watching. If you hit the right nerve, your ideas can spread quickly.
  • It can feel fast. Breakthroughs happen early and often, because no one’s explored the space yet.

Cons:

  • You’re not an expert. You don’t know the terminology, the norms, or the landscape.
  • It’s hard to guide students or propose research topics—you’re unsure what’s feasible, what’s too simple, or what’s completely out of reach.
  • You struggle to make good conjectures. There’s little structure, and few known results to build on.
  • The theory is missing. You're improvising without a reliable framework or classical tools.

The sweet spot

There’s no universal answer. But good research often lives at the boundary between the two: old methods in new contexts, new methods applied to old problems. Like blending Sicilian ingredients with northern technique. You don’t need to abandon your expertise—but you might need to take it on a little vacation.

As for me, I like to keep one foot in the familiar and one in the unknown. It’s a good way to stay grounded—and stay curious. One of my PhD students is working on neural operators to regularize the linear sampling method for inverse problems—a new tool meeting an old friend. Another is following a more traditional path, extending my work on inverse problems in random acoustic media to the elastic setting.

Blog posts about academic life