Dentistry: can we replace dental x-rays with infrared? Can we build optical panographs just using the reflections from a dental mirror? How can we monitor patients' oral health over the long term more often than just an annual visit to the dentist, and does that improve oral health outcomes?

Machine learning: what happens when we replace Euclidean metrics with p-adic ones? Distance is fundamental to so many algorithms (least squares regression; nearest neighbours; anything involving gradient descent). How do those algorithms behave over completely foreign metric spaces?