Matthew Romney is a new faculty member in our department. Welcome to our department!
DE: What type of mathematics are you interested in?
MR: My research deals with metric spaces and their structure. A metric space is anything for which there is a notion of distance. Aside from the obvious examples (such as Euclidean n-space or the n-sphere), there is a huge variety of other metric spaces of interest arising in many applications. For example, certain problems in computer science can be framed as questions about embeddings of metric spaces. The basic problem in my work (often called the uniformization problem) is to decide when a given space can be mapped onto a “canonical space” (one with simple or nice geometry) by a map that doesn’t distort shape by more than a bounded amount (for example, by a conformal or quasiconformal map). I like concrete, specific problems, and especially ones that are suited for computer exploration.
DE: Do you have a favorite open problem?
MR: One of my favorites is Koebe’s conjecture (1908): every domain in the complex sphere can be mapped conformally onto a circle domain, meaning a domain whose complementary components are either geometric circles or points. The simply connected case is Riemann’s mapping theorem, typically covered in a complex analysis course. Koebe himself proved the finitely connected case, and the countably connected case (currently the best result) is due to He and Schramm (1993). I like the conjecture for its simplicity, enough to make one think that a solution is just around the corner. For example, one might try to approximate an arbitrary domain with a sequence of finitely connected domains, apply Koebe’s result to get a corresponding sequence of conformal maps onto circle domains, then pass to a limit. However, this approach fails badly for a surprising reason: while one does in fact get a limiting map, it doesn’t necessarily map to a circle domain! So it takes some time to really appreciate the subtlety of the problem.
DE: What types of courses are you most excited to teach?
MR: Most of my teaching has been in the area of analysis (real analysis, complex analysis, differential equations), though I always love to teach something new. I’m excited to be teaching Numerical Analysis this fall and have the chance to delve more into the computational side of the subject.
DE: What have you liked the most about Hawaiʻi so far?
MR: This might be cliché, but I would definitely say the people. My family and I have felt so welcome here, by the department, our neighbors, the community. First arriving in Hawaii, we were welcomed at the airport by people we hadn’t yet met in person. When another family saw we had rented a car to get through the first couple days, they let us borrow their own. Living in faculty housing has been great, with so many families in a similar life stage all around us.
Also, needless to say, the nature here is spectacular.
DE: Can you tell us something about yourself that might be surprising?
MR: Aside from mathematics, I’m a big fan of learning languages. I’ve spoken Spanish for many years and actually speak Spanish at home. Before coming to Hawaii, I had the opportunity to do postdocs in Finland and Germany, and I became fairly proficient at those languages while there.
