In Search of a New Landscape in the World of Numbers

Masato Kurihara (Professor, Department of Mathematics)

Mathematics is a field of study with a history so long that one could say it began with human history itself. Yet, it remains a science where research is actively pursued and remarkable advancements are still being made today.

Researching mathematics is entirely different from studying for university entrance exams. Mathematical research can be likened to mountain climbing. As you ascend step by step, you may reach a point where the landscape opens up, and you suddenly understand the true nature of things you previously found complex. I believe some of you may have experienced this even in high school or your first year of university—learning something new can provide a clearer perspective on concepts you have already studied. When you climb a little higher from a world of chaos, wondering about its structure, and discover an underlying order, you will surely feel that "mathematics is beautiful." However, tackling a mountain no one has climbed before—which is what research entails—requires effort, such as devising your own new tools and equipment.

My recent research focuses on number theory, particularly a field known as Iwasawa theory. This field has developed around the core theory created by Professor Kenkichi Iwasawa (Princeton University, 1917–1998). The proof of the famous Fermat's Last Theorem by Wiles—stating that for any integer n greater than 2, the equation x^{n}+y^{n}=z^{n} has no positive integer solutions—was ultimately finalized using Iwasawa theory. Today, it is a subject of research worldwide.

For example, Iwasawa theory conjectures that the number of rational solutions on a curve such as the one on the left, known as an elliptic curve, can be determined by the behavior of a function called the p-adic zeta function. I am using new ideas to study cases where elliptic curves have supersingular reduction, a situation that researchers have previously been unable to handle.