Many fields of mathematics are hundreds, or even thousands, of years old. By contrast, the Katsura Group researches operator algebra theory, which is a relatively new field, arising in 1929. It was originated by von Neumann, one of the 20th Century's most influential scientists, to describe quantum mechanics mathematically.
Q"Operating, in ordinary language, means acting on something. In my specialty, operator algebra, what's acted on is an abstract mathematical object called a Hilbert space. The subject that acts on the Hilbert space is called an operator. With operators, if you do one operation, then a different operation, and think of the two as a combined operation, a multiplicative structure arises. Also, the object called a Hilbert space involves an additive structure, and you can use that to define addition between operators. In mathematics, structures where addition and multiplication are conceivable are called algebras. In operator algebra theory, we investigate the various structures of algebras formed by operators."
The mathematical features of operator algebra theory can be expressed in three words: infinite, topology, and non-commutativity. Infinite means the object has an infinite size, and this results in various mysterious phenomena that don't arise in the finite world.
Also, in operator algebra theory, we don't exclude the infinite nature of objects as being pathological; instead, we try to tame it using various tools and methods. The most powerful tool is our second keyword, topology. In operator algebra theory, we try to control infinity using topology, which varies depending on situations. Different topologies lead different concepts, C*-algebras and von Neumann algebras.
Our last keyword, non-commutativity, expresses the phenomenon where considering a product in a different order changes the result. In ordinary arithmetic, products don't depend on order. But for operators, products may well depend on order, like matrix multiplication.
Q"I also do research on the borderline between set theory and operator algebra theory. Set theory involves research on infinite subjects, or infinity itself, more directly than operator algebra theory. One famous example is called the Hilbert Hotel."
The Hilbert Hotel has an infinite number of rooms. Even if it's full when a new customer arrives, the hotel can move the person staying in the Room 1 to Room 2, the person staying in Room 2 to Room 3, and so on, to make Room 1 vacant. Using this method, even if an infinite number of new customers arrive, the hotel can move the guest in Room 1 to Room 2, the guest in Room 2 to Room 4, the guest in Room 3 to Room 6, the guest in Room 4 to Room 8, and the guest in Room N to Room 2N, making all odd-numbered rooms vacant, so an infinite number of people can stay. Moreover, another phenomenon occurs: Even if an infinite number of buses each containing an infinite number of customers arrives, the hotel can arrange things so that all of the new customers can stay.
Research students in the Katsura Group also study the relationship between tiling and C* algebra, and tackle many topics in set theory and logic. In this way, the Group works to open up new frontiers in mathematics.
Q"I think that, in a great many cases, interesting topics lie on the borderline between related fields. Of course, it's important that students do specialized studies. But I'd also like our students to take an interest in other aspects of mathematics, along with techniques from various disciplines. Those could include physics, chemistry, and engineering, but also many other fields."