Mathematics, like many subjects, usually progresses incrementally. The frontier of understanding inches along year by year, and contributing to this steady expansion of human knowledge is one of the joys of research.
At some points along the frontier, though, the inching can slow to a complete halt, and it seems the subject can go no further. It is as if-to use a mining metaphor-the researchers at the coalface have struck an entirely unyielding surface of rock. This is where the really difficult problems in mathematics are found, and it usually takes something special before we can make progress again.
Enter Thomas Creutzig, associate professor in the Department of Mathematical and Statistical Sciences. Creutzig works in the field of representation theory, a far-reaching branch of mathematics that relates abstract algebraic structures to more concrete objects, like the vector spaces encountered by first-year undergraduates.
Theory is not easy. The particular flavour of representation theory that Creutzig studies is especially difficult because the spaces he works with are infinite dimensional, very unlike the three-dimensional world we physically inhabit.
From the incremental to the infinite
Imagine a square grid of numbers-a matrix, as mathematicians say. If you add up all the numbers along the diagonal from the top-left corner to the bottom-right, you get what is called the trace of the matrix. Traces are very important, appearing all over mathematics, especially in representation theory. The problem that researchers in Creutzig's field encounter is something akin to the idea that the grid is infinitely large. And because one cannot typically add an infinite numbers, so the trace becomes a tricky notion.
Representation theorists can cope with infinite dimensions as long as there is a property called rigidity. Proving rigidity has always been extremely difficult, and it is exactly here that progress had halted in Creutzig's field.
Creutzig and his collaborators have recently created an impressive new tool for attacking the problem. While their paper primarily proves a longstanding conjecture in mathematics, it is a game-changer for rigidity, because it cleverly moves the problem from a place where rigidity is difficult to study to somewhere where it is already known.
The results are published in the prestigious journal Inventiones Mathematicae. Creutzig has subsequently used the paper to prove a whole swath of rigidity theorems, and he believes the tool will lead to proofs of rigidity in many more situations.
He has big plans for future projects, delightfully enthusiastic to exploit this powerful new tool to go where no-one else has yet succeeded. But far from wishing to hide his ideas until they are fully formed, Creutzig is eager to share them with other mathematicians.
The paper, "W-algebras as coset vertex algebras," was published in Inventiones Mathematicae (doi: 10.1007/s00222-019-00884-3).