Looking for mathematics’ common ground

Jakob Zimmermann received his doctoral degree in mathematics from Uppsala University in 2018. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Senior Lecturer Vanessa Miemietz at the University of East Anglia, Norwich, Great Britain.

The planned project is within a relatively newly created area of mathematics – higher representation theory. Traditionally, representation is a function that converts input mathematical concepts and functions into other functions and concepts in the hope that they will become easier to deal with. Sometimes the result is not necessarily simpler, but the benefit may instead be completely new insights.

This is what has happened in category theory, a branch of modern mathematics that emerged after the Second World War, in which mathematical concepts are upgraded to more general concepts through a process called categorification. The long-term hope is to create a generalization of large parts of mathematics. By formulating mathematics at a more abstract level, one can reveal a common structure behind several sub-areas that otherwise do not appear to be related.

Higher representation theory emerged from attempts to apply categorification ideas to specific problems in algebra and geometry. One way was to create completely new objects, called 2-categories, which proved useful in studies of knots, among other things – simple closed curves in the three-dimensional space. 2-categories have become one of the most important ingredients in all applications of categorification. Out of attempts to find an abstract basis for them all, 2-representation theory is now being intensively developed.

One of the goals of the project is to deepen the understanding of a specific class of 2-categories. In the long run, this may lead to the development of abstract representation theory for general 2-categories and studies of the theory’s applications in other areas of mathematics.

Photo: Alma Kirlic