Mathematics at a higher level of abstraction
One way of explaining the concept of representation theory is to explore the Rubik’s Cube. Photo: Getty Images
Mateusz Stroinski will receive his doctoral degree in mathematics from Uppsala University in 2025. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Christoph Schweigert, University of Hamburg, Germany.
Mateusz Stroinski, doctoral student in mathematics.
This year, 16 mathematicians have been awarded support as part of the Mathematics Program funded by the Knut och Alice Wallenberg Foundation. One of them is Mateusz Stroinski.
The higher representation theory aims to generalise large parts of mathematics. One way of explaining the concept of representation theory is to explore the Rubik’s Cube. The idea is that everything about the cube can be understood by studying all the ways it can be turned. Altogether, this is called the cube’s symmetry group. The cube itself is a representation of this group.
Around 1900, the pioneers of representation theory, Issai Schur and Emmy Noether, realised that many mathematical objects can be studied very effectively through their symmetry groups and representations. This perspective laid the foundation for the development of the standard model of particle physics and has also been fruitful in many other fields, such as theoretical physics, chemistry, computer science and programming, in addition to mathematics.
Breakthrough in category theory
Another way to think about the Rubik’s Cube is to create a directed graph whose nodes are the different colours on the cube, and the arrows are the twists between them. The graph and all its arrows form a category. The breakthrough in category theory came in the middle of the twentieth century. It is now an independent field that is attempting to abstract the whole of mathematics in terms of categories, independently of what their objects and arrows represent. By formulating mathematics at a more abstract level, a common structure behind otherwise apparently unrelated subfields can be revealed.
The aim of this project is to further develop higher representation theory by generalising classical representation theory concepts, ideas and techniques. At the same time, the ambition is to develop techniques for solving interesting and important problems for certain categories within representation theory and mathematical physics.
Annica Hulth
Stöd från matematikprogrammet
- This year, 16 mathematicians have been granted support within the mathematics programme, which is funded by the Knut and Alice Wallenberg Foundation. The Royal Swedish Academy of Sciences, which evaluates the candidates.
- Over the years 2014–2029, the program provides SEK 650 million to allow Swedish researchers to receive international postdoctoral positions, as well as the international recruitment of visiting professors and of foreign researchers to postdoctoral positions at Swedish universities.