Hunting for broken symmetries

Professor Genkai Zhang will receive funding from Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.

The planned project focuses on representation theory, which is a study of abstract algebraic and geometric structures by representing them by simpler, more familiar linear structures in analytic setup. The ideas and methods are not only used in mathematics, but also in other areas such as theoretical physics. Among other things, representation theory is very instrumental in the Standard Model of particle physics. 

Here, the key concept is symmetry. In our everyday lives, we usually think about symmetry as a mirror image, but there are many other symmetries that are less obvious and yet very vital to the advancement of mathematics. One can think of the situation when two children are playing at seesaw: The two kinds of diagonal and up-down symmetries can be better unified in a bigger symmetry group, whereas the symmetry is broken if we consider only one of them. The subject of this project comprises the rules for symmetry breaking (called branching rules by mathematicians) in representation theory.

Symmetries can be hard to discover. One way of finding them is to study their effects, i.e. their representations on spaces of states that are manifestations of symmetries. Such a study can be more manageable if the representation is restricted to a smaller group of symmetries; the rules for these restrictions are the branching rules. Within the project, researchers will look for and study branching rules for certain restricted representations using methods from analysis and geometry.