Groups are fundamental objects in mathematics. After learning basic conceps, we take the perspective of studying in which ways a given group can act on a vector space, which is called a representation. Symmetries of a vector space (possibly with additional structures, or fixing some figures etc) are an important source for groups, but conversely we can understand more about the structure of a group by classifying all representations.

Symmetries are fundamental concepts in physics. We will discuss two main ways in which symmetry groups and their representation theory enters crucially in mathematicsl physics: On one hand, we discuss symmetries of differential equations like the wave equation on a symmertic body, such as a sphere. As it turns out, the representation theory completely determines which eigevalues (frequencies) appear and how the eigenspaces are structured. In this way we recover the sperical harmonics, hydrogen orbitals etc. On the other hand, the gauge symmetry groups are crucial in particle physics and their representations determine the particle spectra of the theory.

Literature:

Alperin, Bell: Groups and Representations

Additional Material:

The Überton Project, in which we developed an audio plugin for the sound of high-dimensional bodies and spaces.