Non-invertible symmetries

We will discuss recent developments on a novel kind of global symmetry known as non-invertible symmetry. It is implemented by conserved operators that do not have an inverse, going outside the paradigm set by Wigner's theorem. Nonetheless, they lead to new conservation laws, novel topological phases of quantum matter, and insights into particle phenomenology. We will start with the example in the Ising lattice model, and proceed to discuss applications in pion decay and axion physics.