Boltzmann Lecture: Symmetry and Unification: can physics be reduced to a single formula?

The progress of modern physics towards General Relativity and the Standard Model of Particle Physics can be attributed to a large extent to a continual 'enhancement of symmetry', which is also hoped to herald a further unification of the laws of physics. However, attempts to merge the two fundamental theories into a single unified theory of quantum gravity have stalled, due to mathematical difficulties and the absence of clear observational evidence in favour of any of the ansätze proposed so far.
This talk will re-trace some of the history that has led us to this point, and discuss open issues and perspectives for future progress.