  A wide variety of complex phases in quantum materials are driven by
electron-electron interactions, which are enhanced through density of states
peaks. A well known example occurs at van Hove singularities where the Fermi
surface undergoes a topological transition. Here we show that higher order
singularities, where multiple disconnected leaves of Fermi surface touch all at
once, naturally occur at points of high symmetry in the Brillouin zone. Such
multicritical singularities can lead to stronger divergences in the density of
states than canonical van Hove singularities, and critically boost the
formation of complex quantum phases via interactions. As a concrete example of
the power of these Fermi surface topological transitions, we demonstrate how
they can be used in the analysis of experimental data on Sr$_3$Ru$_2$O$_7$.
Understanding the related mechanisms opens up new avenues in material design of
complex quantum phases.
