The class number formula is a precise relationship between two cornerstones of number theory -- the zeta function of a number field and its class group. A mysterious transcendental quantity, the regulator, serves as a bridge that links these objects. A central open question in number theory is Beilinson's vast conjectural generalization of this formula. After explaining and motivating Beilinson's conjecture, we state some new cases obtained jointly with Aaron Pollack. The key behind them is to find a setting where one can simultaneously realize Beilinson's regulator pairing in representation theory and algebraic geometry. We mention some diverse ingredients involved in the proof, such as the relative Fourier-Mukai transform and explicit solutions of systems of vector-valued PDEs, as well as some next steps.