In this talk we give an introduction to the topic of Einstein metrics on spheres. In particular, we prove the existence of three non-round Einstein metrics with positive scalar curvature on $S^{10}.$ Previously, the only even-dimensional spheres known to admit non-round Einstein metrics were $S^6$ and $S^8.$ This talk is based on joint work with Jan Nienhaus.

The Lagrange multipliers method relates critical points on a submanifold with those on an enlarged space. In derived algebraic geometry, we are allowed to consider a more general type of functions called shifted functions and thus a shifted version of the Lagrange multipliers method. If we start with quasi-smooth derived stacks, the Borisov-Joyce-Oh-Thomas virtual Lagrangian cycle of the critical locus coincides with the cosection localized virtual fundamental cycle of the enlarged space. This immediately implies the quantum Lefschetz principle of Chang-Li and an analogous result for branched covers. Based on a joint work with Hyeonjun Park.