Eisenstein congruences play an important role in modern number theory. We survey some topics related to these congruences, starting from the example of Ramanujan’s Delta function modulo 691. This paper does not contain any new results, except Theorem 2.4.
This is an introduction to a particular class of auxiliary complex Monge–Ampère equations which have been instrumental in $L^\infty$ estimates for fully non-linear equations and various questions in complex geometry. The essential comparison inequalities are reviewed and shown to apply in many contexts. Adapted to symplectic geometry, with the auxiliary equation given now by a real Monge–Ampère equation, the method gives an improvement of an earlier theorem of Tosatti–Weinkove–Yau, reducing Donaldson’s conjecture on the Calabi–Yau equation with a taming symplectic form from an exponential bound to an $L^1$ bound.