A conjectural generalization of n! result to arbitrary groups

We relate the n! conjecture (by Garsia and Haiman) to the geometry of principal nilpotent pairs, and state a conjecture generalizing the n! conjecture to arbitrary semisimple algebraic groups. We also show, using Borel's fixed point theorem, how to reduce the n! conjecture to staircase partitions. Finally we study the interplay between characteristic p and the n! conjecture for box partitions.Comment: Main conjectures has changed, 28 page

Four conjectures in Nonlinear Analysis

In this chapter, I formulate four challenging conjectures in Nonlinear Analysis. More precisely: a conjecture on the Monge-Amp\`ere equation; a conjecture on an eigenvalue problem; a conjecture on a non-local problem; a conjecture on disconnectedness versus infinitely many solutions.Comment: arXiv admin note: text overlap with arXiv:1504.01010, arXiv:1409.5919, arXiv:1612.0819

Note on the paper of Fu and Wong on strictly pseudoconvex domains with K\"ahler--Einstein Bergman metrics

It is shown that the Ramadanov conjecture implies the Cheng conjecture. In particular it follows that the Cheng conjecture holds in dimension two