Ivan Cheltsov: What are higher-dimensional analogues of D_n, E_6, E_7 and E_8 singularities?
Exceptional and weakly-exceptional singularities were introduced by Slava Shokurov in his famous paper on 3-fold log flips. Exceptional two-dimensional singularities are exactly singularities of type E_6, E_7 and E_8. Weakly-exceptional two-dimensional singularities are exactly singularities of type D and E_6, E_7 and E_8. Exceptional quotient singularities of dimension up to 7 have been classified by Yura Prokhorov, Dima Markushevich, me and Costya Shramov. Weakly-exceptional quotient singularities of dimension 3 and 4 have been classified by my PhD student Dima Sakovich. In my talk I will sketch a geometric weakly-exceptionality criterion for five-dimensional and six-dimentional quotient singularities. I will also explain how this is related to Fano orbifolds and Kahler-Einstein metrics on them. This is a joint work with Costya Shramov.