**Omprokash Das **

TIFR

November 19, 2015

**Adjunction and Inversion of Adjunction properties of MMP singularities and F-singularities in positi**:
In this talk I will explain the `Adjunction' and `Inversion of
Adjunction' properties of the singularities of the Minimal Model Program (MMP) and F-singularities in char p>0. Minimal model program was started in early 1900. The surface case was solved by the Italian school of Iitaka and Castelnuovo around 1920. Higher dimensional cases were done 60 years later in 1980s by Mori, Kawamata, Shokurov, Kollar, Hacon, McKernan, Birkar and Cascini. Unlike the surface
case, higher dimensional MMP depends heavily on the characteristic of the ground field k. While some of the main conjectures of MMP were solved in arbitrary dimensions in characteristic 0 (by Hacon and McKernan, 2006), it remained widely open in char p>0 until very recently in 2013 it was settled on dimension 3 by Hacon, Xu (HX13) and Birkar (Bir13). Hacon and Xu used the F-singularity techniques coming from the `Tight Closure' theory in commutative algebra to replace the Cohomological Vanishing theorems in char p>0. Their method