Frank Gounelas: Invariants of Fano varieties in families

Fano varieties are a class of well behaved varieties that look a lot like projective space and play a very important role in the minimal model program. In the first half of this talk I'll introduce smooth complex Fano varieties and discuss some classical results about rational curves on them, how invariants like the Picard rank and index are constant in families of Fanos and in the last minutes highlight how some of this analysis breaks down in positive characteristic. In the second half, I'll discuss joint work with A. Javanpeykar giving a geometric proof of invariance of the Picard rank via rational curves and the Brauer group, and attempts at showing invariance of the index.