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The rationality problem for smooth Fano threefolds over algebraically closed fields is basically solved.
In this talk I will discuss rationality  of forms of these  Fanos over nonclosed fields of characteristic 0.
I will concentrate on the case where the  Picard number  equals 1.

The talk is based on joint work with Alexander Kuznetsov (in progress).

There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the character ratio:

The spin groups and the Clifford algebras have played a very important role in Differential Geometry and Physics. In the search for a unified spinorial approach to special Riemannian holonomy, we found a suitable notion for twisted pure spinor which generalizes that of a classical pure spinor developed by Cartan. Along the way, we realized that parallel twisted pure spinors, besides satisfying the corresponding twisted Dirac equation, satisfy a curvature identity analogous to the second Seiberg-Witten equation in 4-dimensions.