Abstract：

In this talk, I will discuss the motivic decomposition of a smooth hyperplane section in twisted Milnor Hypersurfaces. The key feature of our result is the appearance of a spectrum of a particular field in the decomposition. A critical ingredient is in the non-triviality of the (monodromy) Galois action on the equivariant Chow group. The steps of our proof can be likened to several theorems in Hodge theory of complex algebraic geometry. This is a joint with Kirill Zainoulline.