This is joint work with C. Florentino. The theory of motifs is an attempt to establish a universal cohomology theory for algebraic varieties. This has been completed in the case of smooth projective varieties, but the open case remains unsolved.
In this seminar, I will talk about our attempt to adapt our previous work on Hodge structures of character varieties to the simplest type of motifs, the so called naive motifs. I will also try to convey some recent ideas on equivariant Chow motifs to solve these problems.