Grzegorz Banaszak Title: The algebraic Sato-Tate group and Sato Tate conjecture Abstract: Let K be a number field and let A be an abelian variety over K. In an effort of proper setting of the Sato-Tate conjecture concerning the equidistribution of Frobenius elements in the representation of the Galois group $G_K$ on the Tate module of A, one of attempts is the introduction of the algebraic Sato-Tate group AST_{K}(A). Maximal compact subgroups of AST_{K}(A)(C) are expected to be the key tool for the statement of the Sato-Tate conjecture for A. At the lecture, following an idea of J-P. Serre, an explicit construction of AST_{K}(A) will be presented based on P. Deligne's motivic category for absolute Hodge cycles. I will discuss the arithmetic properties of AST_{K}(A) along with explicit computations of AST_{K}(A) for some families of abelian varieties. I will also explain how this construction extends to absolute Hodge cycles motives in the Deligne's motivic category for absolute Hodge cycles. In addition I will show conditions for the Sato-Tate conjecture to be stated only for the connected component of identity of the Sato-Tate group. This is joint work with Kiran Kedlaya.