Title:  Decomposition of the diagonal

Summary:  The  construction and study of algebraic subvarieties of an  algebraic variety are a    central problem of algebraic geometry, which, over the complex numbers, can be studied  via  Hodge theory. For any algebraic variety  X,  their is the  diagonal of X,  which provides an algebraic subvariety of the   product X^{^2}. The question of  decomposing ``à la Künneth" the  cycle of the  diagonal   gives rise to  diverse obstructions. I will discuss the   decomposition of the  diagonal in the sense of  Bloch and  Srinivas and its  applications, for  example to  rationality questions.