Appears in collection : Summer School 2020: Motivic, Equivariant and Non-commutative Homotopy Theory

Notes: https://nextcloud.ihes.fr/index.php/s/kjpdembGjpLYSos In this lecture, we will explain the connection between the (motivic) theory of vanishing cycles and the construction of motivic realizations of singularity categories. We review the results obtained in collaboration with Blanc-Toen-Vezzosi and discuss some of the recent progresses in the field.