Trace Methods and Singular Support

In this thesis we explore the use of categorical methods in Algebraic Geometry. The notion of dualizable objects and traces in symmetric monoidal categories provide a good framework to study trace formulas for different sheaf theories. These techniques allow us to give a novel proof of the Lefschetz trace formula in Stable Motivic Homotopy theory. Moreover, following ideas of Beilinson and Lu, Zheng, we provide a general definition of Singular Support for any sheaf theory satisfying a six-functor formalism.